Research ArticleWhat Really Caused the ROKS Cheonan Warship Sinking

Hwang Su Kim1 and Mauro Caresta2

1 Department of Physics Kyungsung University Pusan 608-736 Republic of Korea2Department of Engineering University of Cambridge Cambridge CB2 1PZ UK

Correspondence should be addressed to Hwang Su Kim jwaksackr

Received 25 May 2014 Revised 8 August 2014 Accepted 27 August 2014 Published 20 November 2014

Academic Editor Emil Manoach

Copyright copy 2014 H S Kim and M Caresta This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is concerned with the sinking of the Korean naval warship (ROKS Cheonan) and the reported spectra of the seismicsignals recorded at the time of the incident The spectra of seismic signals show prominently amplitude peaks at around 85Hzand its harmonics These frequencies were explained with the vibrations of a water column due to an underwater explosion Thisexplanation is highly doubtful and concerns about its validity have already been raised in the scientific community In this workan alternative explanation is presented it is shown that the recorded seismic spectra are consistent with the natural frequenciesof vibrations of a large submarine with a length of around 113m This finding raises the possibility that the ROKS Cheonan sunkbecause of the collision with a large submarine rather than the explosion of a torpedo or an underwater mine

1 Introduction

Several years have passed since the incident of the Koreannaval warship (ROKS Cheonan) sinking that occurred onthe 26th of March 2010 [1 2] The Cheonan warship wassplit largely into two parts as shown in Figure 1 and sankoff Korearsquos west coast near Baekryeong Island in the YellowSea The seismic signals were recorded at the time of theincident in many stations including Baekryeong Island Theanalysis of the seismic signals followed soon after the incidentas reported in [3] Based on the official government report of asummary of its investigation on 20May 2010 the warship wassunk by a North Korean torpedo fired by a midget submarine(JIGrsquos report [1] printed in September 2010) Neverthelesscontroversy over the scientific legitimacy of the report hasbeen aroused among scientists [4ndash8] The doubts raised byeach of the reference are summarized in Appendix A

Thisworkwasmotivated from the report of [3 9] showingthe spectra of the seismic signals generated from theCheonanincident the spectral Figure 7 in [3] shows very interestingspectral peaks at around 85Hz and its harmonics Theauthor in [3] has explained these characteristic frequenciesas those of the hydroacoustic reverberation waves generatedby an underwater explosion This explanation is based onthe theory in the report of [10] that underwater explosion

generates reverberation waves in the column of seawaterwith characteristic odd harmonic series of a fundamentalfrequency by 119891

1= 1198884H where 119888 is the wave speed in water

(sim15 kms) andH is the water column depth (for1198911= 85Hz

H = 44m) However contrary to this prediction Figure 7in [3] clearly shows a series including 2119891

0asymp 177Hz and

41198910asymp 344Hz which is not an odd harmonic series Thus

it is very questionable for the validity of the analysis result in[3] Recently in another paper based on the same underwaterexplosion theory as in [9] where the explosive source wasindicated as a South Korean land control mine rather than aNorth Korean torpedo this problem was recognized and theauthors stated that ldquothese unusual unexplained spectral peaksat 17Hz and its second harmonic frequency at 34Hz can beattributed to the nonlinear deformation of the hull structureby shock waves along with overlapping readings from laterarriving T-phasesrdquo

This statement in our opinion is a speculation and noproof or simulations to support it were shown in [9] besidesthe corresponding amplitude matching of the recorded spec-tra had not been attempted in [3 9]

We think that it is worthwhile to search an alterna-tive theory to explain more properly the spectra includingamplitudes of the seismic signals For instance it will beshown in this work that a possible explanation for the sinking

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2014 Article ID 514346 10 pageshttpdxdoiorg1011552014514346

2 Advances in Acoustics and Vibration

8832m

(a)

Lostpart

99m

32m

(b)

Figure 1 (a) ROKS warship (b) top view Sketch of the partdamaged during the incident

of the ROKS Cheonan warship is a collision with a largesubmarine The reason for this explanation relies on the factthe frequencies observed in the spectra of the seismic dataare consistent with the natural frequencies of vibrations ofa large submarine as it will be shown later in the paperThe collision of the ship with the submarine would generatea large force transmitted to the submarine hull with theconsequent structure born sound radiated in the sea-waterThe sound waves then transmit into the earth crust as seismicwaves and can be recorded at seismic detecting stations Thepurpose of this paper is to explore this possibility to gaininsight into the reality of the cause of the Cheonan sinkingFor this aim both a simplified and more accurate model of asubmarine hull will be presented in the following sections Asimplifiedmodel will be used first to find themain dimensionof the submarine hull amore realisticmodelwill be then usedto properly match the seismic signature

2 Theoretical Formulation and Calculation

21 Natural Vibrations of a Submarine Hull A model of asubmarine is presented according to the approach of [1112] where the submarine was modeled as having a maincylindrical hull with internal bulkheads and ring stiffenersThe cylindrical shell is closed by truncated conical shellswhich are closed by circular plates at each end as shown inFigure 2(a) Hereafter we call this the realistic model of asubmarine however we also consider a simplified versionas shown in Figure 2(b) where the hull is modeled as acylindrical shell with ring stiffeners closed by circular platesat ends of the shell This is because the effect of the end coneson the structural response is minimal [11] and the purposeof using the simplified model is to get a closed form solutionfor the main parameters of the hull and get a first matchingwith the recorded seismic frequencies The detailed model asin [12] will be used later to calculate amore realistic structuraland acoustic response of the submarine

At first let us consider a cylindrical shell of length 119871with shear diaphragm boundary conditions to model the

Truncated cone

End plates

Bulkheads

Stiffeners(a)

(b)

L

ha

u zw r 120579

Figure 2 (a) Schematic diagram of the more realistic model ofa submarine for the free-free cylindrical shell with bulkheadsstiffeners and closed by plates following conical end caps (b)Cylindrical coordinate system (119911 120579 and 119903) and the correspondingdisplacements (119906 ] and 119908) for a thin walled shell The figureindicates the simplified model of a submarine for the cylindricalshell stiffeners and closed by plates at ends of the shell

main part of the hull (Figure 2(b)) The equations of motionand the solutions and the boundary condition equation aregiven in Appendix BThe solution of the free vibration of thecylindrical shell satisfying the Flugge differential equations ofmotion can be written as given by Caresta et al [12 13]

119906(119899) (119911 120579 119905) = 119880(119899) cos (119899120579) cos (119896119911) 119890minus119895119908119905

V(119899) (119911 120579 119905) = 119881(119899) sin (119899120579) sin (119896119911) 119890minus119895119908119905

119908(119899) (119911 120579 119905) = 119882(119899) cos (119899120579) sin (119896119911) 119890minus119895119908119905

(1)

Here 119906(119899) V(119899) and 119908(119899) are the orthogonal components ofshell displacements in the 119911-axial the 120579-circumferential andthe 119903-radial directions respectively 119899 is the circumferentialmode number (119899 = 0 1 2 ) 119895 is the imaginary unit 120596 is theangular frequency and 119896 is the axial wave number The sheardiaphragm boundary conditions are given by

V(119899) (119911 120579 119905) = 0

119908(119899) (119911 120579 119905) = 0

119873119911= 119872119911= 0

119911 = 0 119871

(2)

where 119873119911is the membrane force and the119872

119911is the bending

moment With these boundary conditions the axial wavenumber 119896 is given by

119896119898=119898120587

119871 119898 = 0 1 2 (3)

Substituting (1) and (3) into the Flugge equations gives theeigenvalue equation in the matrix form

[

[

11987111minus Ω2 119871

1211987113

11987121

11987122minus Ω2 119871

23

11987131

11987132

11987133minus Ω2

]

]

[[

[

119880(119899)

119881(119899)

119882(119899)

]]

]

= 0 (4)

Advances in Acoustics and Vibration 3

In (4) Ω is the dimensionless frequency parameter given byΩ = 120596radic120574119886119888119871 The elements of the matrix are given by

11987111= (119896119886)

2 +(1 minus 120592)

21198992 (1 + 1205732) (5a)

11987112= minus

(1 + 120592)

2119899119896119886 (5b)

11987113= minus120592119896119886 minus 1205732 (119896119886)

3 minus(1 minus 120592)

21198992119896119886 (5c)

11987121= 11987112 (5d)

11987122=(1 minus 120592)

2(119896119886)2 (1 + 31205732) + 1198992 (1 + 120583) (5e)

11987123= 119899 (1 + 120583 + 120594) minus 1205941198993 + 1205732

(3 minus 120592)

2119899 (119896119886)

2 (5f)

11987131= 11987113 (5g)

11987132= 11987123 (5h)

11987133= (1 + 120583 + 2120594) minus 21205941198992 + 1205732 [(119896119886)

2 + 1198992]2

+ 1 minus 21198992

(5i)

where

Ω2 =1205741198862

1198882119871

1205962 120574 = (1 +119860

ℎ119887+119898eq

120588ℎ)

1198882119871=

119864

120588 (1 minus 1205922) 1205732 =

ℎ2

121198862

120583 =(1 minus 1205922)119860

ℎ119887 120594 =

(1 minus 1205922)119860119911119890

ℎ119887119886

(5j)

In the previous equations 119886 is the mean radius of thecylindrical shell ℎ is the shell thickness119860 is the cross-sectionarea of the ring stiffener 119887 is the stiffener spacing and 119911

119890is

the distance between the shell mid surface and the center of aring 119864 120588 and 120592 are Yongrsquos modulus density and Poissonrsquosratio of the shell respectively 119888

119871is the longitudinal wave

speed in the shell and 119898eq is the equivalent distributed masson the shell to take into account the onboard equipmentand ballast tanks For the each set of (119899 119898) the matrixequation of (4) gives three real eigenvalues of Ω2 and thecorresponding eigenvectors (119880(119899) 119881(119899) and119882(119899)) under theusual normalization condition |119880(119899)|2 + |119881(119899)|2 + |119882(119899)|2 = 1

22 Axisymmetric Modes of Vibrations If 119899 = 0 the wavemodes are independent of 120579 and if we do not consider thetorsional modes we can set V(119899)(119911 120579 119905) = 0 The matrixequation (4) simplifies in a 2 times 2matrix equation

[11987111minus Ω2 119871

12

11987121

11987122minus Ω2

] [119880(119899)

119882(119899)] = 0 (6a)

where

11987111= (119896119886)

2 11987112= minus120592119896119886 minus 1205732 (119896119886)

3 (6b)

11987121= 11987112 119871

22= (1 + 120583 + 2120594) + 1205732 (119896119886)

4 + 1 (6c)

Under the condition (119896119886) lt 1 the off-diagonal elementsbecome small values and then the solutions of (6a) approx-imately give for the nonzero of (119880(119899) 119881(119899))

11987111minus Ω2 asymp 0 119871

22minus Ω2 asymp 0 (7)

The first equation (7) with (3) and the first relation in (5j) and11987111= (119896119886)2in (6b) give

119891(0)119898asymp119898

2119871

119888119871

radic120574 where 119898 = 1 2 (8)

It is very important to note that (8) is a harmonic series witha fundamental frequency of 119891(0)

1 which can be used to match

the characteristic frequencies of the seismic signals reportedin [3 9] The validity of (8) in the approximation can be seenup to the wave number 119898 = 8 from the solution of (6a)The first 4 frequencies and the corresponding normalizedeigenvectors of (6a) are listed in Table 1 Using (8) and (5j)we can obtain analytically the length 119871 of a submarine giving119891(0)1= 85Hz as to match the recorded seismic signatureThe

submarine hull is assumed to be made of steel with physicalproperties of 120588 asymp 7800 kgm3 119864 asymp 21 times 1011Nm2 and120592 asymp 03 119888

119871= 5450 kms It is widely known that the typical

dimension of submarines has the ratio of 2119886119871 =sim01 andℎ119886 = sim001 In 120574 = (1 + 119860ℎ119887 + 119898eq120588ℎ) 119860 = 008m times015m and 119887 = 05m can be used as in [12] The stiffenersparameters 119860 and 119887 give minor contribution in 120574 so it is notnecessary to know accurate values 119898eq is supposed to beless than the value estimated by the relation 119898eq = 120588119891119881119904119878119904where 119881

119904and 119878

119904are the volume and the surface area of

submarine respectively 120588119891is the density of seawater about

1025 kgm3 If 119881119904= 1205871198862119871 and 119878

119904= 2120587119886119871 119898eq = 120588

1198911198862

Inserting these values in (8) gives 119871 = 113m This valueis the typical length of a large submarine In summary wehave 119871 = 113m 119886 = 565m (radius of the cylindricalmodel) ℎ = 00565m (thickness of the hull) and 119898eq =

2896 kgm2 giving radic120574 = 284 This model of submarineaccording to (8) has the natural frequencies as a harmonicseries of 85Hz In this result the equivalent mass is foundto be 119898eq = 2896 kgm2 and it seems a little too highFor a submarine with a length of 113m considering about400 tons added at the ends to simulate the ballast tanks amore realistic value should be around 119898eq =sim2150 kgm

2 Ifthis distributed mass is used radic120574 = 251 and 119891(0)

1becomes

96Hz However by considering the realistic model of thesubmarine with the conical end caps and the fluid loadingwith119898eq = sim2150 kgm

2 the match of the frequencies is stillachieved as can be seen in Table 1 Therefore we will keepusing the simplified model of a submarine in vacuum with119898eq = 2896 kgm

2 for a quick guide It has been noted that theadjusting dimensional parameters of length 119871 radius 119886 hull

4 Advances in Acoustics and Vibration

Table 1 Comparison of the natural frequency of the simplifiedmodel structure (119898eq = 2896 kgm

2) with the resonant frequencyof the realistic model structure (119898eq = 2150 kgm

2) for the 119899 = 0mode and the 119899 = 1 mode vibrations 119871 = 113m (119880119882) for 119899 = 0and (119880119881119882) for 119899 = 1 are the normalized eigenvectors

(119899119898) Simplified model Realistic modelNatural frequency (Hz) (119880119882) Resonant frequency (Hz)

(0 1) 85 (0999 0022) 86(0 2) 171 (0999 0046) 174(0 3) 255 (0997 0046) 257(0 4) 340 (0994 0105) 355

(119880119881119882)(1 1) 09 (0104 0705 0702) 14(1 2) 33 (0175 0701 0691) 35(1 3) 66 (0210 0698 0684) 63(1 4) 104 (0219 0696 0684) 93(1 5) 143 (0212 0693 0689) 124(1 6) 182 (0197 0686 0700) 147

thickness ℎ distributed mass 119898eq and so forth within 10can still give harmonics of 85Hz for the natural frequenciesvibrations of the models This result means the range of theuncertainty of the dimension of this model structure is lessthan 10 The solution of the second equation in (7) for thesecond branch of the eigenvalues gives under the conditionof small (119896119886) and small value of 1205732 sim 10minus5

Ω2 asymp (1 + 120583 + 2120594) (9)

Here 120583 asymp 120594 = 0387 and we get 119891 = 816Hz The accuratecalculation of the corresponding frequencies gives 816Hz(119898 = 1) 817Hz (119898 = 2) 818 (119898 = 3) and 850HzSince these frequencies are greater than the cut-off frequency119891119888= 40Hz of the seismic signals reported for this event we

will not consider this branch of frequencies any more

23 Asymmetric Modes of Vibration For the 119899 = 1 bendingmodes the solution of Ω2 in (4) gives three branches ofeigenvalues and the corresponding three eigenvectors Thefrequency for each 119898 can be calculated from the eigenvaluesof Ω2 The lowest frequency branch represents mainly radialmotion and is of interest here the first 6 frequencies and thecorresponding normalized eigenvectors are listed in Table 1The corresponding resonant frequencies calculated with therealistic model are also listed in Table 1 The frequenciesof the second and the third branches of Ω2 are greaterthan the frequency range investigated and they will not beconsidered in this study The vibrations with 119899 gt 1 areout of consideration in this work because those give littlecontribution to the response as it will be shown later It shouldbe noted that inTable 1 the difference between the frequenciescalculated using the simplifiedmodel with119898eq = 2896 kgm

2

and the realistic model with 119898eq = 2150 kgm2 is smallsupporting the usefulness of the simplified model

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)

10minus8

10minus9

10minus10

10minus11

10minus12

n = 0 1

n = 0

n = 1

Figure 3 Frequency response function for an axial force applied tothe submarine end

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)10

minus8

10minus9

10minus10

10minus11

10minus12

10minus13

10minus14

n = 0 1

n = 0

n = 1

Figure 4 Frequency response function for a radial force applied tothe submarine end

3 Structural and Acoustic Responses

31 Structural Response Figures 3 and 4 show the struc-tural frequency response function (FRF) of the completesubmarine model for the axial and radial displacement atthe driving point as applying a point unit force at one endof the cylindrical hull in axial direction For the reciprocityprinciple the FRFs shown can be also interpreted as theresponse for the radial and axial displacement in responseto a radial force applied at the same point The calculationsin these plots were carried out using the modeling approachpresented in [12] The contribution of the 119899 = 0 and 119899 = 1modes is also shown in the same figure modes with 119899 gt1 were excluded by the simulation since it did not give asignificant contribution to the response It has to be noted

Advances in Acoustics and Vibration 5

that those results are for a point force in case the excitationforce is well distributed around the hull such as in the caseof a front collision the contribution on the response wouldbe given only by the 119899 = 0 modes The natural frequenciescorresponding to the sharp peaks in the FRF are summarizedin Table 1

The striking feature in Figure 3 is the resonant frequenciesof the 119899 = 0 mode being around 86Hz 170Hz 250Hzand 360Hz The contribution of the 119899 = 1 modes is alsoimportant and an increase in response is observed at around350Hz where the waves of Class II (the second branch ofW2) cut on as it was also observed in [12] These resultsshow a qualitative similarity between the seismic signatureand the natural frequencies of a large submarine validatingthe possibility of a collision discussed before The result alsosuggest a collision with the front part of the submarines witha main excitation of the axisymmetric (119899 = 0) modes ofvibration It can be seen in Figure 4 that the contribution of119899 = 0 to the radial displacement is much lower than the axialdisplacements as shown in Figure 3 This aspect is consistingwith the fact of low ratios of119882119880 in the eigenvectors (119880119882)as in Table 1 Also it can be noted that the 119899 = 1 modes havea mainly radial displacement This aspect can be expected bynoting the ratios of119882119880 in the eigenvectors (119880 119881 and119882)in Table 1

It has to bementioned that the purpose of this calculationis to show the FRF of the submarine model and its naturalfrequencies and a possible qualitative correlation with theseismic data recorded In reality the exact impact point onthe hull the actual force transmitted to the submarine and itsfrequency content would surely affect the quantitative resultsbut an exact reconstruction of the impact it is surely a hugeand uncertain task and is out of the scopes of this work

32 Radiated Sound at Far Field The sound pressure radiatedat far field was also calculated using the model in [12] andthe corresponding wave displacement is shown in Figure 5normalized as a displacement per unit force and per kmThefar field location is chosen to be in front of the submarineas a standard point for discussion The trend of the figure isnot much different from the seismic spectra reported in [3 9]as discussed in Section 4 However this matching does notmean that the front of the submarine was toward to the BARstation where Korea Meteorological Administration Station(379771N 1247142E) in Baekryong Island is [9] from wherethe seismic data analyzed in [3 9] was obtainedThe reason isthat the waves from the vibration source of the conical shellsof the submarine will propagate spherically in the far fieldthrough mediums The path of this spherically propagatedwaves arrived at the BAR station was suggested in Figure 6of [3] (see Figure 6 in this paper)

4 Discussion

Figure 7 of [3] showed the frequency spectrum of the p-wave signals in 119885-direction which was the Fourier transformof the first arrived p-waves at the BAR recording system inone second time window Also Figure 3(b) of [9] showed the

minus15

minus10

minus5

0

5

10

15

20

25

Frequency (Hz)

Disp

(dB

re10minus15

mN

km

)

100

101

n = 0 1

n = 0

n = 1

Figure 5 Far field wave displacement for axial force acting on thesubmarine end

113 km

004

7km

Bakryeong Island

Crust

Mantle

Moho boundary

Zd

WS

N

E

RS

BAR

o

s

o

Seawater

43

km

Figure 6 s The source of the vibration of signals RS the recordingsystem at BAR station Table 4 of [3] lists that the sound velocity insea water is 15 kms the velocities of p-wave in crust and mantleare 63 kms and 795 kms respectively while those of s-wave are350 kms and 441 kms The densities of crust and mantle are258 kgm3 and 330 kgm3 respectively

normalized spectra of seismic signals in a time window of 15seconds for each119885 EW (east-west direction) and NS (north-south direction) separately using the sameBAR recordings ofthe seismic signals In this case the analyzed signals containnot only the first arrived p-wave but the later arrived s-waveRayleigh Love waves and so forth The Fourier transformsof all these waves gives the spectrum which is the sum ofthe spectrum of each signal travelled via several independentpaths As expected the spectrum in each direction of 119885 EWand NS in Figure 3 of [9] showed similar patterns to eachother For the comparison of the spectrum in Figure 5 withthese reported spectra the data of the spectra peaks in Figure7 of [3] and in Figure 3(b) of [9] are summarized in Tables 2and 3 respectively The frequencies of the peaks of Figure 5are compared with the peaks of the reported spectra of

6 Advances in Acoustics and Vibration

Table 2The amplitude data of the peaks in the frequency spectrumreported in Figure 7 of [3] 119860-column is for meter unit scale 119861-column is for nm unit 119862-column is for the relative amplitudesdivided by 85Hz amplitude and 119863-column is for the relativeamplitudes of pressure converted from 119862-column displacementamplitudes

119891 (Hz) 119860 (m) 119861 (nm) 119862 119863 Callowast

85 10minus822 60 10 10+ 10177 10minus888 13 022 045 023260 10minus940 04 007 020 004346 10minus1017 007 001 005 0016+52 Pa lowastCal the estimated peak ratios fromFigure 5 which are comparablewith those in 119862-column

Figure 7 of [3] in Table 2 and with the peaks of Figure 3(b)of [9] in Table 3

In [9] the original unit scale was not specified in thespectra but supposed to be in Pascal Thus the differencebetween those in D column in Table 2 and those in theaverage in Table 3 may indicate uncertainty of amplitudes inthe recorded seismic spectra On the other hand the peakratios in Figure 5 are estimated to be 10 for 86Hz 023 for170Hz 004 for 250Hz and 0016 for 360Hz as listed inCal-column in Table 2 These ratios are close to those inTable 2 C-column implying good fitting for both frequenciesand amplitudes ratios too with the reported spectra Wealso noted that the frequencies of natural vibrations of thehull frame of ROKS Cheonan were calculated and listedin ⟨Table III-6-2⟩ (p160) in [1] as 232Hz 474Hz 771Hz1041Hz and 1340Hz in the bending motion only (No axialvibration is expected for the hull frame of the warship)Clearly these natural frequencies could not be correlatedwith the characteristic harmonic frequencies (85Hz and themultiples) of the seismic signals recorded

In summary the main peaks at 85Hz and its multiplesin the reported spectra of the seismic signals are reasonablyconsistent with the natural frequencies of the submarine witha length of around 113m This result suggests a possibilityof the collision between the Cheonan warship and thesubmarine Then one might wonder about the damage thatthe submarine would have following a collision We believethat the submarine would be negligibly damaged by thecollision since the thickness of the hull of a large submarineis supposed to be more than 6 cm with and made of highstrength steels while the hull thickness of the ROKSCheonanis known to be about 12 cm with steels and aluminum alloysin the upper parts Besides we have found that it is no difficultwith this collision theory discussed in this paper to illustratedamage aspects including deformations on the recovered bowand the stern of the split Cheonan warship from the sinking

5 Conclusion

This study shows that the characteristic frequencies atspectral peaks of the seismic signals recorded at the timeof the Cheonan incident are consistent with the naturalfrequencies of vibrations of a large submarine with a length

Table 3 The amplitude data of the peaks in the normalizedfrequency spectra for 119885 EW and NS seismic signals reported inFigure 3(b) of [9]

119891 (Hz) 119885 EW NS Avlowast

85 10 10 10 10170 030 038 022 030255 013 017 010 013340 003 003 002 003lowastAv is for the average of peak amplitudes in 119885 EW and NS

of around 113m The matching is particularly good with theaxisymmetric (119899 = 0) modes of vibrations suggesting afront collision with the submarine that might have caused theROKS Cheonan sinking The authors hope this work may bethe starting point for new investigations to shine some lighton the mysterious causes of the Cheonan sinking that causedthe death of 46 people and for which a clear and definitiveexplanation has not been given yet

Appendices

A Doubts about the JIGrsquos Report

Several doubts have been raised about the JIGrsquos report andthey are explained in what follows The first issue of thecontroversy over the JIGrsquos report [1] is on the validity of com-position analysis results of the Al-containing white powderabsorbed These materials were found from the followingthree locations

(A) the split ROKS Cheonan recovered in April of 2010(B) the remnants of a torpedo retrieved near the incident

site on 15 May 2010(C) the explosion products from the small-scale explo-

sion experiment using a highly aluminized explosivesource of 15 g in a tank filled with 45 tons of seawater

The composition analysis was performed with dataobtained from SEM (scanning electron microscopy) EDS(energy dispersive spectrometer) and XRD (X-ray diffrac-tion) for all three cases The analysis was based on theirknowledge or assumption (1) XRD data of an amorphousaluminum oxide will not show any noticeable diffractionpeaks (2)When the aluminum is exposed to moisture acidsand bases as in seawater environment for a long time it formsnaturally white corrosion products the major components ofthese corrosion products are aluminum hydroxide (Al(OH)

3

bayerite) along with boehmite (AlO(OH)) and Al2O3 all of

which are known to be crystalline rather than amorphousIn the JIG analysis the EDS data of A B and C samplesshowed commonly prominent intensity peaks of oxygen andaluminumwith ratio in I(O) I(Al) = 09 1 JIG regarded thisresult as evidence that the absorbed materials of A B andC samples were aluminum oxides The XRD data of the Aand B samples did not show any noticeable X-ray diffractionpeaks of an aluminum oxide crystal and the XRD data of Csample showed prominent aluminumBragg diffraction peaks

Advances in Acoustics and Vibration 7

and other weak peaks These weak peaks were analyzed tobe irrelevant to aluminum oxide crystals However JIG didnot give explicitly the reason why the strong Al Bragg peaksappeared in theXRDdata and they thought that theXRDdataofC sample also did not showanynoticeableX-ray diffractionpeaks relevant to the aluminum oxide These results led JIGto the conclusion that the A B and C samples were almost all(sim100) an amorphous aluminum oxide by (1) Because theproduct was an amorphous material these materials couldnot be natural corrosion products of aluminum which areknown to be crystalline by (2) Then JIG concluded that allthese results were clear evidence that the adsorbed materialsof A and B were amorphous aluminum oxides which are theexplosion product of the torpedowhose remnantswere foundnear the incident site We think the steps stated above wereJIGrsquos chain logic to draw the conclusion

The second issue about JIGrsquos report is whether the torpedoremnants were genuine or not Inside the rear section ofthe torpedo remnants JIG found Korean handwrite mark-ing ldquo1bun (No 1 in English in blue ink)rdquo similar to themarking of a North Korean test torpedo obtained in 2003as seen in ⟨Figure Summary-4⟩ and ⟨Figure Summary-5⟩of JIGrsquos report JIG declared promptly ldquowe found conclusiveevidencerdquo for the cause of the incident JIG said theseevidences confirmed that the recovered torpedo parts weremanufactured in North Korea and concluded that ROKSCheonan was split and sunk due to shockwave and bubbleeffects generated by the underwater explosion of a torpedomanufactured byNorth Korea However in JIGrsquos compositionanalysis data Lee [4] indentified the XRD weak peaks forC sample in fact came from diffraction of a small amountof well crystallized 120572-Al

2O3 contrary to JIGrsquos claim that

the weak peaks were irrelevant to aluminum oxide crystalsOne can see clearly weak 120572-Al

2O3Bragg diffraction peaks in

⟨Figure Appendix v-5-3⟩ p280 in JIGrsquos report [1] We thinkthis finding broke JIGrsquos chain logic

This result means that at least the C sample was quitedifferent from the A and B samples and in turn JIGrsquosconclusion above is very questionable as Lee has concludedLee and Yang then reported their own experimental resultsin another study [5] They presented the XRD data in Figure2 of [5] for the sample of an Al-powder that underwentmelting followed by rapid quenching This figure showedprominent Al Bragg diffraction peaks and weak 120572-Al

2O3

Bragg diffraction peaks They said that ldquothis clearly indicatesAl-powder oxides partially not entirely during the heatingand rapid quenchingrdquo This oxidation was supposed to occuron the surface thin layers of each grain ofAl powderTheXRDdata in Figure 2 of [5] showed the similar pattern as in ⟨FigureAppendix V-5-3⟩ of JIGrsquos report for C sample Neverthelessthe EDS data in Figure 1 of [5] showsmarkedly different from⟨Figure Appendix V-5-2⟩ of JIGrsquos report for C sampleThat isthe peak ratio of I(O)I(Al) in Figure 1 of [5] was measured as025 whereas ⟨Figure Appendix v-5-2⟩ showed very differentvalues of 081 On the other hand the authors in [5] noticedthat the high value of around 08-09 for I(O)I(Al) in EDSdata was expected for aluminumhydroxide such as Al(OH)

3

They also showed the simulation results of Al(OH)3and

Al2O3in Figure 4 of [5] supporting their expectation (It

should be noted that the EDS intensity data comes from thesurface thin layers of the sample That is the EDS data ofFigure 1 of [5] came from the surface thin layers of oxidizedgrains with 120572-Al

2O3of the heat treated Al-powder sam-

ple) From these results they concluded that JIGrsquos adsorbedmaterials taken from the warship and the torpedo remnantswere not associated with any explosion and that JIGrsquos EDSdata of their test-explosion sample (Figure Appendix v-5-2) were likely fabricated The South Korean government hasadamantly denied any fabrication or any major problemswith its interpretation of the data Later on independentlythe A and B samples were analyzed to be a basaluminite[Al4(SO4)(OH)

10sdot4-5H

2O] materials by Professor Jeong GY

in Earth amp Environmental Science from Andong NationalUniversity in Korea using several instruments including aTEM He said that the materials appeared to be formed fora long time such as in seawater implying they had nothing todo with explosions Unfortunately he was not able to test theC sample

Cyranoski [6] reported in Nature online the summaryof the controversy over JIGrsquos report Some important pointsraised in the article are as follows

(a) An expert investigator was placed on the JIG by theopposition partymdashShin Sang-chul a former officerin the South Korean navy who had also worked at ashipbuilding companymdashsuggested before the reportwas even released that an accidental collision witha US warship and not North Korea was to blameTheUnited States and South Korea had been carryingout military exercises in the area at the time Now heis insisting on the collision with a submarine with alength of around 60m not by a US warship

(b) The reportrsquos claim that a torpedo-induced water col-umn sank theCheonan contradicted earlier testimonyfrom survivors that they did not see that water col-umn They also testified neither seeing any explosionflash from the underwater nor smelling of explosivesat the time of the incident

Lee and Suh in [7] reported in Policy Forum 10-039ldquoInconsistencies in South Korearsquos Cheonan Reportrdquo In thearticle we found particularly interesting the following point

ldquoIf the bottom of the ship was hit by a bubble it shouldshow a spherical concave deformation resembling the shapeof a bubble as the JIGrsquos own simulation suggests in theAppendix of the report but it does not The bottom of thefront part of the ship is pushed up in an angular shape asthe yellow line shows in ⟨Figure III-1-7⟩ of the report moreconsistent with a collision with a hard object Suh has furtherargued that contrary to common reports that the Cheonanwas split in half as also seen in JIGrsquos simulation it actuallybroke into two larger pieces as well as a third smaller piece(see Figure 1 in this paper)rdquo Shin Sang-chul clearly pointedout if the warship was split by a bubble jet generated fromthe non-contact underwater explosion the top hull of therecovered broken warship should show upward bending butit does not seem the case as can be seen in ⟨Figure II-3-3 and4⟩ and ⟨Figure III-1-8⟩ of the report

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

2 Advances in Acoustics and Vibration

8832m

(a)

Lostpart

99m

32m

(b)

Figure 1 (a) ROKS warship (b) top view Sketch of the partdamaged during the incident

of the ROKS Cheonan warship is a collision with a largesubmarine The reason for this explanation relies on the factthe frequencies observed in the spectra of the seismic dataare consistent with the natural frequencies of vibrations ofa large submarine as it will be shown later in the paperThe collision of the ship with the submarine would generatea large force transmitted to the submarine hull with theconsequent structure born sound radiated in the sea-waterThe sound waves then transmit into the earth crust as seismicwaves and can be recorded at seismic detecting stations Thepurpose of this paper is to explore this possibility to gaininsight into the reality of the cause of the Cheonan sinkingFor this aim both a simplified and more accurate model of asubmarine hull will be presented in the following sections Asimplifiedmodel will be used first to find themain dimensionof the submarine hull amore realisticmodelwill be then usedto properly match the seismic signature

2 Theoretical Formulation and Calculation

21 Natural Vibrations of a Submarine Hull A model of asubmarine is presented according to the approach of [1112] where the submarine was modeled as having a maincylindrical hull with internal bulkheads and ring stiffenersThe cylindrical shell is closed by truncated conical shellswhich are closed by circular plates at each end as shown inFigure 2(a) Hereafter we call this the realistic model of asubmarine however we also consider a simplified versionas shown in Figure 2(b) where the hull is modeled as acylindrical shell with ring stiffeners closed by circular platesat ends of the shell This is because the effect of the end coneson the structural response is minimal [11] and the purposeof using the simplified model is to get a closed form solutionfor the main parameters of the hull and get a first matchingwith the recorded seismic frequencies The detailed model asin [12] will be used later to calculate amore realistic structuraland acoustic response of the submarine

At first let us consider a cylindrical shell of length 119871with shear diaphragm boundary conditions to model the

Truncated cone

End plates

Bulkheads

Stiffeners(a)

(b)

L

ha

u zw r 120579

Figure 2 (a) Schematic diagram of the more realistic model ofa submarine for the free-free cylindrical shell with bulkheadsstiffeners and closed by plates following conical end caps (b)Cylindrical coordinate system (119911 120579 and 119903) and the correspondingdisplacements (119906 ] and 119908) for a thin walled shell The figureindicates the simplified model of a submarine for the cylindricalshell stiffeners and closed by plates at ends of the shell

main part of the hull (Figure 2(b)) The equations of motionand the solutions and the boundary condition equation aregiven in Appendix BThe solution of the free vibration of thecylindrical shell satisfying the Flugge differential equations ofmotion can be written as given by Caresta et al [12 13]

119906(119899) (119911 120579 119905) = 119880(119899) cos (119899120579) cos (119896119911) 119890minus119895119908119905

V(119899) (119911 120579 119905) = 119881(119899) sin (119899120579) sin (119896119911) 119890minus119895119908119905

119908(119899) (119911 120579 119905) = 119882(119899) cos (119899120579) sin (119896119911) 119890minus119895119908119905

(1)

Here 119906(119899) V(119899) and 119908(119899) are the orthogonal components ofshell displacements in the 119911-axial the 120579-circumferential andthe 119903-radial directions respectively 119899 is the circumferentialmode number (119899 = 0 1 2 ) 119895 is the imaginary unit 120596 is theangular frequency and 119896 is the axial wave number The sheardiaphragm boundary conditions are given by

V(119899) (119911 120579 119905) = 0

119908(119899) (119911 120579 119905) = 0

119873119911= 119872119911= 0

119911 = 0 119871

(2)

where 119873119911is the membrane force and the119872

119911is the bending

moment With these boundary conditions the axial wavenumber 119896 is given by

119896119898=119898120587

119871 119898 = 0 1 2 (3)

Substituting (1) and (3) into the Flugge equations gives theeigenvalue equation in the matrix form

[

[

11987111minus Ω2 119871

1211987113

11987121

11987122minus Ω2 119871

23

11987131

11987132

11987133minus Ω2

]

]

[[

[

119880(119899)

119881(119899)

119882(119899)

]]

]

= 0 (4)

Advances in Acoustics and Vibration 3

In (4) Ω is the dimensionless frequency parameter given byΩ = 120596radic120574119886119888119871 The elements of the matrix are given by

11987111= (119896119886)

2 +(1 minus 120592)

21198992 (1 + 1205732) (5a)

11987112= minus

(1 + 120592)

2119899119896119886 (5b)

11987113= minus120592119896119886 minus 1205732 (119896119886)

3 minus(1 minus 120592)

21198992119896119886 (5c)

11987121= 11987112 (5d)

11987122=(1 minus 120592)

2(119896119886)2 (1 + 31205732) + 1198992 (1 + 120583) (5e)

11987123= 119899 (1 + 120583 + 120594) minus 1205941198993 + 1205732

(3 minus 120592)

2119899 (119896119886)

2 (5f)

11987131= 11987113 (5g)

11987132= 11987123 (5h)

11987133= (1 + 120583 + 2120594) minus 21205941198992 + 1205732 [(119896119886)

2 + 1198992]2

+ 1 minus 21198992

(5i)

where

Ω2 =1205741198862

1198882119871

1205962 120574 = (1 +119860

ℎ119887+119898eq

120588ℎ)

1198882119871=

119864

120588 (1 minus 1205922) 1205732 =

ℎ2

121198862

120583 =(1 minus 1205922)119860

ℎ119887 120594 =

(1 minus 1205922)119860119911119890

ℎ119887119886

(5j)

In the previous equations 119886 is the mean radius of thecylindrical shell ℎ is the shell thickness119860 is the cross-sectionarea of the ring stiffener 119887 is the stiffener spacing and 119911

119890is

the distance between the shell mid surface and the center of aring 119864 120588 and 120592 are Yongrsquos modulus density and Poissonrsquosratio of the shell respectively 119888

119871is the longitudinal wave

speed in the shell and 119898eq is the equivalent distributed masson the shell to take into account the onboard equipmentand ballast tanks For the each set of (119899 119898) the matrixequation of (4) gives three real eigenvalues of Ω2 and thecorresponding eigenvectors (119880(119899) 119881(119899) and119882(119899)) under theusual normalization condition |119880(119899)|2 + |119881(119899)|2 + |119882(119899)|2 = 1

22 Axisymmetric Modes of Vibrations If 119899 = 0 the wavemodes are independent of 120579 and if we do not consider thetorsional modes we can set V(119899)(119911 120579 119905) = 0 The matrixequation (4) simplifies in a 2 times 2matrix equation

[11987111minus Ω2 119871

12

11987121

11987122minus Ω2

] [119880(119899)

119882(119899)] = 0 (6a)

where

11987111= (119896119886)

2 11987112= minus120592119896119886 minus 1205732 (119896119886)

3 (6b)

11987121= 11987112 119871

22= (1 + 120583 + 2120594) + 1205732 (119896119886)

4 + 1 (6c)

Under the condition (119896119886) lt 1 the off-diagonal elementsbecome small values and then the solutions of (6a) approx-imately give for the nonzero of (119880(119899) 119881(119899))

11987111minus Ω2 asymp 0 119871

22minus Ω2 asymp 0 (7)

The first equation (7) with (3) and the first relation in (5j) and11987111= (119896119886)2in (6b) give

119891(0)119898asymp119898

2119871

119888119871

radic120574 where 119898 = 1 2 (8)

It is very important to note that (8) is a harmonic series witha fundamental frequency of 119891(0)

1 which can be used to match

the characteristic frequencies of the seismic signals reportedin [3 9] The validity of (8) in the approximation can be seenup to the wave number 119898 = 8 from the solution of (6a)The first 4 frequencies and the corresponding normalizedeigenvectors of (6a) are listed in Table 1 Using (8) and (5j)we can obtain analytically the length 119871 of a submarine giving119891(0)1= 85Hz as to match the recorded seismic signatureThe

submarine hull is assumed to be made of steel with physicalproperties of 120588 asymp 7800 kgm3 119864 asymp 21 times 1011Nm2 and120592 asymp 03 119888

119871= 5450 kms It is widely known that the typical

dimension of submarines has the ratio of 2119886119871 =sim01 andℎ119886 = sim001 In 120574 = (1 + 119860ℎ119887 + 119898eq120588ℎ) 119860 = 008m times015m and 119887 = 05m can be used as in [12] The stiffenersparameters 119860 and 119887 give minor contribution in 120574 so it is notnecessary to know accurate values 119898eq is supposed to beless than the value estimated by the relation 119898eq = 120588119891119881119904119878119904where 119881

119904and 119878

119904are the volume and the surface area of

submarine respectively 120588119891is the density of seawater about

1025 kgm3 If 119881119904= 1205871198862119871 and 119878

119904= 2120587119886119871 119898eq = 120588

1198911198862

Inserting these values in (8) gives 119871 = 113m This valueis the typical length of a large submarine In summary wehave 119871 = 113m 119886 = 565m (radius of the cylindricalmodel) ℎ = 00565m (thickness of the hull) and 119898eq =

2896 kgm2 giving radic120574 = 284 This model of submarineaccording to (8) has the natural frequencies as a harmonicseries of 85Hz In this result the equivalent mass is foundto be 119898eq = 2896 kgm2 and it seems a little too highFor a submarine with a length of 113m considering about400 tons added at the ends to simulate the ballast tanks amore realistic value should be around 119898eq =sim2150 kgm

2 Ifthis distributed mass is used radic120574 = 251 and 119891(0)

1becomes

96Hz However by considering the realistic model of thesubmarine with the conical end caps and the fluid loadingwith119898eq = sim2150 kgm

2 the match of the frequencies is stillachieved as can be seen in Table 1 Therefore we will keepusing the simplified model of a submarine in vacuum with119898eq = 2896 kgm

2 for a quick guide It has been noted that theadjusting dimensional parameters of length 119871 radius 119886 hull

4 Advances in Acoustics and Vibration

Table 1 Comparison of the natural frequency of the simplifiedmodel structure (119898eq = 2896 kgm

2) with the resonant frequencyof the realistic model structure (119898eq = 2150 kgm

2) for the 119899 = 0mode and the 119899 = 1 mode vibrations 119871 = 113m (119880119882) for 119899 = 0and (119880119881119882) for 119899 = 1 are the normalized eigenvectors

(119899119898) Simplified model Realistic modelNatural frequency (Hz) (119880119882) Resonant frequency (Hz)

(0 1) 85 (0999 0022) 86(0 2) 171 (0999 0046) 174(0 3) 255 (0997 0046) 257(0 4) 340 (0994 0105) 355

(119880119881119882)(1 1) 09 (0104 0705 0702) 14(1 2) 33 (0175 0701 0691) 35(1 3) 66 (0210 0698 0684) 63(1 4) 104 (0219 0696 0684) 93(1 5) 143 (0212 0693 0689) 124(1 6) 182 (0197 0686 0700) 147

thickness ℎ distributed mass 119898eq and so forth within 10can still give harmonics of 85Hz for the natural frequenciesvibrations of the models This result means the range of theuncertainty of the dimension of this model structure is lessthan 10 The solution of the second equation in (7) for thesecond branch of the eigenvalues gives under the conditionof small (119896119886) and small value of 1205732 sim 10minus5

Ω2 asymp (1 + 120583 + 2120594) (9)

Here 120583 asymp 120594 = 0387 and we get 119891 = 816Hz The accuratecalculation of the corresponding frequencies gives 816Hz(119898 = 1) 817Hz (119898 = 2) 818 (119898 = 3) and 850HzSince these frequencies are greater than the cut-off frequency119891119888= 40Hz of the seismic signals reported for this event we

will not consider this branch of frequencies any more

23 Asymmetric Modes of Vibration For the 119899 = 1 bendingmodes the solution of Ω2 in (4) gives three branches ofeigenvalues and the corresponding three eigenvectors Thefrequency for each 119898 can be calculated from the eigenvaluesof Ω2 The lowest frequency branch represents mainly radialmotion and is of interest here the first 6 frequencies and thecorresponding normalized eigenvectors are listed in Table 1The corresponding resonant frequencies calculated with therealistic model are also listed in Table 1 The frequenciesof the second and the third branches of Ω2 are greaterthan the frequency range investigated and they will not beconsidered in this study The vibrations with 119899 gt 1 areout of consideration in this work because those give littlecontribution to the response as it will be shown later It shouldbe noted that inTable 1 the difference between the frequenciescalculated using the simplifiedmodel with119898eq = 2896 kgm

2

and the realistic model with 119898eq = 2150 kgm2 is smallsupporting the usefulness of the simplified model

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)

10minus8

10minus9

10minus10

10minus11

10minus12

n = 0 1

n = 0

n = 1

Figure 3 Frequency response function for an axial force applied tothe submarine end

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)10

minus8

10minus9

10minus10

10minus11

10minus12

10minus13

10minus14

n = 0 1

n = 0

n = 1

Figure 4 Frequency response function for a radial force applied tothe submarine end

3 Structural and Acoustic Responses

31 Structural Response Figures 3 and 4 show the struc-tural frequency response function (FRF) of the completesubmarine model for the axial and radial displacement atthe driving point as applying a point unit force at one endof the cylindrical hull in axial direction For the reciprocityprinciple the FRFs shown can be also interpreted as theresponse for the radial and axial displacement in responseto a radial force applied at the same point The calculationsin these plots were carried out using the modeling approachpresented in [12] The contribution of the 119899 = 0 and 119899 = 1modes is also shown in the same figure modes with 119899 gt1 were excluded by the simulation since it did not give asignificant contribution to the response It has to be noted

Advances in Acoustics and Vibration 5

that those results are for a point force in case the excitationforce is well distributed around the hull such as in the caseof a front collision the contribution on the response wouldbe given only by the 119899 = 0 modes The natural frequenciescorresponding to the sharp peaks in the FRF are summarizedin Table 1

The striking feature in Figure 3 is the resonant frequenciesof the 119899 = 0 mode being around 86Hz 170Hz 250Hzand 360Hz The contribution of the 119899 = 1 modes is alsoimportant and an increase in response is observed at around350Hz where the waves of Class II (the second branch ofW2) cut on as it was also observed in [12] These resultsshow a qualitative similarity between the seismic signatureand the natural frequencies of a large submarine validatingthe possibility of a collision discussed before The result alsosuggest a collision with the front part of the submarines witha main excitation of the axisymmetric (119899 = 0) modes ofvibration It can be seen in Figure 4 that the contribution of119899 = 0 to the radial displacement is much lower than the axialdisplacements as shown in Figure 3 This aspect is consistingwith the fact of low ratios of119882119880 in the eigenvectors (119880119882)as in Table 1 Also it can be noted that the 119899 = 1 modes havea mainly radial displacement This aspect can be expected bynoting the ratios of119882119880 in the eigenvectors (119880 119881 and119882)in Table 1

It has to bementioned that the purpose of this calculationis to show the FRF of the submarine model and its naturalfrequencies and a possible qualitative correlation with theseismic data recorded In reality the exact impact point onthe hull the actual force transmitted to the submarine and itsfrequency content would surely affect the quantitative resultsbut an exact reconstruction of the impact it is surely a hugeand uncertain task and is out of the scopes of this work

32 Radiated Sound at Far Field The sound pressure radiatedat far field was also calculated using the model in [12] andthe corresponding wave displacement is shown in Figure 5normalized as a displacement per unit force and per kmThefar field location is chosen to be in front of the submarineas a standard point for discussion The trend of the figure isnot much different from the seismic spectra reported in [3 9]as discussed in Section 4 However this matching does notmean that the front of the submarine was toward to the BARstation where Korea Meteorological Administration Station(379771N 1247142E) in Baekryong Island is [9] from wherethe seismic data analyzed in [3 9] was obtainedThe reason isthat the waves from the vibration source of the conical shellsof the submarine will propagate spherically in the far fieldthrough mediums The path of this spherically propagatedwaves arrived at the BAR station was suggested in Figure 6of [3] (see Figure 6 in this paper)

4 Discussion

Figure 7 of [3] showed the frequency spectrum of the p-wave signals in 119885-direction which was the Fourier transformof the first arrived p-waves at the BAR recording system inone second time window Also Figure 3(b) of [9] showed the

minus15

minus10

minus5

0

5

10

15

20

25

Frequency (Hz)

Disp

(dB

re10minus15

mN

km

)

100

101

n = 0 1

n = 0

n = 1

Figure 5 Far field wave displacement for axial force acting on thesubmarine end

113 km

004

7km

Bakryeong Island

Crust

Mantle

Moho boundary

Zd

WS

N

E

RS

BAR

o

s

o

Seawater

43

km

Figure 6 s The source of the vibration of signals RS the recordingsystem at BAR station Table 4 of [3] lists that the sound velocity insea water is 15 kms the velocities of p-wave in crust and mantleare 63 kms and 795 kms respectively while those of s-wave are350 kms and 441 kms The densities of crust and mantle are258 kgm3 and 330 kgm3 respectively

normalized spectra of seismic signals in a time window of 15seconds for each119885 EW (east-west direction) and NS (north-south direction) separately using the sameBAR recordings ofthe seismic signals In this case the analyzed signals containnot only the first arrived p-wave but the later arrived s-waveRayleigh Love waves and so forth The Fourier transformsof all these waves gives the spectrum which is the sum ofthe spectrum of each signal travelled via several independentpaths As expected the spectrum in each direction of 119885 EWand NS in Figure 3 of [9] showed similar patterns to eachother For the comparison of the spectrum in Figure 5 withthese reported spectra the data of the spectra peaks in Figure7 of [3] and in Figure 3(b) of [9] are summarized in Tables 2and 3 respectively The frequencies of the peaks of Figure 5are compared with the peaks of the reported spectra of

6 Advances in Acoustics and Vibration

Table 2The amplitude data of the peaks in the frequency spectrumreported in Figure 7 of [3] 119860-column is for meter unit scale 119861-column is for nm unit 119862-column is for the relative amplitudesdivided by 85Hz amplitude and 119863-column is for the relativeamplitudes of pressure converted from 119862-column displacementamplitudes

119891 (Hz) 119860 (m) 119861 (nm) 119862 119863 Callowast

85 10minus822 60 10 10+ 10177 10minus888 13 022 045 023260 10minus940 04 007 020 004346 10minus1017 007 001 005 0016+52 Pa lowastCal the estimated peak ratios fromFigure 5 which are comparablewith those in 119862-column

Figure 7 of [3] in Table 2 and with the peaks of Figure 3(b)of [9] in Table 3

In [9] the original unit scale was not specified in thespectra but supposed to be in Pascal Thus the differencebetween those in D column in Table 2 and those in theaverage in Table 3 may indicate uncertainty of amplitudes inthe recorded seismic spectra On the other hand the peakratios in Figure 5 are estimated to be 10 for 86Hz 023 for170Hz 004 for 250Hz and 0016 for 360Hz as listed inCal-column in Table 2 These ratios are close to those inTable 2 C-column implying good fitting for both frequenciesand amplitudes ratios too with the reported spectra Wealso noted that the frequencies of natural vibrations of thehull frame of ROKS Cheonan were calculated and listedin ⟨Table III-6-2⟩ (p160) in [1] as 232Hz 474Hz 771Hz1041Hz and 1340Hz in the bending motion only (No axialvibration is expected for the hull frame of the warship)Clearly these natural frequencies could not be correlatedwith the characteristic harmonic frequencies (85Hz and themultiples) of the seismic signals recorded

In summary the main peaks at 85Hz and its multiplesin the reported spectra of the seismic signals are reasonablyconsistent with the natural frequencies of the submarine witha length of around 113m This result suggests a possibilityof the collision between the Cheonan warship and thesubmarine Then one might wonder about the damage thatthe submarine would have following a collision We believethat the submarine would be negligibly damaged by thecollision since the thickness of the hull of a large submarineis supposed to be more than 6 cm with and made of highstrength steels while the hull thickness of the ROKSCheonanis known to be about 12 cm with steels and aluminum alloysin the upper parts Besides we have found that it is no difficultwith this collision theory discussed in this paper to illustratedamage aspects including deformations on the recovered bowand the stern of the split Cheonan warship from the sinking

5 Conclusion

This study shows that the characteristic frequencies atspectral peaks of the seismic signals recorded at the timeof the Cheonan incident are consistent with the naturalfrequencies of vibrations of a large submarine with a length

Table 3 The amplitude data of the peaks in the normalizedfrequency spectra for 119885 EW and NS seismic signals reported inFigure 3(b) of [9]

119891 (Hz) 119885 EW NS Avlowast

85 10 10 10 10170 030 038 022 030255 013 017 010 013340 003 003 002 003lowastAv is for the average of peak amplitudes in 119885 EW and NS

of around 113m The matching is particularly good with theaxisymmetric (119899 = 0) modes of vibrations suggesting afront collision with the submarine that might have caused theROKS Cheonan sinking The authors hope this work may bethe starting point for new investigations to shine some lighton the mysterious causes of the Cheonan sinking that causedthe death of 46 people and for which a clear and definitiveexplanation has not been given yet

Appendices

A Doubts about the JIGrsquos Report

Several doubts have been raised about the JIGrsquos report andthey are explained in what follows The first issue of thecontroversy over the JIGrsquos report [1] is on the validity of com-position analysis results of the Al-containing white powderabsorbed These materials were found from the followingthree locations

(A) the split ROKS Cheonan recovered in April of 2010(B) the remnants of a torpedo retrieved near the incident

site on 15 May 2010(C) the explosion products from the small-scale explo-

sion experiment using a highly aluminized explosivesource of 15 g in a tank filled with 45 tons of seawater

The composition analysis was performed with dataobtained from SEM (scanning electron microscopy) EDS(energy dispersive spectrometer) and XRD (X-ray diffrac-tion) for all three cases The analysis was based on theirknowledge or assumption (1) XRD data of an amorphousaluminum oxide will not show any noticeable diffractionpeaks (2)When the aluminum is exposed to moisture acidsand bases as in seawater environment for a long time it formsnaturally white corrosion products the major components ofthese corrosion products are aluminum hydroxide (Al(OH)

3

bayerite) along with boehmite (AlO(OH)) and Al2O3 all of

which are known to be crystalline rather than amorphousIn the JIG analysis the EDS data of A B and C samplesshowed commonly prominent intensity peaks of oxygen andaluminumwith ratio in I(O) I(Al) = 09 1 JIG regarded thisresult as evidence that the absorbed materials of A B andC samples were aluminum oxides The XRD data of the Aand B samples did not show any noticeable X-ray diffractionpeaks of an aluminum oxide crystal and the XRD data of Csample showed prominent aluminumBragg diffraction peaks

Advances in Acoustics and Vibration 7

and other weak peaks These weak peaks were analyzed tobe irrelevant to aluminum oxide crystals However JIG didnot give explicitly the reason why the strong Al Bragg peaksappeared in theXRDdata and they thought that theXRDdataofC sample also did not showanynoticeableX-ray diffractionpeaks relevant to the aluminum oxide These results led JIGto the conclusion that the A B and C samples were almost all(sim100) an amorphous aluminum oxide by (1) Because theproduct was an amorphous material these materials couldnot be natural corrosion products of aluminum which areknown to be crystalline by (2) Then JIG concluded that allthese results were clear evidence that the adsorbed materialsof A and B were amorphous aluminum oxides which are theexplosion product of the torpedowhose remnantswere foundnear the incident site We think the steps stated above wereJIGrsquos chain logic to draw the conclusion

The second issue about JIGrsquos report is whether the torpedoremnants were genuine or not Inside the rear section ofthe torpedo remnants JIG found Korean handwrite mark-ing ldquo1bun (No 1 in English in blue ink)rdquo similar to themarking of a North Korean test torpedo obtained in 2003as seen in ⟨Figure Summary-4⟩ and ⟨Figure Summary-5⟩of JIGrsquos report JIG declared promptly ldquowe found conclusiveevidencerdquo for the cause of the incident JIG said theseevidences confirmed that the recovered torpedo parts weremanufactured in North Korea and concluded that ROKSCheonan was split and sunk due to shockwave and bubbleeffects generated by the underwater explosion of a torpedomanufactured byNorth Korea However in JIGrsquos compositionanalysis data Lee [4] indentified the XRD weak peaks forC sample in fact came from diffraction of a small amountof well crystallized 120572-Al

2O3 contrary to JIGrsquos claim that

the weak peaks were irrelevant to aluminum oxide crystalsOne can see clearly weak 120572-Al

2O3Bragg diffraction peaks in

⟨Figure Appendix v-5-3⟩ p280 in JIGrsquos report [1] We thinkthis finding broke JIGrsquos chain logic

This result means that at least the C sample was quitedifferent from the A and B samples and in turn JIGrsquosconclusion above is very questionable as Lee has concludedLee and Yang then reported their own experimental resultsin another study [5] They presented the XRD data in Figure2 of [5] for the sample of an Al-powder that underwentmelting followed by rapid quenching This figure showedprominent Al Bragg diffraction peaks and weak 120572-Al

2O3

Bragg diffraction peaks They said that ldquothis clearly indicatesAl-powder oxides partially not entirely during the heatingand rapid quenchingrdquo This oxidation was supposed to occuron the surface thin layers of each grain ofAl powderTheXRDdata in Figure 2 of [5] showed the similar pattern as in ⟨FigureAppendix V-5-3⟩ of JIGrsquos report for C sample Neverthelessthe EDS data in Figure 1 of [5] showsmarkedly different from⟨Figure Appendix V-5-2⟩ of JIGrsquos report for C sampleThat isthe peak ratio of I(O)I(Al) in Figure 1 of [5] was measured as025 whereas ⟨Figure Appendix v-5-2⟩ showed very differentvalues of 081 On the other hand the authors in [5] noticedthat the high value of around 08-09 for I(O)I(Al) in EDSdata was expected for aluminumhydroxide such as Al(OH)

3

They also showed the simulation results of Al(OH)3and

Al2O3in Figure 4 of [5] supporting their expectation (It

should be noted that the EDS intensity data comes from thesurface thin layers of the sample That is the EDS data ofFigure 1 of [5] came from the surface thin layers of oxidizedgrains with 120572-Al

2O3of the heat treated Al-powder sam-

ple) From these results they concluded that JIGrsquos adsorbedmaterials taken from the warship and the torpedo remnantswere not associated with any explosion and that JIGrsquos EDSdata of their test-explosion sample (Figure Appendix v-5-2) were likely fabricated The South Korean government hasadamantly denied any fabrication or any major problemswith its interpretation of the data Later on independentlythe A and B samples were analyzed to be a basaluminite[Al4(SO4)(OH)

10sdot4-5H

2O] materials by Professor Jeong GY

in Earth amp Environmental Science from Andong NationalUniversity in Korea using several instruments including aTEM He said that the materials appeared to be formed fora long time such as in seawater implying they had nothing todo with explosions Unfortunately he was not able to test theC sample

Cyranoski [6] reported in Nature online the summaryof the controversy over JIGrsquos report Some important pointsraised in the article are as follows

(a) An expert investigator was placed on the JIG by theopposition partymdashShin Sang-chul a former officerin the South Korean navy who had also worked at ashipbuilding companymdashsuggested before the reportwas even released that an accidental collision witha US warship and not North Korea was to blameTheUnited States and South Korea had been carryingout military exercises in the area at the time Now heis insisting on the collision with a submarine with alength of around 60m not by a US warship

(b) The reportrsquos claim that a torpedo-induced water col-umn sank theCheonan contradicted earlier testimonyfrom survivors that they did not see that water col-umn They also testified neither seeing any explosionflash from the underwater nor smelling of explosivesat the time of the incident

Lee and Suh in [7] reported in Policy Forum 10-039ldquoInconsistencies in South Korearsquos Cheonan Reportrdquo In thearticle we found particularly interesting the following point

ldquoIf the bottom of the ship was hit by a bubble it shouldshow a spherical concave deformation resembling the shapeof a bubble as the JIGrsquos own simulation suggests in theAppendix of the report but it does not The bottom of thefront part of the ship is pushed up in an angular shape asthe yellow line shows in ⟨Figure III-1-7⟩ of the report moreconsistent with a collision with a hard object Suh has furtherargued that contrary to common reports that the Cheonanwas split in half as also seen in JIGrsquos simulation it actuallybroke into two larger pieces as well as a third smaller piece(see Figure 1 in this paper)rdquo Shin Sang-chul clearly pointedout if the warship was split by a bubble jet generated fromthe non-contact underwater explosion the top hull of therecovered broken warship should show upward bending butit does not seem the case as can be seen in ⟨Figure II-3-3 and4⟩ and ⟨Figure III-1-8⟩ of the report

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

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International Journal of

Advances in Acoustics and Vibration 3

In (4) Ω is the dimensionless frequency parameter given byΩ = 120596radic120574119886119888119871 The elements of the matrix are given by

11987111= (119896119886)

2 +(1 minus 120592)

21198992 (1 + 1205732) (5a)

11987112= minus

(1 + 120592)

2119899119896119886 (5b)

11987113= minus120592119896119886 minus 1205732 (119896119886)

3 minus(1 minus 120592)

21198992119896119886 (5c)

11987121= 11987112 (5d)

11987122=(1 minus 120592)

2(119896119886)2 (1 + 31205732) + 1198992 (1 + 120583) (5e)

11987123= 119899 (1 + 120583 + 120594) minus 1205941198993 + 1205732

(3 minus 120592)

2119899 (119896119886)

2 (5f)

11987131= 11987113 (5g)

11987132= 11987123 (5h)

11987133= (1 + 120583 + 2120594) minus 21205941198992 + 1205732 [(119896119886)

2 + 1198992]2

+ 1 minus 21198992

(5i)

where

Ω2 =1205741198862

1198882119871

1205962 120574 = (1 +119860

ℎ119887+119898eq

120588ℎ)

1198882119871=

119864

120588 (1 minus 1205922) 1205732 =

ℎ2

121198862

120583 =(1 minus 1205922)119860

ℎ119887 120594 =

(1 minus 1205922)119860119911119890

ℎ119887119886

(5j)

In the previous equations 119886 is the mean radius of thecylindrical shell ℎ is the shell thickness119860 is the cross-sectionarea of the ring stiffener 119887 is the stiffener spacing and 119911

119890is

the distance between the shell mid surface and the center of aring 119864 120588 and 120592 are Yongrsquos modulus density and Poissonrsquosratio of the shell respectively 119888

119871is the longitudinal wave

speed in the shell and 119898eq is the equivalent distributed masson the shell to take into account the onboard equipmentand ballast tanks For the each set of (119899 119898) the matrixequation of (4) gives three real eigenvalues of Ω2 and thecorresponding eigenvectors (119880(119899) 119881(119899) and119882(119899)) under theusual normalization condition |119880(119899)|2 + |119881(119899)|2 + |119882(119899)|2 = 1

22 Axisymmetric Modes of Vibrations If 119899 = 0 the wavemodes are independent of 120579 and if we do not consider thetorsional modes we can set V(119899)(119911 120579 119905) = 0 The matrixequation (4) simplifies in a 2 times 2matrix equation

[11987111minus Ω2 119871

12

11987121

11987122minus Ω2

] [119880(119899)

119882(119899)] = 0 (6a)

where

11987111= (119896119886)

2 11987112= minus120592119896119886 minus 1205732 (119896119886)

3 (6b)

11987121= 11987112 119871

22= (1 + 120583 + 2120594) + 1205732 (119896119886)

4 + 1 (6c)

Under the condition (119896119886) lt 1 the off-diagonal elementsbecome small values and then the solutions of (6a) approx-imately give for the nonzero of (119880(119899) 119881(119899))

11987111minus Ω2 asymp 0 119871

22minus Ω2 asymp 0 (7)

The first equation (7) with (3) and the first relation in (5j) and11987111= (119896119886)2in (6b) give

119891(0)119898asymp119898

2119871

119888119871

radic120574 where 119898 = 1 2 (8)

It is very important to note that (8) is a harmonic series witha fundamental frequency of 119891(0)

1 which can be used to match

the characteristic frequencies of the seismic signals reportedin [3 9] The validity of (8) in the approximation can be seenup to the wave number 119898 = 8 from the solution of (6a)The first 4 frequencies and the corresponding normalizedeigenvectors of (6a) are listed in Table 1 Using (8) and (5j)we can obtain analytically the length 119871 of a submarine giving119891(0)1= 85Hz as to match the recorded seismic signatureThe

submarine hull is assumed to be made of steel with physicalproperties of 120588 asymp 7800 kgm3 119864 asymp 21 times 1011Nm2 and120592 asymp 03 119888

119871= 5450 kms It is widely known that the typical

dimension of submarines has the ratio of 2119886119871 =sim01 andℎ119886 = sim001 In 120574 = (1 + 119860ℎ119887 + 119898eq120588ℎ) 119860 = 008m times015m and 119887 = 05m can be used as in [12] The stiffenersparameters 119860 and 119887 give minor contribution in 120574 so it is notnecessary to know accurate values 119898eq is supposed to beless than the value estimated by the relation 119898eq = 120588119891119881119904119878119904where 119881

119904and 119878

119904are the volume and the surface area of

submarine respectively 120588119891is the density of seawater about

1025 kgm3 If 119881119904= 1205871198862119871 and 119878

119904= 2120587119886119871 119898eq = 120588

1198911198862

Inserting these values in (8) gives 119871 = 113m This valueis the typical length of a large submarine In summary wehave 119871 = 113m 119886 = 565m (radius of the cylindricalmodel) ℎ = 00565m (thickness of the hull) and 119898eq =

2896 kgm2 giving radic120574 = 284 This model of submarineaccording to (8) has the natural frequencies as a harmonicseries of 85Hz In this result the equivalent mass is foundto be 119898eq = 2896 kgm2 and it seems a little too highFor a submarine with a length of 113m considering about400 tons added at the ends to simulate the ballast tanks amore realistic value should be around 119898eq =sim2150 kgm

2 Ifthis distributed mass is used radic120574 = 251 and 119891(0)

1becomes

96Hz However by considering the realistic model of thesubmarine with the conical end caps and the fluid loadingwith119898eq = sim2150 kgm

2 the match of the frequencies is stillachieved as can be seen in Table 1 Therefore we will keepusing the simplified model of a submarine in vacuum with119898eq = 2896 kgm

2 for a quick guide It has been noted that theadjusting dimensional parameters of length 119871 radius 119886 hull

4 Advances in Acoustics and Vibration

Table 1 Comparison of the natural frequency of the simplifiedmodel structure (119898eq = 2896 kgm

2) with the resonant frequencyof the realistic model structure (119898eq = 2150 kgm

2) for the 119899 = 0mode and the 119899 = 1 mode vibrations 119871 = 113m (119880119882) for 119899 = 0and (119880119881119882) for 119899 = 1 are the normalized eigenvectors

(119899119898) Simplified model Realistic modelNatural frequency (Hz) (119880119882) Resonant frequency (Hz)

(0 1) 85 (0999 0022) 86(0 2) 171 (0999 0046) 174(0 3) 255 (0997 0046) 257(0 4) 340 (0994 0105) 355

(119880119881119882)(1 1) 09 (0104 0705 0702) 14(1 2) 33 (0175 0701 0691) 35(1 3) 66 (0210 0698 0684) 63(1 4) 104 (0219 0696 0684) 93(1 5) 143 (0212 0693 0689) 124(1 6) 182 (0197 0686 0700) 147

thickness ℎ distributed mass 119898eq and so forth within 10can still give harmonics of 85Hz for the natural frequenciesvibrations of the models This result means the range of theuncertainty of the dimension of this model structure is lessthan 10 The solution of the second equation in (7) for thesecond branch of the eigenvalues gives under the conditionof small (119896119886) and small value of 1205732 sim 10minus5

Ω2 asymp (1 + 120583 + 2120594) (9)

Here 120583 asymp 120594 = 0387 and we get 119891 = 816Hz The accuratecalculation of the corresponding frequencies gives 816Hz(119898 = 1) 817Hz (119898 = 2) 818 (119898 = 3) and 850HzSince these frequencies are greater than the cut-off frequency119891119888= 40Hz of the seismic signals reported for this event we

will not consider this branch of frequencies any more

23 Asymmetric Modes of Vibration For the 119899 = 1 bendingmodes the solution of Ω2 in (4) gives three branches ofeigenvalues and the corresponding three eigenvectors Thefrequency for each 119898 can be calculated from the eigenvaluesof Ω2 The lowest frequency branch represents mainly radialmotion and is of interest here the first 6 frequencies and thecorresponding normalized eigenvectors are listed in Table 1The corresponding resonant frequencies calculated with therealistic model are also listed in Table 1 The frequenciesof the second and the third branches of Ω2 are greaterthan the frequency range investigated and they will not beconsidered in this study The vibrations with 119899 gt 1 areout of consideration in this work because those give littlecontribution to the response as it will be shown later It shouldbe noted that inTable 1 the difference between the frequenciescalculated using the simplifiedmodel with119898eq = 2896 kgm

2

and the realistic model with 119898eq = 2150 kgm2 is smallsupporting the usefulness of the simplified model

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)

10minus8

10minus9

10minus10

10minus11

10minus12

n = 0 1

n = 0

n = 1

Figure 3 Frequency response function for an axial force applied tothe submarine end

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)10

minus8

10minus9

10minus10

10minus11

10minus12

10minus13

10minus14

n = 0 1

n = 0

n = 1

Figure 4 Frequency response function for a radial force applied tothe submarine end

3 Structural and Acoustic Responses

31 Structural Response Figures 3 and 4 show the struc-tural frequency response function (FRF) of the completesubmarine model for the axial and radial displacement atthe driving point as applying a point unit force at one endof the cylindrical hull in axial direction For the reciprocityprinciple the FRFs shown can be also interpreted as theresponse for the radial and axial displacement in responseto a radial force applied at the same point The calculationsin these plots were carried out using the modeling approachpresented in [12] The contribution of the 119899 = 0 and 119899 = 1modes is also shown in the same figure modes with 119899 gt1 were excluded by the simulation since it did not give asignificant contribution to the response It has to be noted

Advances in Acoustics and Vibration 5

that those results are for a point force in case the excitationforce is well distributed around the hull such as in the caseof a front collision the contribution on the response wouldbe given only by the 119899 = 0 modes The natural frequenciescorresponding to the sharp peaks in the FRF are summarizedin Table 1

The striking feature in Figure 3 is the resonant frequenciesof the 119899 = 0 mode being around 86Hz 170Hz 250Hzand 360Hz The contribution of the 119899 = 1 modes is alsoimportant and an increase in response is observed at around350Hz where the waves of Class II (the second branch ofW2) cut on as it was also observed in [12] These resultsshow a qualitative similarity between the seismic signatureand the natural frequencies of a large submarine validatingthe possibility of a collision discussed before The result alsosuggest a collision with the front part of the submarines witha main excitation of the axisymmetric (119899 = 0) modes ofvibration It can be seen in Figure 4 that the contribution of119899 = 0 to the radial displacement is much lower than the axialdisplacements as shown in Figure 3 This aspect is consistingwith the fact of low ratios of119882119880 in the eigenvectors (119880119882)as in Table 1 Also it can be noted that the 119899 = 1 modes havea mainly radial displacement This aspect can be expected bynoting the ratios of119882119880 in the eigenvectors (119880 119881 and119882)in Table 1

It has to bementioned that the purpose of this calculationis to show the FRF of the submarine model and its naturalfrequencies and a possible qualitative correlation with theseismic data recorded In reality the exact impact point onthe hull the actual force transmitted to the submarine and itsfrequency content would surely affect the quantitative resultsbut an exact reconstruction of the impact it is surely a hugeand uncertain task and is out of the scopes of this work

32 Radiated Sound at Far Field The sound pressure radiatedat far field was also calculated using the model in [12] andthe corresponding wave displacement is shown in Figure 5normalized as a displacement per unit force and per kmThefar field location is chosen to be in front of the submarineas a standard point for discussion The trend of the figure isnot much different from the seismic spectra reported in [3 9]as discussed in Section 4 However this matching does notmean that the front of the submarine was toward to the BARstation where Korea Meteorological Administration Station(379771N 1247142E) in Baekryong Island is [9] from wherethe seismic data analyzed in [3 9] was obtainedThe reason isthat the waves from the vibration source of the conical shellsof the submarine will propagate spherically in the far fieldthrough mediums The path of this spherically propagatedwaves arrived at the BAR station was suggested in Figure 6of [3] (see Figure 6 in this paper)

4 Discussion

Figure 7 of [3] showed the frequency spectrum of the p-wave signals in 119885-direction which was the Fourier transformof the first arrived p-waves at the BAR recording system inone second time window Also Figure 3(b) of [9] showed the

minus15

minus10

minus5

0

5

10

15

20

25

Frequency (Hz)

Disp

(dB

re10minus15

mN

km

)

100

101

n = 0 1

n = 0

n = 1

Figure 5 Far field wave displacement for axial force acting on thesubmarine end

113 km

004

7km

Bakryeong Island

Crust

Mantle

Moho boundary

Zd

WS

N

E

RS

BAR

o

s

o

Seawater

43

km

Figure 6 s The source of the vibration of signals RS the recordingsystem at BAR station Table 4 of [3] lists that the sound velocity insea water is 15 kms the velocities of p-wave in crust and mantleare 63 kms and 795 kms respectively while those of s-wave are350 kms and 441 kms The densities of crust and mantle are258 kgm3 and 330 kgm3 respectively

normalized spectra of seismic signals in a time window of 15seconds for each119885 EW (east-west direction) and NS (north-south direction) separately using the sameBAR recordings ofthe seismic signals In this case the analyzed signals containnot only the first arrived p-wave but the later arrived s-waveRayleigh Love waves and so forth The Fourier transformsof all these waves gives the spectrum which is the sum ofthe spectrum of each signal travelled via several independentpaths As expected the spectrum in each direction of 119885 EWand NS in Figure 3 of [9] showed similar patterns to eachother For the comparison of the spectrum in Figure 5 withthese reported spectra the data of the spectra peaks in Figure7 of [3] and in Figure 3(b) of [9] are summarized in Tables 2and 3 respectively The frequencies of the peaks of Figure 5are compared with the peaks of the reported spectra of

6 Advances in Acoustics and Vibration

Table 2The amplitude data of the peaks in the frequency spectrumreported in Figure 7 of [3] 119860-column is for meter unit scale 119861-column is for nm unit 119862-column is for the relative amplitudesdivided by 85Hz amplitude and 119863-column is for the relativeamplitudes of pressure converted from 119862-column displacementamplitudes

119891 (Hz) 119860 (m) 119861 (nm) 119862 119863 Callowast

85 10minus822 60 10 10+ 10177 10minus888 13 022 045 023260 10minus940 04 007 020 004346 10minus1017 007 001 005 0016+52 Pa lowastCal the estimated peak ratios fromFigure 5 which are comparablewith those in 119862-column

Figure 7 of [3] in Table 2 and with the peaks of Figure 3(b)of [9] in Table 3

In [9] the original unit scale was not specified in thespectra but supposed to be in Pascal Thus the differencebetween those in D column in Table 2 and those in theaverage in Table 3 may indicate uncertainty of amplitudes inthe recorded seismic spectra On the other hand the peakratios in Figure 5 are estimated to be 10 for 86Hz 023 for170Hz 004 for 250Hz and 0016 for 360Hz as listed inCal-column in Table 2 These ratios are close to those inTable 2 C-column implying good fitting for both frequenciesand amplitudes ratios too with the reported spectra Wealso noted that the frequencies of natural vibrations of thehull frame of ROKS Cheonan were calculated and listedin ⟨Table III-6-2⟩ (p160) in [1] as 232Hz 474Hz 771Hz1041Hz and 1340Hz in the bending motion only (No axialvibration is expected for the hull frame of the warship)Clearly these natural frequencies could not be correlatedwith the characteristic harmonic frequencies (85Hz and themultiples) of the seismic signals recorded

In summary the main peaks at 85Hz and its multiplesin the reported spectra of the seismic signals are reasonablyconsistent with the natural frequencies of the submarine witha length of around 113m This result suggests a possibilityof the collision between the Cheonan warship and thesubmarine Then one might wonder about the damage thatthe submarine would have following a collision We believethat the submarine would be negligibly damaged by thecollision since the thickness of the hull of a large submarineis supposed to be more than 6 cm with and made of highstrength steels while the hull thickness of the ROKSCheonanis known to be about 12 cm with steels and aluminum alloysin the upper parts Besides we have found that it is no difficultwith this collision theory discussed in this paper to illustratedamage aspects including deformations on the recovered bowand the stern of the split Cheonan warship from the sinking

5 Conclusion

This study shows that the characteristic frequencies atspectral peaks of the seismic signals recorded at the timeof the Cheonan incident are consistent with the naturalfrequencies of vibrations of a large submarine with a length

Table 3 The amplitude data of the peaks in the normalizedfrequency spectra for 119885 EW and NS seismic signals reported inFigure 3(b) of [9]

119891 (Hz) 119885 EW NS Avlowast

85 10 10 10 10170 030 038 022 030255 013 017 010 013340 003 003 002 003lowastAv is for the average of peak amplitudes in 119885 EW and NS

of around 113m The matching is particularly good with theaxisymmetric (119899 = 0) modes of vibrations suggesting afront collision with the submarine that might have caused theROKS Cheonan sinking The authors hope this work may bethe starting point for new investigations to shine some lighton the mysterious causes of the Cheonan sinking that causedthe death of 46 people and for which a clear and definitiveexplanation has not been given yet

Appendices

A Doubts about the JIGrsquos Report

Several doubts have been raised about the JIGrsquos report andthey are explained in what follows The first issue of thecontroversy over the JIGrsquos report [1] is on the validity of com-position analysis results of the Al-containing white powderabsorbed These materials were found from the followingthree locations

(A) the split ROKS Cheonan recovered in April of 2010(B) the remnants of a torpedo retrieved near the incident

site on 15 May 2010(C) the explosion products from the small-scale explo-

sion experiment using a highly aluminized explosivesource of 15 g in a tank filled with 45 tons of seawater

The composition analysis was performed with dataobtained from SEM (scanning electron microscopy) EDS(energy dispersive spectrometer) and XRD (X-ray diffrac-tion) for all three cases The analysis was based on theirknowledge or assumption (1) XRD data of an amorphousaluminum oxide will not show any noticeable diffractionpeaks (2)When the aluminum is exposed to moisture acidsand bases as in seawater environment for a long time it formsnaturally white corrosion products the major components ofthese corrosion products are aluminum hydroxide (Al(OH)

3

bayerite) along with boehmite (AlO(OH)) and Al2O3 all of

which are known to be crystalline rather than amorphousIn the JIG analysis the EDS data of A B and C samplesshowed commonly prominent intensity peaks of oxygen andaluminumwith ratio in I(O) I(Al) = 09 1 JIG regarded thisresult as evidence that the absorbed materials of A B andC samples were aluminum oxides The XRD data of the Aand B samples did not show any noticeable X-ray diffractionpeaks of an aluminum oxide crystal and the XRD data of Csample showed prominent aluminumBragg diffraction peaks

Advances in Acoustics and Vibration 7

and other weak peaks These weak peaks were analyzed tobe irrelevant to aluminum oxide crystals However JIG didnot give explicitly the reason why the strong Al Bragg peaksappeared in theXRDdata and they thought that theXRDdataofC sample also did not showanynoticeableX-ray diffractionpeaks relevant to the aluminum oxide These results led JIGto the conclusion that the A B and C samples were almost all(sim100) an amorphous aluminum oxide by (1) Because theproduct was an amorphous material these materials couldnot be natural corrosion products of aluminum which areknown to be crystalline by (2) Then JIG concluded that allthese results were clear evidence that the adsorbed materialsof A and B were amorphous aluminum oxides which are theexplosion product of the torpedowhose remnantswere foundnear the incident site We think the steps stated above wereJIGrsquos chain logic to draw the conclusion

The second issue about JIGrsquos report is whether the torpedoremnants were genuine or not Inside the rear section ofthe torpedo remnants JIG found Korean handwrite mark-ing ldquo1bun (No 1 in English in blue ink)rdquo similar to themarking of a North Korean test torpedo obtained in 2003as seen in ⟨Figure Summary-4⟩ and ⟨Figure Summary-5⟩of JIGrsquos report JIG declared promptly ldquowe found conclusiveevidencerdquo for the cause of the incident JIG said theseevidences confirmed that the recovered torpedo parts weremanufactured in North Korea and concluded that ROKSCheonan was split and sunk due to shockwave and bubbleeffects generated by the underwater explosion of a torpedomanufactured byNorth Korea However in JIGrsquos compositionanalysis data Lee [4] indentified the XRD weak peaks forC sample in fact came from diffraction of a small amountof well crystallized 120572-Al

2O3 contrary to JIGrsquos claim that

the weak peaks were irrelevant to aluminum oxide crystalsOne can see clearly weak 120572-Al

2O3Bragg diffraction peaks in

⟨Figure Appendix v-5-3⟩ p280 in JIGrsquos report [1] We thinkthis finding broke JIGrsquos chain logic

This result means that at least the C sample was quitedifferent from the A and B samples and in turn JIGrsquosconclusion above is very questionable as Lee has concludedLee and Yang then reported their own experimental resultsin another study [5] They presented the XRD data in Figure2 of [5] for the sample of an Al-powder that underwentmelting followed by rapid quenching This figure showedprominent Al Bragg diffraction peaks and weak 120572-Al

2O3

Bragg diffraction peaks They said that ldquothis clearly indicatesAl-powder oxides partially not entirely during the heatingand rapid quenchingrdquo This oxidation was supposed to occuron the surface thin layers of each grain ofAl powderTheXRDdata in Figure 2 of [5] showed the similar pattern as in ⟨FigureAppendix V-5-3⟩ of JIGrsquos report for C sample Neverthelessthe EDS data in Figure 1 of [5] showsmarkedly different from⟨Figure Appendix V-5-2⟩ of JIGrsquos report for C sampleThat isthe peak ratio of I(O)I(Al) in Figure 1 of [5] was measured as025 whereas ⟨Figure Appendix v-5-2⟩ showed very differentvalues of 081 On the other hand the authors in [5] noticedthat the high value of around 08-09 for I(O)I(Al) in EDSdata was expected for aluminumhydroxide such as Al(OH)

3

They also showed the simulation results of Al(OH)3and

Al2O3in Figure 4 of [5] supporting their expectation (It

should be noted that the EDS intensity data comes from thesurface thin layers of the sample That is the EDS data ofFigure 1 of [5] came from the surface thin layers of oxidizedgrains with 120572-Al

2O3of the heat treated Al-powder sam-

ple) From these results they concluded that JIGrsquos adsorbedmaterials taken from the warship and the torpedo remnantswere not associated with any explosion and that JIGrsquos EDSdata of their test-explosion sample (Figure Appendix v-5-2) were likely fabricated The South Korean government hasadamantly denied any fabrication or any major problemswith its interpretation of the data Later on independentlythe A and B samples were analyzed to be a basaluminite[Al4(SO4)(OH)

10sdot4-5H

2O] materials by Professor Jeong GY

in Earth amp Environmental Science from Andong NationalUniversity in Korea using several instruments including aTEM He said that the materials appeared to be formed fora long time such as in seawater implying they had nothing todo with explosions Unfortunately he was not able to test theC sample

Cyranoski [6] reported in Nature online the summaryof the controversy over JIGrsquos report Some important pointsraised in the article are as follows

(a) An expert investigator was placed on the JIG by theopposition partymdashShin Sang-chul a former officerin the South Korean navy who had also worked at ashipbuilding companymdashsuggested before the reportwas even released that an accidental collision witha US warship and not North Korea was to blameTheUnited States and South Korea had been carryingout military exercises in the area at the time Now heis insisting on the collision with a submarine with alength of around 60m not by a US warship

(b) The reportrsquos claim that a torpedo-induced water col-umn sank theCheonan contradicted earlier testimonyfrom survivors that they did not see that water col-umn They also testified neither seeing any explosionflash from the underwater nor smelling of explosivesat the time of the incident

Lee and Suh in [7] reported in Policy Forum 10-039ldquoInconsistencies in South Korearsquos Cheonan Reportrdquo In thearticle we found particularly interesting the following point

ldquoIf the bottom of the ship was hit by a bubble it shouldshow a spherical concave deformation resembling the shapeof a bubble as the JIGrsquos own simulation suggests in theAppendix of the report but it does not The bottom of thefront part of the ship is pushed up in an angular shape asthe yellow line shows in ⟨Figure III-1-7⟩ of the report moreconsistent with a collision with a hard object Suh has furtherargued that contrary to common reports that the Cheonanwas split in half as also seen in JIGrsquos simulation it actuallybroke into two larger pieces as well as a third smaller piece(see Figure 1 in this paper)rdquo Shin Sang-chul clearly pointedout if the warship was split by a bubble jet generated fromthe non-contact underwater explosion the top hull of therecovered broken warship should show upward bending butit does not seem the case as can be seen in ⟨Figure II-3-3 and4⟩ and ⟨Figure III-1-8⟩ of the report

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 Advances in Acoustics and Vibration

Table 1 Comparison of the natural frequency of the simplifiedmodel structure (119898eq = 2896 kgm

2) with the resonant frequencyof the realistic model structure (119898eq = 2150 kgm

2) for the 119899 = 0mode and the 119899 = 1 mode vibrations 119871 = 113m (119880119882) for 119899 = 0and (119880119881119882) for 119899 = 1 are the normalized eigenvectors

(119899119898) Simplified model Realistic modelNatural frequency (Hz) (119880119882) Resonant frequency (Hz)

(0 1) 85 (0999 0022) 86(0 2) 171 (0999 0046) 174(0 3) 255 (0997 0046) 257(0 4) 340 (0994 0105) 355

(119880119881119882)(1 1) 09 (0104 0705 0702) 14(1 2) 33 (0175 0701 0691) 35(1 3) 66 (0210 0698 0684) 63(1 4) 104 (0219 0696 0684) 93(1 5) 143 (0212 0693 0689) 124(1 6) 182 (0197 0686 0700) 147

thickness ℎ distributed mass 119898eq and so forth within 10can still give harmonics of 85Hz for the natural frequenciesvibrations of the models This result means the range of theuncertainty of the dimension of this model structure is lessthan 10 The solution of the second equation in (7) for thesecond branch of the eigenvalues gives under the conditionof small (119896119886) and small value of 1205732 sim 10minus5

Ω2 asymp (1 + 120583 + 2120594) (9)

Here 120583 asymp 120594 = 0387 and we get 119891 = 816Hz The accuratecalculation of the corresponding frequencies gives 816Hz(119898 = 1) 817Hz (119898 = 2) 818 (119898 = 3) and 850HzSince these frequencies are greater than the cut-off frequency119891119888= 40Hz of the seismic signals reported for this event we

will not consider this branch of frequencies any more

23 Asymmetric Modes of Vibration For the 119899 = 1 bendingmodes the solution of Ω2 in (4) gives three branches ofeigenvalues and the corresponding three eigenvectors Thefrequency for each 119898 can be calculated from the eigenvaluesof Ω2 The lowest frequency branch represents mainly radialmotion and is of interest here the first 6 frequencies and thecorresponding normalized eigenvectors are listed in Table 1The corresponding resonant frequencies calculated with therealistic model are also listed in Table 1 The frequenciesof the second and the third branches of Ω2 are greaterthan the frequency range investigated and they will not beconsidered in this study The vibrations with 119899 gt 1 areout of consideration in this work because those give littlecontribution to the response as it will be shown later It shouldbe noted that inTable 1 the difference between the frequenciescalculated using the simplifiedmodel with119898eq = 2896 kgm

2

and the realistic model with 119898eq = 2150 kgm2 is smallsupporting the usefulness of the simplified model

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)

10minus8

10minus9

10minus10

10minus11

10minus12

n = 0 1

n = 0

n = 1

Figure 3 Frequency response function for an axial force applied tothe submarine end

0 5 10 15 20 25 30 35 40Frequency (Hz)

FRF

axia

l disp

(mN

)10

minus8

10minus9

10minus10

10minus11

10minus12

10minus13

10minus14

n = 0 1

n = 0

n = 1

Figure 4 Frequency response function for a radial force applied tothe submarine end

3 Structural and Acoustic Responses

31 Structural Response Figures 3 and 4 show the struc-tural frequency response function (FRF) of the completesubmarine model for the axial and radial displacement atthe driving point as applying a point unit force at one endof the cylindrical hull in axial direction For the reciprocityprinciple the FRFs shown can be also interpreted as theresponse for the radial and axial displacement in responseto a radial force applied at the same point The calculationsin these plots were carried out using the modeling approachpresented in [12] The contribution of the 119899 = 0 and 119899 = 1modes is also shown in the same figure modes with 119899 gt1 were excluded by the simulation since it did not give asignificant contribution to the response It has to be noted

Advances in Acoustics and Vibration 5

that those results are for a point force in case the excitationforce is well distributed around the hull such as in the caseof a front collision the contribution on the response wouldbe given only by the 119899 = 0 modes The natural frequenciescorresponding to the sharp peaks in the FRF are summarizedin Table 1

The striking feature in Figure 3 is the resonant frequenciesof the 119899 = 0 mode being around 86Hz 170Hz 250Hzand 360Hz The contribution of the 119899 = 1 modes is alsoimportant and an increase in response is observed at around350Hz where the waves of Class II (the second branch ofW2) cut on as it was also observed in [12] These resultsshow a qualitative similarity between the seismic signatureand the natural frequencies of a large submarine validatingthe possibility of a collision discussed before The result alsosuggest a collision with the front part of the submarines witha main excitation of the axisymmetric (119899 = 0) modes ofvibration It can be seen in Figure 4 that the contribution of119899 = 0 to the radial displacement is much lower than the axialdisplacements as shown in Figure 3 This aspect is consistingwith the fact of low ratios of119882119880 in the eigenvectors (119880119882)as in Table 1 Also it can be noted that the 119899 = 1 modes havea mainly radial displacement This aspect can be expected bynoting the ratios of119882119880 in the eigenvectors (119880 119881 and119882)in Table 1

It has to bementioned that the purpose of this calculationis to show the FRF of the submarine model and its naturalfrequencies and a possible qualitative correlation with theseismic data recorded In reality the exact impact point onthe hull the actual force transmitted to the submarine and itsfrequency content would surely affect the quantitative resultsbut an exact reconstruction of the impact it is surely a hugeand uncertain task and is out of the scopes of this work

32 Radiated Sound at Far Field The sound pressure radiatedat far field was also calculated using the model in [12] andthe corresponding wave displacement is shown in Figure 5normalized as a displacement per unit force and per kmThefar field location is chosen to be in front of the submarineas a standard point for discussion The trend of the figure isnot much different from the seismic spectra reported in [3 9]as discussed in Section 4 However this matching does notmean that the front of the submarine was toward to the BARstation where Korea Meteorological Administration Station(379771N 1247142E) in Baekryong Island is [9] from wherethe seismic data analyzed in [3 9] was obtainedThe reason isthat the waves from the vibration source of the conical shellsof the submarine will propagate spherically in the far fieldthrough mediums The path of this spherically propagatedwaves arrived at the BAR station was suggested in Figure 6of [3] (see Figure 6 in this paper)

4 Discussion

Figure 7 of [3] showed the frequency spectrum of the p-wave signals in 119885-direction which was the Fourier transformof the first arrived p-waves at the BAR recording system inone second time window Also Figure 3(b) of [9] showed the

minus15

minus10

minus5

0

5

10

15

20

25

Frequency (Hz)

Disp

(dB

re10minus15

mN

km

)

100

101

n = 0 1

n = 0

n = 1

Figure 5 Far field wave displacement for axial force acting on thesubmarine end

113 km

004

7km

Bakryeong Island

Crust

Mantle

Moho boundary

Zd

WS

N

E

RS

BAR

o

s

o

Seawater

43

km

Figure 6 s The source of the vibration of signals RS the recordingsystem at BAR station Table 4 of [3] lists that the sound velocity insea water is 15 kms the velocities of p-wave in crust and mantleare 63 kms and 795 kms respectively while those of s-wave are350 kms and 441 kms The densities of crust and mantle are258 kgm3 and 330 kgm3 respectively

normalized spectra of seismic signals in a time window of 15seconds for each119885 EW (east-west direction) and NS (north-south direction) separately using the sameBAR recordings ofthe seismic signals In this case the analyzed signals containnot only the first arrived p-wave but the later arrived s-waveRayleigh Love waves and so forth The Fourier transformsof all these waves gives the spectrum which is the sum ofthe spectrum of each signal travelled via several independentpaths As expected the spectrum in each direction of 119885 EWand NS in Figure 3 of [9] showed similar patterns to eachother For the comparison of the spectrum in Figure 5 withthese reported spectra the data of the spectra peaks in Figure7 of [3] and in Figure 3(b) of [9] are summarized in Tables 2and 3 respectively The frequencies of the peaks of Figure 5are compared with the peaks of the reported spectra of

6 Advances in Acoustics and Vibration

Table 2The amplitude data of the peaks in the frequency spectrumreported in Figure 7 of [3] 119860-column is for meter unit scale 119861-column is for nm unit 119862-column is for the relative amplitudesdivided by 85Hz amplitude and 119863-column is for the relativeamplitudes of pressure converted from 119862-column displacementamplitudes

119891 (Hz) 119860 (m) 119861 (nm) 119862 119863 Callowast

85 10minus822 60 10 10+ 10177 10minus888 13 022 045 023260 10minus940 04 007 020 004346 10minus1017 007 001 005 0016+52 Pa lowastCal the estimated peak ratios fromFigure 5 which are comparablewith those in 119862-column

Figure 7 of [3] in Table 2 and with the peaks of Figure 3(b)of [9] in Table 3

In [9] the original unit scale was not specified in thespectra but supposed to be in Pascal Thus the differencebetween those in D column in Table 2 and those in theaverage in Table 3 may indicate uncertainty of amplitudes inthe recorded seismic spectra On the other hand the peakratios in Figure 5 are estimated to be 10 for 86Hz 023 for170Hz 004 for 250Hz and 0016 for 360Hz as listed inCal-column in Table 2 These ratios are close to those inTable 2 C-column implying good fitting for both frequenciesand amplitudes ratios too with the reported spectra Wealso noted that the frequencies of natural vibrations of thehull frame of ROKS Cheonan were calculated and listedin ⟨Table III-6-2⟩ (p160) in [1] as 232Hz 474Hz 771Hz1041Hz and 1340Hz in the bending motion only (No axialvibration is expected for the hull frame of the warship)Clearly these natural frequencies could not be correlatedwith the characteristic harmonic frequencies (85Hz and themultiples) of the seismic signals recorded

In summary the main peaks at 85Hz and its multiplesin the reported spectra of the seismic signals are reasonablyconsistent with the natural frequencies of the submarine witha length of around 113m This result suggests a possibilityof the collision between the Cheonan warship and thesubmarine Then one might wonder about the damage thatthe submarine would have following a collision We believethat the submarine would be negligibly damaged by thecollision since the thickness of the hull of a large submarineis supposed to be more than 6 cm with and made of highstrength steels while the hull thickness of the ROKSCheonanis known to be about 12 cm with steels and aluminum alloysin the upper parts Besides we have found that it is no difficultwith this collision theory discussed in this paper to illustratedamage aspects including deformations on the recovered bowand the stern of the split Cheonan warship from the sinking

5 Conclusion

This study shows that the characteristic frequencies atspectral peaks of the seismic signals recorded at the timeof the Cheonan incident are consistent with the naturalfrequencies of vibrations of a large submarine with a length

Table 3 The amplitude data of the peaks in the normalizedfrequency spectra for 119885 EW and NS seismic signals reported inFigure 3(b) of [9]

119891 (Hz) 119885 EW NS Avlowast

85 10 10 10 10170 030 038 022 030255 013 017 010 013340 003 003 002 003lowastAv is for the average of peak amplitudes in 119885 EW and NS

of around 113m The matching is particularly good with theaxisymmetric (119899 = 0) modes of vibrations suggesting afront collision with the submarine that might have caused theROKS Cheonan sinking The authors hope this work may bethe starting point for new investigations to shine some lighton the mysterious causes of the Cheonan sinking that causedthe death of 46 people and for which a clear and definitiveexplanation has not been given yet

Appendices

A Doubts about the JIGrsquos Report

Several doubts have been raised about the JIGrsquos report andthey are explained in what follows The first issue of thecontroversy over the JIGrsquos report [1] is on the validity of com-position analysis results of the Al-containing white powderabsorbed These materials were found from the followingthree locations

(A) the split ROKS Cheonan recovered in April of 2010(B) the remnants of a torpedo retrieved near the incident

site on 15 May 2010(C) the explosion products from the small-scale explo-

sion experiment using a highly aluminized explosivesource of 15 g in a tank filled with 45 tons of seawater

The composition analysis was performed with dataobtained from SEM (scanning electron microscopy) EDS(energy dispersive spectrometer) and XRD (X-ray diffrac-tion) for all three cases The analysis was based on theirknowledge or assumption (1) XRD data of an amorphousaluminum oxide will not show any noticeable diffractionpeaks (2)When the aluminum is exposed to moisture acidsand bases as in seawater environment for a long time it formsnaturally white corrosion products the major components ofthese corrosion products are aluminum hydroxide (Al(OH)

3

bayerite) along with boehmite (AlO(OH)) and Al2O3 all of

which are known to be crystalline rather than amorphousIn the JIG analysis the EDS data of A B and C samplesshowed commonly prominent intensity peaks of oxygen andaluminumwith ratio in I(O) I(Al) = 09 1 JIG regarded thisresult as evidence that the absorbed materials of A B andC samples were aluminum oxides The XRD data of the Aand B samples did not show any noticeable X-ray diffractionpeaks of an aluminum oxide crystal and the XRD data of Csample showed prominent aluminumBragg diffraction peaks

Advances in Acoustics and Vibration 7

and other weak peaks These weak peaks were analyzed tobe irrelevant to aluminum oxide crystals However JIG didnot give explicitly the reason why the strong Al Bragg peaksappeared in theXRDdata and they thought that theXRDdataofC sample also did not showanynoticeableX-ray diffractionpeaks relevant to the aluminum oxide These results led JIGto the conclusion that the A B and C samples were almost all(sim100) an amorphous aluminum oxide by (1) Because theproduct was an amorphous material these materials couldnot be natural corrosion products of aluminum which areknown to be crystalline by (2) Then JIG concluded that allthese results were clear evidence that the adsorbed materialsof A and B were amorphous aluminum oxides which are theexplosion product of the torpedowhose remnantswere foundnear the incident site We think the steps stated above wereJIGrsquos chain logic to draw the conclusion

The second issue about JIGrsquos report is whether the torpedoremnants were genuine or not Inside the rear section ofthe torpedo remnants JIG found Korean handwrite mark-ing ldquo1bun (No 1 in English in blue ink)rdquo similar to themarking of a North Korean test torpedo obtained in 2003as seen in ⟨Figure Summary-4⟩ and ⟨Figure Summary-5⟩of JIGrsquos report JIG declared promptly ldquowe found conclusiveevidencerdquo for the cause of the incident JIG said theseevidences confirmed that the recovered torpedo parts weremanufactured in North Korea and concluded that ROKSCheonan was split and sunk due to shockwave and bubbleeffects generated by the underwater explosion of a torpedomanufactured byNorth Korea However in JIGrsquos compositionanalysis data Lee [4] indentified the XRD weak peaks forC sample in fact came from diffraction of a small amountof well crystallized 120572-Al

2O3 contrary to JIGrsquos claim that

the weak peaks were irrelevant to aluminum oxide crystalsOne can see clearly weak 120572-Al

2O3Bragg diffraction peaks in

⟨Figure Appendix v-5-3⟩ p280 in JIGrsquos report [1] We thinkthis finding broke JIGrsquos chain logic

This result means that at least the C sample was quitedifferent from the A and B samples and in turn JIGrsquosconclusion above is very questionable as Lee has concludedLee and Yang then reported their own experimental resultsin another study [5] They presented the XRD data in Figure2 of [5] for the sample of an Al-powder that underwentmelting followed by rapid quenching This figure showedprominent Al Bragg diffraction peaks and weak 120572-Al

2O3

Bragg diffraction peaks They said that ldquothis clearly indicatesAl-powder oxides partially not entirely during the heatingand rapid quenchingrdquo This oxidation was supposed to occuron the surface thin layers of each grain ofAl powderTheXRDdata in Figure 2 of [5] showed the similar pattern as in ⟨FigureAppendix V-5-3⟩ of JIGrsquos report for C sample Neverthelessthe EDS data in Figure 1 of [5] showsmarkedly different from⟨Figure Appendix V-5-2⟩ of JIGrsquos report for C sampleThat isthe peak ratio of I(O)I(Al) in Figure 1 of [5] was measured as025 whereas ⟨Figure Appendix v-5-2⟩ showed very differentvalues of 081 On the other hand the authors in [5] noticedthat the high value of around 08-09 for I(O)I(Al) in EDSdata was expected for aluminumhydroxide such as Al(OH)

3

They also showed the simulation results of Al(OH)3and

Al2O3in Figure 4 of [5] supporting their expectation (It

should be noted that the EDS intensity data comes from thesurface thin layers of the sample That is the EDS data ofFigure 1 of [5] came from the surface thin layers of oxidizedgrains with 120572-Al

2O3of the heat treated Al-powder sam-

ple) From these results they concluded that JIGrsquos adsorbedmaterials taken from the warship and the torpedo remnantswere not associated with any explosion and that JIGrsquos EDSdata of their test-explosion sample (Figure Appendix v-5-2) were likely fabricated The South Korean government hasadamantly denied any fabrication or any major problemswith its interpretation of the data Later on independentlythe A and B samples were analyzed to be a basaluminite[Al4(SO4)(OH)

10sdot4-5H

2O] materials by Professor Jeong GY

in Earth amp Environmental Science from Andong NationalUniversity in Korea using several instruments including aTEM He said that the materials appeared to be formed fora long time such as in seawater implying they had nothing todo with explosions Unfortunately he was not able to test theC sample

Cyranoski [6] reported in Nature online the summaryof the controversy over JIGrsquos report Some important pointsraised in the article are as follows

(a) An expert investigator was placed on the JIG by theopposition partymdashShin Sang-chul a former officerin the South Korean navy who had also worked at ashipbuilding companymdashsuggested before the reportwas even released that an accidental collision witha US warship and not North Korea was to blameTheUnited States and South Korea had been carryingout military exercises in the area at the time Now heis insisting on the collision with a submarine with alength of around 60m not by a US warship

(b) The reportrsquos claim that a torpedo-induced water col-umn sank theCheonan contradicted earlier testimonyfrom survivors that they did not see that water col-umn They also testified neither seeing any explosionflash from the underwater nor smelling of explosivesat the time of the incident

Lee and Suh in [7] reported in Policy Forum 10-039ldquoInconsistencies in South Korearsquos Cheonan Reportrdquo In thearticle we found particularly interesting the following point

ldquoIf the bottom of the ship was hit by a bubble it shouldshow a spherical concave deformation resembling the shapeof a bubble as the JIGrsquos own simulation suggests in theAppendix of the report but it does not The bottom of thefront part of the ship is pushed up in an angular shape asthe yellow line shows in ⟨Figure III-1-7⟩ of the report moreconsistent with a collision with a hard object Suh has furtherargued that contrary to common reports that the Cheonanwas split in half as also seen in JIGrsquos simulation it actuallybroke into two larger pieces as well as a third smaller piece(see Figure 1 in this paper)rdquo Shin Sang-chul clearly pointedout if the warship was split by a bubble jet generated fromthe non-contact underwater explosion the top hull of therecovered broken warship should show upward bending butit does not seem the case as can be seen in ⟨Figure II-3-3 and4⟩ and ⟨Figure III-1-8⟩ of the report

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

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RotatingMachinery

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VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 5

that those results are for a point force in case the excitationforce is well distributed around the hull such as in the caseof a front collision the contribution on the response wouldbe given only by the 119899 = 0 modes The natural frequenciescorresponding to the sharp peaks in the FRF are summarizedin Table 1

The striking feature in Figure 3 is the resonant frequenciesof the 119899 = 0 mode being around 86Hz 170Hz 250Hzand 360Hz The contribution of the 119899 = 1 modes is alsoimportant and an increase in response is observed at around350Hz where the waves of Class II (the second branch ofW2) cut on as it was also observed in [12] These resultsshow a qualitative similarity between the seismic signatureand the natural frequencies of a large submarine validatingthe possibility of a collision discussed before The result alsosuggest a collision with the front part of the submarines witha main excitation of the axisymmetric (119899 = 0) modes ofvibration It can be seen in Figure 4 that the contribution of119899 = 0 to the radial displacement is much lower than the axialdisplacements as shown in Figure 3 This aspect is consistingwith the fact of low ratios of119882119880 in the eigenvectors (119880119882)as in Table 1 Also it can be noted that the 119899 = 1 modes havea mainly radial displacement This aspect can be expected bynoting the ratios of119882119880 in the eigenvectors (119880 119881 and119882)in Table 1

It has to bementioned that the purpose of this calculationis to show the FRF of the submarine model and its naturalfrequencies and a possible qualitative correlation with theseismic data recorded In reality the exact impact point onthe hull the actual force transmitted to the submarine and itsfrequency content would surely affect the quantitative resultsbut an exact reconstruction of the impact it is surely a hugeand uncertain task and is out of the scopes of this work

32 Radiated Sound at Far Field The sound pressure radiatedat far field was also calculated using the model in [12] andthe corresponding wave displacement is shown in Figure 5normalized as a displacement per unit force and per kmThefar field location is chosen to be in front of the submarineas a standard point for discussion The trend of the figure isnot much different from the seismic spectra reported in [3 9]as discussed in Section 4 However this matching does notmean that the front of the submarine was toward to the BARstation where Korea Meteorological Administration Station(379771N 1247142E) in Baekryong Island is [9] from wherethe seismic data analyzed in [3 9] was obtainedThe reason isthat the waves from the vibration source of the conical shellsof the submarine will propagate spherically in the far fieldthrough mediums The path of this spherically propagatedwaves arrived at the BAR station was suggested in Figure 6of [3] (see Figure 6 in this paper)

4 Discussion

Figure 7 of [3] showed the frequency spectrum of the p-wave signals in 119885-direction which was the Fourier transformof the first arrived p-waves at the BAR recording system inone second time window Also Figure 3(b) of [9] showed the

minus15

minus10

minus5

0

5

10

15

20

25

Frequency (Hz)

Disp

(dB

re10minus15

mN

km

)

100

101

n = 0 1

n = 0

n = 1

Figure 5 Far field wave displacement for axial force acting on thesubmarine end

113 km

004

7km

Bakryeong Island

Crust

Mantle

Moho boundary

Zd

WS

N

E

RS

BAR

o

s

o

Seawater

43

km

Figure 6 s The source of the vibration of signals RS the recordingsystem at BAR station Table 4 of [3] lists that the sound velocity insea water is 15 kms the velocities of p-wave in crust and mantleare 63 kms and 795 kms respectively while those of s-wave are350 kms and 441 kms The densities of crust and mantle are258 kgm3 and 330 kgm3 respectively

normalized spectra of seismic signals in a time window of 15seconds for each119885 EW (east-west direction) and NS (north-south direction) separately using the sameBAR recordings ofthe seismic signals In this case the analyzed signals containnot only the first arrived p-wave but the later arrived s-waveRayleigh Love waves and so forth The Fourier transformsof all these waves gives the spectrum which is the sum ofthe spectrum of each signal travelled via several independentpaths As expected the spectrum in each direction of 119885 EWand NS in Figure 3 of [9] showed similar patterns to eachother For the comparison of the spectrum in Figure 5 withthese reported spectra the data of the spectra peaks in Figure7 of [3] and in Figure 3(b) of [9] are summarized in Tables 2and 3 respectively The frequencies of the peaks of Figure 5are compared with the peaks of the reported spectra of

6 Advances in Acoustics and Vibration

Table 2The amplitude data of the peaks in the frequency spectrumreported in Figure 7 of [3] 119860-column is for meter unit scale 119861-column is for nm unit 119862-column is for the relative amplitudesdivided by 85Hz amplitude and 119863-column is for the relativeamplitudes of pressure converted from 119862-column displacementamplitudes

119891 (Hz) 119860 (m) 119861 (nm) 119862 119863 Callowast

85 10minus822 60 10 10+ 10177 10minus888 13 022 045 023260 10minus940 04 007 020 004346 10minus1017 007 001 005 0016+52 Pa lowastCal the estimated peak ratios fromFigure 5 which are comparablewith those in 119862-column

Figure 7 of [3] in Table 2 and with the peaks of Figure 3(b)of [9] in Table 3

In [9] the original unit scale was not specified in thespectra but supposed to be in Pascal Thus the differencebetween those in D column in Table 2 and those in theaverage in Table 3 may indicate uncertainty of amplitudes inthe recorded seismic spectra On the other hand the peakratios in Figure 5 are estimated to be 10 for 86Hz 023 for170Hz 004 for 250Hz and 0016 for 360Hz as listed inCal-column in Table 2 These ratios are close to those inTable 2 C-column implying good fitting for both frequenciesand amplitudes ratios too with the reported spectra Wealso noted that the frequencies of natural vibrations of thehull frame of ROKS Cheonan were calculated and listedin ⟨Table III-6-2⟩ (p160) in [1] as 232Hz 474Hz 771Hz1041Hz and 1340Hz in the bending motion only (No axialvibration is expected for the hull frame of the warship)Clearly these natural frequencies could not be correlatedwith the characteristic harmonic frequencies (85Hz and themultiples) of the seismic signals recorded

In summary the main peaks at 85Hz and its multiplesin the reported spectra of the seismic signals are reasonablyconsistent with the natural frequencies of the submarine witha length of around 113m This result suggests a possibilityof the collision between the Cheonan warship and thesubmarine Then one might wonder about the damage thatthe submarine would have following a collision We believethat the submarine would be negligibly damaged by thecollision since the thickness of the hull of a large submarineis supposed to be more than 6 cm with and made of highstrength steels while the hull thickness of the ROKSCheonanis known to be about 12 cm with steels and aluminum alloysin the upper parts Besides we have found that it is no difficultwith this collision theory discussed in this paper to illustratedamage aspects including deformations on the recovered bowand the stern of the split Cheonan warship from the sinking

5 Conclusion

This study shows that the characteristic frequencies atspectral peaks of the seismic signals recorded at the timeof the Cheonan incident are consistent with the naturalfrequencies of vibrations of a large submarine with a length

Table 3 The amplitude data of the peaks in the normalizedfrequency spectra for 119885 EW and NS seismic signals reported inFigure 3(b) of [9]

119891 (Hz) 119885 EW NS Avlowast

85 10 10 10 10170 030 038 022 030255 013 017 010 013340 003 003 002 003lowastAv is for the average of peak amplitudes in 119885 EW and NS

of around 113m The matching is particularly good with theaxisymmetric (119899 = 0) modes of vibrations suggesting afront collision with the submarine that might have caused theROKS Cheonan sinking The authors hope this work may bethe starting point for new investigations to shine some lighton the mysterious causes of the Cheonan sinking that causedthe death of 46 people and for which a clear and definitiveexplanation has not been given yet

Appendices

A Doubts about the JIGrsquos Report

Several doubts have been raised about the JIGrsquos report andthey are explained in what follows The first issue of thecontroversy over the JIGrsquos report [1] is on the validity of com-position analysis results of the Al-containing white powderabsorbed These materials were found from the followingthree locations

(A) the split ROKS Cheonan recovered in April of 2010(B) the remnants of a torpedo retrieved near the incident

site on 15 May 2010(C) the explosion products from the small-scale explo-

sion experiment using a highly aluminized explosivesource of 15 g in a tank filled with 45 tons of seawater

The composition analysis was performed with dataobtained from SEM (scanning electron microscopy) EDS(energy dispersive spectrometer) and XRD (X-ray diffrac-tion) for all three cases The analysis was based on theirknowledge or assumption (1) XRD data of an amorphousaluminum oxide will not show any noticeable diffractionpeaks (2)When the aluminum is exposed to moisture acidsand bases as in seawater environment for a long time it formsnaturally white corrosion products the major components ofthese corrosion products are aluminum hydroxide (Al(OH)

3

bayerite) along with boehmite (AlO(OH)) and Al2O3 all of

which are known to be crystalline rather than amorphousIn the JIG analysis the EDS data of A B and C samplesshowed commonly prominent intensity peaks of oxygen andaluminumwith ratio in I(O) I(Al) = 09 1 JIG regarded thisresult as evidence that the absorbed materials of A B andC samples were aluminum oxides The XRD data of the Aand B samples did not show any noticeable X-ray diffractionpeaks of an aluminum oxide crystal and the XRD data of Csample showed prominent aluminumBragg diffraction peaks

Advances in Acoustics and Vibration 7

and other weak peaks These weak peaks were analyzed tobe irrelevant to aluminum oxide crystals However JIG didnot give explicitly the reason why the strong Al Bragg peaksappeared in theXRDdata and they thought that theXRDdataofC sample also did not showanynoticeableX-ray diffractionpeaks relevant to the aluminum oxide These results led JIGto the conclusion that the A B and C samples were almost all(sim100) an amorphous aluminum oxide by (1) Because theproduct was an amorphous material these materials couldnot be natural corrosion products of aluminum which areknown to be crystalline by (2) Then JIG concluded that allthese results were clear evidence that the adsorbed materialsof A and B were amorphous aluminum oxides which are theexplosion product of the torpedowhose remnantswere foundnear the incident site We think the steps stated above wereJIGrsquos chain logic to draw the conclusion

The second issue about JIGrsquos report is whether the torpedoremnants were genuine or not Inside the rear section ofthe torpedo remnants JIG found Korean handwrite mark-ing ldquo1bun (No 1 in English in blue ink)rdquo similar to themarking of a North Korean test torpedo obtained in 2003as seen in ⟨Figure Summary-4⟩ and ⟨Figure Summary-5⟩of JIGrsquos report JIG declared promptly ldquowe found conclusiveevidencerdquo for the cause of the incident JIG said theseevidences confirmed that the recovered torpedo parts weremanufactured in North Korea and concluded that ROKSCheonan was split and sunk due to shockwave and bubbleeffects generated by the underwater explosion of a torpedomanufactured byNorth Korea However in JIGrsquos compositionanalysis data Lee [4] indentified the XRD weak peaks forC sample in fact came from diffraction of a small amountof well crystallized 120572-Al

2O3 contrary to JIGrsquos claim that

the weak peaks were irrelevant to aluminum oxide crystalsOne can see clearly weak 120572-Al

2O3Bragg diffraction peaks in

⟨Figure Appendix v-5-3⟩ p280 in JIGrsquos report [1] We thinkthis finding broke JIGrsquos chain logic

This result means that at least the C sample was quitedifferent from the A and B samples and in turn JIGrsquosconclusion above is very questionable as Lee has concludedLee and Yang then reported their own experimental resultsin another study [5] They presented the XRD data in Figure2 of [5] for the sample of an Al-powder that underwentmelting followed by rapid quenching This figure showedprominent Al Bragg diffraction peaks and weak 120572-Al

2O3

Bragg diffraction peaks They said that ldquothis clearly indicatesAl-powder oxides partially not entirely during the heatingand rapid quenchingrdquo This oxidation was supposed to occuron the surface thin layers of each grain ofAl powderTheXRDdata in Figure 2 of [5] showed the similar pattern as in ⟨FigureAppendix V-5-3⟩ of JIGrsquos report for C sample Neverthelessthe EDS data in Figure 1 of [5] showsmarkedly different from⟨Figure Appendix V-5-2⟩ of JIGrsquos report for C sampleThat isthe peak ratio of I(O)I(Al) in Figure 1 of [5] was measured as025 whereas ⟨Figure Appendix v-5-2⟩ showed very differentvalues of 081 On the other hand the authors in [5] noticedthat the high value of around 08-09 for I(O)I(Al) in EDSdata was expected for aluminumhydroxide such as Al(OH)

3

They also showed the simulation results of Al(OH)3and

Al2O3in Figure 4 of [5] supporting their expectation (It

should be noted that the EDS intensity data comes from thesurface thin layers of the sample That is the EDS data ofFigure 1 of [5] came from the surface thin layers of oxidizedgrains with 120572-Al

2O3of the heat treated Al-powder sam-

ple) From these results they concluded that JIGrsquos adsorbedmaterials taken from the warship and the torpedo remnantswere not associated with any explosion and that JIGrsquos EDSdata of their test-explosion sample (Figure Appendix v-5-2) were likely fabricated The South Korean government hasadamantly denied any fabrication or any major problemswith its interpretation of the data Later on independentlythe A and B samples were analyzed to be a basaluminite[Al4(SO4)(OH)

10sdot4-5H

2O] materials by Professor Jeong GY

in Earth amp Environmental Science from Andong NationalUniversity in Korea using several instruments including aTEM He said that the materials appeared to be formed fora long time such as in seawater implying they had nothing todo with explosions Unfortunately he was not able to test theC sample

Cyranoski [6] reported in Nature online the summaryof the controversy over JIGrsquos report Some important pointsraised in the article are as follows

(a) An expert investigator was placed on the JIG by theopposition partymdashShin Sang-chul a former officerin the South Korean navy who had also worked at ashipbuilding companymdashsuggested before the reportwas even released that an accidental collision witha US warship and not North Korea was to blameTheUnited States and South Korea had been carryingout military exercises in the area at the time Now heis insisting on the collision with a submarine with alength of around 60m not by a US warship

(b) The reportrsquos claim that a torpedo-induced water col-umn sank theCheonan contradicted earlier testimonyfrom survivors that they did not see that water col-umn They also testified neither seeing any explosionflash from the underwater nor smelling of explosivesat the time of the incident

Lee and Suh in [7] reported in Policy Forum 10-039ldquoInconsistencies in South Korearsquos Cheonan Reportrdquo In thearticle we found particularly interesting the following point

ldquoIf the bottom of the ship was hit by a bubble it shouldshow a spherical concave deformation resembling the shapeof a bubble as the JIGrsquos own simulation suggests in theAppendix of the report but it does not The bottom of thefront part of the ship is pushed up in an angular shape asthe yellow line shows in ⟨Figure III-1-7⟩ of the report moreconsistent with a collision with a hard object Suh has furtherargued that contrary to common reports that the Cheonanwas split in half as also seen in JIGrsquos simulation it actuallybroke into two larger pieces as well as a third smaller piece(see Figure 1 in this paper)rdquo Shin Sang-chul clearly pointedout if the warship was split by a bubble jet generated fromthe non-contact underwater explosion the top hull of therecovered broken warship should show upward bending butit does not seem the case as can be seen in ⟨Figure II-3-3 and4⟩ and ⟨Figure III-1-8⟩ of the report

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

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RoboticsJournal of

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Active and Passive Electronic Components

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RotatingMachinery

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VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Advances in Acoustics and Vibration

Table 2The amplitude data of the peaks in the frequency spectrumreported in Figure 7 of [3] 119860-column is for meter unit scale 119861-column is for nm unit 119862-column is for the relative amplitudesdivided by 85Hz amplitude and 119863-column is for the relativeamplitudes of pressure converted from 119862-column displacementamplitudes

119891 (Hz) 119860 (m) 119861 (nm) 119862 119863 Callowast

85 10minus822 60 10 10+ 10177 10minus888 13 022 045 023260 10minus940 04 007 020 004346 10minus1017 007 001 005 0016+52 Pa lowastCal the estimated peak ratios fromFigure 5 which are comparablewith those in 119862-column

Figure 7 of [3] in Table 2 and with the peaks of Figure 3(b)of [9] in Table 3

In [9] the original unit scale was not specified in thespectra but supposed to be in Pascal Thus the differencebetween those in D column in Table 2 and those in theaverage in Table 3 may indicate uncertainty of amplitudes inthe recorded seismic spectra On the other hand the peakratios in Figure 5 are estimated to be 10 for 86Hz 023 for170Hz 004 for 250Hz and 0016 for 360Hz as listed inCal-column in Table 2 These ratios are close to those inTable 2 C-column implying good fitting for both frequenciesand amplitudes ratios too with the reported spectra Wealso noted that the frequencies of natural vibrations of thehull frame of ROKS Cheonan were calculated and listedin ⟨Table III-6-2⟩ (p160) in [1] as 232Hz 474Hz 771Hz1041Hz and 1340Hz in the bending motion only (No axialvibration is expected for the hull frame of the warship)Clearly these natural frequencies could not be correlatedwith the characteristic harmonic frequencies (85Hz and themultiples) of the seismic signals recorded

In summary the main peaks at 85Hz and its multiplesin the reported spectra of the seismic signals are reasonablyconsistent with the natural frequencies of the submarine witha length of around 113m This result suggests a possibilityof the collision between the Cheonan warship and thesubmarine Then one might wonder about the damage thatthe submarine would have following a collision We believethat the submarine would be negligibly damaged by thecollision since the thickness of the hull of a large submarineis supposed to be more than 6 cm with and made of highstrength steels while the hull thickness of the ROKSCheonanis known to be about 12 cm with steels and aluminum alloysin the upper parts Besides we have found that it is no difficultwith this collision theory discussed in this paper to illustratedamage aspects including deformations on the recovered bowand the stern of the split Cheonan warship from the sinking

5 Conclusion

This study shows that the characteristic frequencies atspectral peaks of the seismic signals recorded at the timeof the Cheonan incident are consistent with the naturalfrequencies of vibrations of a large submarine with a length

Table 3 The amplitude data of the peaks in the normalizedfrequency spectra for 119885 EW and NS seismic signals reported inFigure 3(b) of [9]

119891 (Hz) 119885 EW NS Avlowast

85 10 10 10 10170 030 038 022 030255 013 017 010 013340 003 003 002 003lowastAv is for the average of peak amplitudes in 119885 EW and NS

of around 113m The matching is particularly good with theaxisymmetric (119899 = 0) modes of vibrations suggesting afront collision with the submarine that might have caused theROKS Cheonan sinking The authors hope this work may bethe starting point for new investigations to shine some lighton the mysterious causes of the Cheonan sinking that causedthe death of 46 people and for which a clear and definitiveexplanation has not been given yet

Appendices

A Doubts about the JIGrsquos Report

Several doubts have been raised about the JIGrsquos report andthey are explained in what follows The first issue of thecontroversy over the JIGrsquos report [1] is on the validity of com-position analysis results of the Al-containing white powderabsorbed These materials were found from the followingthree locations

(A) the split ROKS Cheonan recovered in April of 2010(B) the remnants of a torpedo retrieved near the incident

site on 15 May 2010(C) the explosion products from the small-scale explo-

sion experiment using a highly aluminized explosivesource of 15 g in a tank filled with 45 tons of seawater

The composition analysis was performed with dataobtained from SEM (scanning electron microscopy) EDS(energy dispersive spectrometer) and XRD (X-ray diffrac-tion) for all three cases The analysis was based on theirknowledge or assumption (1) XRD data of an amorphousaluminum oxide will not show any noticeable diffractionpeaks (2)When the aluminum is exposed to moisture acidsand bases as in seawater environment for a long time it formsnaturally white corrosion products the major components ofthese corrosion products are aluminum hydroxide (Al(OH)

3

bayerite) along with boehmite (AlO(OH)) and Al2O3 all of

which are known to be crystalline rather than amorphousIn the JIG analysis the EDS data of A B and C samplesshowed commonly prominent intensity peaks of oxygen andaluminumwith ratio in I(O) I(Al) = 09 1 JIG regarded thisresult as evidence that the absorbed materials of A B andC samples were aluminum oxides The XRD data of the Aand B samples did not show any noticeable X-ray diffractionpeaks of an aluminum oxide crystal and the XRD data of Csample showed prominent aluminumBragg diffraction peaks

Advances in Acoustics and Vibration 7

and other weak peaks These weak peaks were analyzed tobe irrelevant to aluminum oxide crystals However JIG didnot give explicitly the reason why the strong Al Bragg peaksappeared in theXRDdata and they thought that theXRDdataofC sample also did not showanynoticeableX-ray diffractionpeaks relevant to the aluminum oxide These results led JIGto the conclusion that the A B and C samples were almost all(sim100) an amorphous aluminum oxide by (1) Because theproduct was an amorphous material these materials couldnot be natural corrosion products of aluminum which areknown to be crystalline by (2) Then JIG concluded that allthese results were clear evidence that the adsorbed materialsof A and B were amorphous aluminum oxides which are theexplosion product of the torpedowhose remnantswere foundnear the incident site We think the steps stated above wereJIGrsquos chain logic to draw the conclusion

The second issue about JIGrsquos report is whether the torpedoremnants were genuine or not Inside the rear section ofthe torpedo remnants JIG found Korean handwrite mark-ing ldquo1bun (No 1 in English in blue ink)rdquo similar to themarking of a North Korean test torpedo obtained in 2003as seen in ⟨Figure Summary-4⟩ and ⟨Figure Summary-5⟩of JIGrsquos report JIG declared promptly ldquowe found conclusiveevidencerdquo for the cause of the incident JIG said theseevidences confirmed that the recovered torpedo parts weremanufactured in North Korea and concluded that ROKSCheonan was split and sunk due to shockwave and bubbleeffects generated by the underwater explosion of a torpedomanufactured byNorth Korea However in JIGrsquos compositionanalysis data Lee [4] indentified the XRD weak peaks forC sample in fact came from diffraction of a small amountof well crystallized 120572-Al

2O3 contrary to JIGrsquos claim that

the weak peaks were irrelevant to aluminum oxide crystalsOne can see clearly weak 120572-Al

2O3Bragg diffraction peaks in

⟨Figure Appendix v-5-3⟩ p280 in JIGrsquos report [1] We thinkthis finding broke JIGrsquos chain logic

This result means that at least the C sample was quitedifferent from the A and B samples and in turn JIGrsquosconclusion above is very questionable as Lee has concludedLee and Yang then reported their own experimental resultsin another study [5] They presented the XRD data in Figure2 of [5] for the sample of an Al-powder that underwentmelting followed by rapid quenching This figure showedprominent Al Bragg diffraction peaks and weak 120572-Al

2O3

Bragg diffraction peaks They said that ldquothis clearly indicatesAl-powder oxides partially not entirely during the heatingand rapid quenchingrdquo This oxidation was supposed to occuron the surface thin layers of each grain ofAl powderTheXRDdata in Figure 2 of [5] showed the similar pattern as in ⟨FigureAppendix V-5-3⟩ of JIGrsquos report for C sample Neverthelessthe EDS data in Figure 1 of [5] showsmarkedly different from⟨Figure Appendix V-5-2⟩ of JIGrsquos report for C sampleThat isthe peak ratio of I(O)I(Al) in Figure 1 of [5] was measured as025 whereas ⟨Figure Appendix v-5-2⟩ showed very differentvalues of 081 On the other hand the authors in [5] noticedthat the high value of around 08-09 for I(O)I(Al) in EDSdata was expected for aluminumhydroxide such as Al(OH)

3

They also showed the simulation results of Al(OH)3and

Al2O3in Figure 4 of [5] supporting their expectation (It

should be noted that the EDS intensity data comes from thesurface thin layers of the sample That is the EDS data ofFigure 1 of [5] came from the surface thin layers of oxidizedgrains with 120572-Al

2O3of the heat treated Al-powder sam-

ple) From these results they concluded that JIGrsquos adsorbedmaterials taken from the warship and the torpedo remnantswere not associated with any explosion and that JIGrsquos EDSdata of their test-explosion sample (Figure Appendix v-5-2) were likely fabricated The South Korean government hasadamantly denied any fabrication or any major problemswith its interpretation of the data Later on independentlythe A and B samples were analyzed to be a basaluminite[Al4(SO4)(OH)

10sdot4-5H

2O] materials by Professor Jeong GY

in Earth amp Environmental Science from Andong NationalUniversity in Korea using several instruments including aTEM He said that the materials appeared to be formed fora long time such as in seawater implying they had nothing todo with explosions Unfortunately he was not able to test theC sample

Cyranoski [6] reported in Nature online the summaryof the controversy over JIGrsquos report Some important pointsraised in the article are as follows

(a) An expert investigator was placed on the JIG by theopposition partymdashShin Sang-chul a former officerin the South Korean navy who had also worked at ashipbuilding companymdashsuggested before the reportwas even released that an accidental collision witha US warship and not North Korea was to blameTheUnited States and South Korea had been carryingout military exercises in the area at the time Now heis insisting on the collision with a submarine with alength of around 60m not by a US warship

(b) The reportrsquos claim that a torpedo-induced water col-umn sank theCheonan contradicted earlier testimonyfrom survivors that they did not see that water col-umn They also testified neither seeing any explosionflash from the underwater nor smelling of explosivesat the time of the incident

Lee and Suh in [7] reported in Policy Forum 10-039ldquoInconsistencies in South Korearsquos Cheonan Reportrdquo In thearticle we found particularly interesting the following point

ldquoIf the bottom of the ship was hit by a bubble it shouldshow a spherical concave deformation resembling the shapeof a bubble as the JIGrsquos own simulation suggests in theAppendix of the report but it does not The bottom of thefront part of the ship is pushed up in an angular shape asthe yellow line shows in ⟨Figure III-1-7⟩ of the report moreconsistent with a collision with a hard object Suh has furtherargued that contrary to common reports that the Cheonanwas split in half as also seen in JIGrsquos simulation it actuallybroke into two larger pieces as well as a third smaller piece(see Figure 1 in this paper)rdquo Shin Sang-chul clearly pointedout if the warship was split by a bubble jet generated fromthe non-contact underwater explosion the top hull of therecovered broken warship should show upward bending butit does not seem the case as can be seen in ⟨Figure II-3-3 and4⟩ and ⟨Figure III-1-8⟩ of the report

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

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AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 7

and other weak peaks These weak peaks were analyzed tobe irrelevant to aluminum oxide crystals However JIG didnot give explicitly the reason why the strong Al Bragg peaksappeared in theXRDdata and they thought that theXRDdataofC sample also did not showanynoticeableX-ray diffractionpeaks relevant to the aluminum oxide These results led JIGto the conclusion that the A B and C samples were almost all(sim100) an amorphous aluminum oxide by (1) Because theproduct was an amorphous material these materials couldnot be natural corrosion products of aluminum which areknown to be crystalline by (2) Then JIG concluded that allthese results were clear evidence that the adsorbed materialsof A and B were amorphous aluminum oxides which are theexplosion product of the torpedowhose remnantswere foundnear the incident site We think the steps stated above wereJIGrsquos chain logic to draw the conclusion

The second issue about JIGrsquos report is whether the torpedoremnants were genuine or not Inside the rear section ofthe torpedo remnants JIG found Korean handwrite mark-ing ldquo1bun (No 1 in English in blue ink)rdquo similar to themarking of a North Korean test torpedo obtained in 2003as seen in ⟨Figure Summary-4⟩ and ⟨Figure Summary-5⟩of JIGrsquos report JIG declared promptly ldquowe found conclusiveevidencerdquo for the cause of the incident JIG said theseevidences confirmed that the recovered torpedo parts weremanufactured in North Korea and concluded that ROKSCheonan was split and sunk due to shockwave and bubbleeffects generated by the underwater explosion of a torpedomanufactured byNorth Korea However in JIGrsquos compositionanalysis data Lee [4] indentified the XRD weak peaks forC sample in fact came from diffraction of a small amountof well crystallized 120572-Al

2O3 contrary to JIGrsquos claim that

the weak peaks were irrelevant to aluminum oxide crystalsOne can see clearly weak 120572-Al

2O3Bragg diffraction peaks in

⟨Figure Appendix v-5-3⟩ p280 in JIGrsquos report [1] We thinkthis finding broke JIGrsquos chain logic

This result means that at least the C sample was quitedifferent from the A and B samples and in turn JIGrsquosconclusion above is very questionable as Lee has concludedLee and Yang then reported their own experimental resultsin another study [5] They presented the XRD data in Figure2 of [5] for the sample of an Al-powder that underwentmelting followed by rapid quenching This figure showedprominent Al Bragg diffraction peaks and weak 120572-Al

2O3

Bragg diffraction peaks They said that ldquothis clearly indicatesAl-powder oxides partially not entirely during the heatingand rapid quenchingrdquo This oxidation was supposed to occuron the surface thin layers of each grain ofAl powderTheXRDdata in Figure 2 of [5] showed the similar pattern as in ⟨FigureAppendix V-5-3⟩ of JIGrsquos report for C sample Neverthelessthe EDS data in Figure 1 of [5] showsmarkedly different from⟨Figure Appendix V-5-2⟩ of JIGrsquos report for C sampleThat isthe peak ratio of I(O)I(Al) in Figure 1 of [5] was measured as025 whereas ⟨Figure Appendix v-5-2⟩ showed very differentvalues of 081 On the other hand the authors in [5] noticedthat the high value of around 08-09 for I(O)I(Al) in EDSdata was expected for aluminumhydroxide such as Al(OH)

3

They also showed the simulation results of Al(OH)3and

Al2O3in Figure 4 of [5] supporting their expectation (It

should be noted that the EDS intensity data comes from thesurface thin layers of the sample That is the EDS data ofFigure 1 of [5] came from the surface thin layers of oxidizedgrains with 120572-Al

2O3of the heat treated Al-powder sam-

ple) From these results they concluded that JIGrsquos adsorbedmaterials taken from the warship and the torpedo remnantswere not associated with any explosion and that JIGrsquos EDSdata of their test-explosion sample (Figure Appendix v-5-2) were likely fabricated The South Korean government hasadamantly denied any fabrication or any major problemswith its interpretation of the data Later on independentlythe A and B samples were analyzed to be a basaluminite[Al4(SO4)(OH)

10sdot4-5H

2O] materials by Professor Jeong GY

in Earth amp Environmental Science from Andong NationalUniversity in Korea using several instruments including aTEM He said that the materials appeared to be formed fora long time such as in seawater implying they had nothing todo with explosions Unfortunately he was not able to test theC sample

Cyranoski [6] reported in Nature online the summaryof the controversy over JIGrsquos report Some important pointsraised in the article are as follows

(a) An expert investigator was placed on the JIG by theopposition partymdashShin Sang-chul a former officerin the South Korean navy who had also worked at ashipbuilding companymdashsuggested before the reportwas even released that an accidental collision witha US warship and not North Korea was to blameTheUnited States and South Korea had been carryingout military exercises in the area at the time Now heis insisting on the collision with a submarine with alength of around 60m not by a US warship

(b) The reportrsquos claim that a torpedo-induced water col-umn sank theCheonan contradicted earlier testimonyfrom survivors that they did not see that water col-umn They also testified neither seeing any explosionflash from the underwater nor smelling of explosivesat the time of the incident

Lee and Suh in [7] reported in Policy Forum 10-039ldquoInconsistencies in South Korearsquos Cheonan Reportrdquo In thearticle we found particularly interesting the following point

ldquoIf the bottom of the ship was hit by a bubble it shouldshow a spherical concave deformation resembling the shapeof a bubble as the JIGrsquos own simulation suggests in theAppendix of the report but it does not The bottom of thefront part of the ship is pushed up in an angular shape asthe yellow line shows in ⟨Figure III-1-7⟩ of the report moreconsistent with a collision with a hard object Suh has furtherargued that contrary to common reports that the Cheonanwas split in half as also seen in JIGrsquos simulation it actuallybroke into two larger pieces as well as a third smaller piece(see Figure 1 in this paper)rdquo Shin Sang-chul clearly pointedout if the warship was split by a bubble jet generated fromthe non-contact underwater explosion the top hull of therecovered broken warship should show upward bending butit does not seem the case as can be seen in ⟨Figure II-3-3 and4⟩ and ⟨Figure III-1-8⟩ of the report

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Advances in Acoustics and Vibration

Kim et al in ldquoForeign Policy In Focusrdquo [8] reported brieflyskeptics on JIGrsquos judgment about the Korean handwritemarking ldquo1bunrdquo on the rear section of the torpedo remnantsJIG claimed this handwriting marking had been writtenby a North Korean in the process of manufacturing thetorpedo and is the conclusive evidence that the torpedo wasmade in North Korea Nevertheless Shin Sang-chul found thetrace that the rusts on the panel of the rear section of thetorpedo remnants had been removed by a sand-paper afterthe rust was removed the marking of Korean words ldquo1bunrdquowas written on the panel He said therefore this markingcould not be ldquoconclusive evidencerdquo that the torpedowasmadein North Korea One can judge whether his judgment iscorrect or not from ⟨Figure Summary-4⟩ of the report Wepersonally think Shin Sang-chul is right It is reminded thatthe remnantswere fragments following the torpedo explosionand had been deposited in the mud of seabed for 50 daysand recovered by the special fishing net [1] Considering allthe several doubts explained above we believe the torpedoremnants they found near the incident site could not be agenuine conclusive evidence for the cause of the incidentFurthermore the conclusion was drawn after only five daysof work (the remnant was found on 15 May 2010 and JIGrsquosreport of a summary of its investigation was released on 20May 2010)

B The Free Vibrations ofa Cylindrical Shell with Shear-DiaphragmBoundary Conditions

The Flugge equations of motion for the vibrations of aring stiffened cylindrical shell are given by Caresta andKessissoglou [12]

1205972119906

1205971199112+(1 minus 120592)

21198862(1 + 1205732)

1205972119906

1205971205792+(1 + 120592)

2119886

1205972]120597119911120597120579

+120592

119886

120597119908

120597119911minus 1205732119886

1205973119908

1205971199113+ 1205732

(1 minus 120592)

2119886

1205973119908

1205971199111205971205792minus120574

1198882119871

1205972119906

1205971199052= 0

(B1)

(1 + 120592)

2119886

1205972119906

120597119911120597120579+(1 minus 120592)

2

1205972]1205971199112

+1 + 120583

11988621205972]1205971205792

+1 + 120583 + 120594

1198862120597119908

120597120579+120594

11988621205973119908

1205971205793

+ 1205732 (3 (1 minus 120592)

2

1205972]1205971199112

minus3 minus 120592

2

1205973119908

1205971199112120597120579) minus

120574

1198882119871

1205972]1205971199052

= 0

(B2)

1205732 (11988621205974119908

1205971199114+ 2

1205974119908

12059711991121205971205792+1

11988621205974119908

1205971205794minus 1198861205973119906

1205971199113

+1 minus 120592

2119886

1205973119906

1205971199111205971205792minus3 minus 120592

2

1205973]1205971199112120597120579

+2

11988621205972119908

1205971205792)

+120592

119886

120597119906

120597119911+120594

11988621205973]1205971205793

+2120594

11988621205972119908

1205971205792+1 + 120583 + 120594

1198862120597]120597120579

+1 + 120583 + 2120594

1198862119908 +

120574

1198882119871

1205972119908

1205971199052= 0

(B3)

Here 119906 V and 119908 are the orthogonal components of shelldisplacements in the 119911-axial the 120579-circumferential and the 119903-radial directions in time 119905 respectively The meanings of thesymbols in the equations are defined in Section 2The generalsolutions for shear diaphragm boundary conditions can bewritten as in [13]

119906 (119911 120579 119905) = 119880 cos (119899120579) cos (119896119899119911) 119890minus119895119908119905

V (119911 120579 119905) = 119881 sin (119899120579) sin (119896119899119911) 119890minus119895119908119905

119908 (119911 120579 119905) = 119882 cos (119899120579) sin (119896119899119911) 119890minus119895119908119905

(B4)

These solutions represent standing waves in both the axialand circumferential directions with 119899 nodal lines in the 120579direction and with nodal cross-sections spaced at a distance2120587119896119899in the 119911 direction 119896

119899is the axial wave number and 119899

is the circumferential mode number 119895 is the imaginary unitThe shear diaphragm boundary conditions at 119911 = 0 119871 aregiven by

119873119911=

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911) = 0 (B5)

119872119911=

119864ℎ3

12 (1 minus 1205922)(120576119911

119886+ 120592120576120579+ 119870119911) = 0 (B6)

] (119911 120579 119905) = 0 (B7)

119908 (119911 120579 119905) = 0 (B8)

The membrane force 119873119911and the bending moment 119872

119911are

expressed as functions of the normal strains in the middlesurface 120576

119911and 120576

120579 as well as the mid-surface changes in

curvature119870119911119870120579 The expressions for the strain and changes

in curvature are given by Leissa [13] as

120576119911=120597119906

120597119911

120576120579=1

119886(120597]120597120579+ 119908)

119870119911= minus

1205972119908

1205971199112

119870120579=1

1198862(120597]120597120579minus1205972119908

1205971205792)

(B9)

As explained in [13] the shear diaphragm boundary condi-tions in (B5) to (B8) can be justified for a rigid thin circularcover plate at each end so that ] and 119908 displacements arerestrained at both end boundaries However the plates byvirtue of their thinness would have very little stiffness in

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 9

the 119911 direction transverse to their planes consequentlythey would generate negligible bending moment 119872

119911and

longitudinal membrane force 119873119911in the shell as the shell

deforms Under these boundary conditions the axial wavenumber becomes 119896

119899= 119898120587119871 where 119898 = 0 1 2 is the

axial mode number The natural frequencies of the shell canbe found substituting the solution (B4) with 119896

119899= 119898120587119871 into

(B3) and arrange in matrix form For a nontrivial solutionthe determinant of the matrix must be zero Expansion ofthe determinant leads to the characteristic equation of theshell In vacuum the dispersion equation is of sixth order in120596 From this equation three different natural frequencies canfound for each value of 119896

119899

B1 Frequency Response Function (FRF) Suppose that theshell is axially excited at one end by a point force of unityamplitude located at (119911

0 1205790) The force can be described in

terms of a Dirac delta function in function of the arc length120590 = 119886120579

119865 (1199110 1205790 119905) = 119865

0120575 (120590 minus 120590

0) (B10)

The equilibrium of the membrane force 119873119911 given by (B5)

evaluated at 119911 = 1199110becomes

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

= 1198650120575 (120590 minus 120590

0) (B11)

The delta function 120575(120590 minus 1205900) can be expanded as a Fourier

series around the circumference as

120575 (120590 minus 1205900) =

1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

) (B12)

With 1198791199110= 2120587119886 (B11) can be rewritten as

119864ℎ

1 minus 1205922(120576119911+ 120592120576120579+ℎ2

12119886119870119911)100381610038161003816100381610038161003816100381610038161003816119911=1199110

=1

1198791199110

+2

1198791199110

infin

sum119899=1

cos(21205871198991205901198791199110

)

(B13)

For every circumferential mode number 119899 the 8 boundaryequations as expressed in [12] for the 3 membrane force3 bending moments transverse shearing and the Kelvin-Kirchoff shear force-together with the equilibrium of theforces under point force excitation can be arranged in thematrix form Ax = F x is the vector of the 8 unknowndisplacement coefficients and F is an 8 times 1 force vector withonly one nonzero term 120576119865

0 where 120576 = 12120587119886 if 119899 = 0 and

120576 = 1120587119886 if 119899 = 0 Structural damping can be introducedusing a complex Young modulus 119864 = 119864(1 minus 119895120578) where 120578(about 002) is the structural loss factor Solving the systemfor each circumferential mode number gives the steady stateshell displacement response at a certain frequency (FRF)

C The Path of the Signals Arriving at the BARRecording System

The optimized path of the signals arriving at the BARstation from the submarine is drawn in Figure 6 In this

figure the waves having the velocity 15 kms in seawaterwith the incident angle 109∘ at o on the boundary of theseabedcrust refracted into the crust with the angle 524∘ incase of p-waves (the velocity 63 kms in crust) by Snellrsquos lawUnder this condition the refracted waves will have maximumamplitudes with the transmission coefficient 0174 Then thewaves reflected on the Moho boundary should continuouslytravel to the BAR recording system This path is drawn aso-o1015840-d in Figure 6 For the case of s-waves with velocity of35 kms in crust the incident angle on the boundary must be199∘ to have the same path of o-o1015840-d For all p and s-wavesthe refraction angle is 90∘ at o1015840 with the zero transmissioncoefficients To have this optimized o-o1015840-d with 1422 kmdistance the depth of the Moho boundary must be sim43 kmIt is generally known that the depth ofMoho boundary undersea is usually 5sim7 km As considering this fact we thinkthe estimated depth of the Moho boundary about 43 km isacceptable in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] JIG (Multinational Civilian-Military Joint InvestigationGroup) Joint Investigation Report On the Attack againstROKS Cheonan Military of National Defenses ROK 2010httpcheonan46gokr100

[2] ROKS Cheonan sinking httpenwikipediaorgwikiROKSCheonan sinking

[3] T-K Hong ldquoSeismic investigation of the 26March 2010 sinkingof the South Korean naval vessel Cheonanhamrdquo Bulletin of theSeismological Society of America vol 101 no 4 pp 1554ndash15622011

[4] S-H Lee ldquoComments on the section ldquoadsorbed materialanalysisrdquo of the Cheonan report made by the South Koreancivil and military joint investigation group (CIV-MIL JIG)rdquohttparxivorgabs10060680v2

[5] S-H Lee and P S Yang ldquoWas the ldquoCritical Evidencerdquo presentedin the South Korean Official Cheonan Report fabricatedrdquohttparxivorgabs10060680v4

[6] D Cyranoski ldquoControversy over South Korearsquos sunken shiprdquoNature July 2010 httpwwwnaturecomnews2010100708fullnews2010343html

[7] S-H Lee and J J Suh ldquoRush to judgment inconsistencies inSouth Korearsquos Cheonan reportrdquo Policy Forum 10-039The Asia-Pacific Journal Japan Focus 2010 httpwwwjapanfocusorg-Seunghun-Lee3382

[8] H-E Kim G Chaffin and P Certo The Cheonan IncidentSkepticism Abounds Foreign Policy in Focus Washington DCUSA 2010

[9] S G Kim and Y Gitterman ldquoUnderwater explosion (UWE)analysis of the ROKS cheonan incidentrdquo Pure and AppliedGeophysics vol 170 no 4 pp 547ndash560 2013

[10] M S Weinstein ldquoSpectra of acoustic and seismic signalsgenerated by underwater explosions during Chase experimentrdquoJournal of Geophysical Research vol 73 pp 5473ndash5476 1968

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Advances in Acoustics and Vibration

[11] M Caresta Structual and acoustic responses of a submergedvessel [Ph D thesis]Mechanical EngineeringUniversity ofNewSouth Wales Sydney Australia 2009

[12] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[13] A W Leissa Vibration of Shells American Institute of PhysicsNew York NY USA 1993

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of