Research Article Floquet-Bloch Theory and Its Application to the ...downloads.hindawi.com/journals/mpe/2015/475364.pdf · Research Article Floquet-Bloch Theory and Its Application

Research ArticleFloquet-Bloch Theory and Its Application to the DispersionCurves of Nonperiodic Layered Systems

Pablo Goacutemez Garciacutea12 and Joseacute-Paulino Fernaacutendez-Aacutelvarez1

1Hydrogeophysics and NDT Modelling Unit University of Oviedo CGonzalo Gutierrez Quiros sn 33600 Mieres Spain2Dynamics Division Applied Mechanics Department Chalmers University of Technology Horsalsvagen 7 41296 Gothenburg Sweden

Correspondence should be addressed to Pablo Gomez Garcıa pggsierragmailcom

Received 20 October 2014 Revised 28 November 2014 Accepted 29 November 2014

Academic Editor Xiao-Qiao He

Copyright copy 2015 P Gomez Garcıa and J-P Fernandez-Alvarez This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

Dispersion curves play a relevant role in nondestructive testingThey provide estimations of the elastic and geometrical parametersfrom experiments and offer a better perspective to explain the wave field behavior inside bodies They are obtained by differentmethods The Floquet-Bloch theory is presented as an alternative to them The method is explained in an intuitive manner it iscompared to other frequently employed techniques like searching root based algorithms or the multichannel analysis of surfacewaves methodology and finally applied to fit the results of a real experiment The Floquet-Bloch strategy computes the solution ona unit cell whose influence is studied here It is implemented in commercially finite element software and increasing the numberof layers of the system does not bring additional numerical difficulties The lateral unboundedness of the layers is implicitly takencare of without having to resort to artificial extensions of the modelling domain designed to produce damping as happens withperfectly matched layers or absorbing regions The study is performed for the single layer case and the results indicate that for unitcell aspect ratios under 02 accurate dispersion curves are obtainedThemethod is finally used to estimate the elastic parameters ofa real steel slab

1 Introduction

Floquet-Bloch (hereafter F-B) theory provides a strategy toanalyze the behavior of systemswith a periodic structure Flo-quetrsquos seminal paper dealt with the solution of 1D partial dif-ferential equations with periodic coefficients [1] In solid statephysics Bloch generalized Floquetrsquos results to 3D systems andobtained the description of the wave function associated withan electron traveling across a periodic crystal lattice [2] Thiswave function is a solution of the Schrodinger equationwith aperiodic potential andBloch showed that it was the product ofa simple planewavemultiplied by a periodic functionwith thesame periodicity of the lattice The mathematical descriptionof these ideas in the context of quantum mechanics can befound in [3 4]

In the literature dealing with wave propagation problemsin mechanical systems the theory is referred to as Floquet-Bloch theory or simply Floquet theory In layered systems

due to the heterogeneity of the relevant elastic properties toparticular geometric features or to both only certain wavemodes can physically propagate inside the structure [5] Eachof these modes can be identified by a determinedmdashgenerallynonlinearmdashfunction relating the time frequency and thespatial frequency (or wave number) These relationships arecalled dispersion curves default and as they summarize allthe oscillatory behavior of the system their calculation is ofparamount importance in NDE applications [6]

Vibrations occur also in objects with periodic structure[7] These problems usually admit a separation between thetime and the spatial dependent parts of the solution For ins-tance the Helmholtz equation is a known example of equ-ation describing the spatial behavior [8] There the physicalperiodic structure of the studied object translates into spatialperiodicity of its coefficients Therefore the F-B theory hasbeen applied to obtain the dispersive properties of differentmechanical periodic systems [8ndash12]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 475364 12 pageshttpdxdoiorg1011552015475364

2 Mathematical Problems in Engineering

Many relevant structures can be assumed to be layeredsystems of infinite extent for example [13 14] in civil engi-neering constructions [15 16] in optics or [17] in electroma-gnetics Therefore theoretical methods and experimentaltechniques to obtain their dispersion curves have beendevised From the theoretical side different matrix tech-niques have been developed to address the calculation Theyinvolve numerical computational methods whose complexityincreases with the number of layers in the system [18 19] ormore recently [6]

In laboratory experiments or field work the dispersioncurves can be obtained using for example the multichannelanalysis of surface waves (MASW) methodTheMASW pro-cedure involves collecting equally spaced measures of vibra-tion along a profile on the system surface using for exampleaccelerometers The resulting 2D space-time discrete imageis Fourier-transformed to the frequency-wave number (120596 119896)

domain and then processed to build the dispersion curves[20 21] The method has some drawbacks inherent to theFourier transform limitations which will be discussed laterThe MASW has been applied successfully in the characteri-zation of pavement systems [13] as a seismic data acquisitiontechnique [20] or for geotechnical characterization [22] TheMASW strategy is here also used to perform a computernumerical simulation of the system closely mimicking thefield setupThe issue of infinite lateral extent is usually tackledby using perfectly matched layers (PML) [23ndash26] as has beendone here or absorbing regions Both techniques presentdrawbacks [26]

In this paper an alternative way to calculate the disper-sion curves of layered systems with infinite lateral extentusing the F-B theory is presentedThemethod has never beenapplied to the dispersion curves calculations of nonperiodiclayered systems Here it is used to obtain the dispersioncurves of a single layer case and to estimate the elastic param-eters of a real steel slab for showing themethodHowever thenovelty in this work is that it can be applied to an arbitrarynumber of layers even if the layers are anisotropic or ortho-tropic with the same complexity levelThe power of the met-hod is that the equations are solved by the finite elementsoftware because the F-B theory only affects the propagationterm which is the same whatever the nature of the layersIt is not necessary to develop the equations for each specificproblem and to generate complex codes to get the dispersionrelations

The F-B theory reduces the problem to calculationsperformed in the so-called unit cell subject to certain specificboundary conditions derived from the F-B theory and elasto-dynamics The influence of the size of the unit cell is ascert-ained The results are first compared with the dispersion cur-ves derived from the Rayleigh equations [5] solved by asearching root numerical method Comparison is also madewith the curves resulting from a FEM computer simulationfollowed by a 2D Fourier transformation Finally a realexperiment was performed on a steel slab employing theMASW method and the empirical dispersion curves werecompared with the analytical and numerical ones

The results show that the F-Bmethod compares favorablywith other methods and fits accurately the empirical data

providing a good alternative to obtain dispersion curves inlayered systemsThe F-B technique can be run on a finite ele-ment package like COMSOL Multiphysics can be applied toan arbitrary number of layers in the system with the samecomplexity level and eliminates issues of infinite lateral ext-ent

2 Floquet-Bloch Theory Explanation

The Floquet-Bloch theory provides a strategy to obtain a setof solutions of a linear ordinary equations system of the form

f1rfloor (119909) = M (119909) f (119909) (1)

where f(119909) is the solution vector and thematrixM is periodicsuch thatM(119909+119871) = M(119909) for a certain period 119871 At first sightit might seem that the solution of such a problem would haveto be also 119871-periodic But Floquet showed that this need notbe soThere exists however a simple relationship between thesolutionrsquos behaviors inside one period and outside it If F is afundamental matrix of solutions then another matrix B canbe found such that

F (119909 + 119871) = BF (119909) (2)

B can be constructed by setting 119909 = 0 in (2) such thatB = Fminus1(0)F(119871) A simplest case is obtained using F(0) = Iso that B = F(119871) As there is not a unique choice for thefundamental matrix F and how it is exactly chosen dependson the problem B is also not unique But its eigenvalues areintrinsic of the problem and under the right transformationcan be used as a propagator or evolution factor 120588 relating thevalue of the solution at a point inside the period with its valueat a point outside of it Only the solution inside a period istherefore needed verifying that

f (119909 + 119871) = 120588f (119909) (3)

Following the classical nomenclature 120588 = exp(119896119865119871) is known

as Floquetmultiplier 119896119865being the complex Floquet exponent

Moreover Floquet found that the solution at any point canalso be factored in two terms

f (119909) = p (119909) exp (119896119865119909) (4)

Here p(119909) is a periodic function playing the role of theeigenvectors if M was a constant matrix and carrying theperiodicity 119871 of the coefficients of the problem The complexexponential distorts the strict periodicity of p(119909) incorporat-ing damping or ever growing effects in the amplitudes of thesolution depending on the value of |120588| This is why solutionsare in general not periodic and also why the Floquetperspective is usually employed to study their stability Asolution will be stable if the Floquet multipliers verify |120588| le 1

[27]

3 Guided Waves in LayersAnalytical Dispersion Curves

The guided wave propagation problem in a homogeneousisotropic and infinite single layer has been widely treated in

Mathematical Problems in Engineering 3

Traction free surfacesPropagating shape y

2b

rarrt

Layer

Propagation direction

infinite extent

x

zInfinite extentplain strain

(a)

Symmetric mode

Antisymmetric mode

(b)

0 5 10 150

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000Ph

ase v

eloci

ty (m

s)

Frequency (Hz) times104

A0

S0

A1

S1

A2

S2

c1

c2cR

(c)

Figure 1 An infinite homogenous isotropic and elastic layer with thickness 119889 = 2119887 as part of an infinite 3D layer under plain strainconsideration (a) Typical deformation for the symmetric and antisymmetric modes (b) and the dispersion curves in (V minus 119891) representationfor a given thickness 119887 = 0045m P and S velocities 119888

1= 5800ms and 119888

2= 3200ms obtained with searching root algorithm (c) The parts

marked with dashed boxes and the intersection points between modes present difficulties with typical searching root algorithms

the literature [5 28] In this paper we follow the theory devel-oped in [5] So consider an infinite (in 119909 direction) homoge-neous isotropic and elastic layer with thickness 2119887 as shownin Figure 1

The conservation of momentum equation plus free trac-tion boundary conditions leads to a system of equations thatproduces a solution when its determinant vanishes In theabsence of body forces the equation reads

120588u (119909 119910 119905) + nabla [C (119909 119910) nabla120576(u (119909 119910 119905))] = 0 (5)

where u is the displacement vector C(119909 119910) is the elasticconstants tensor and nabla

120576(u(119909 119910 119905)) is the strain tensor If u isin

1198622 then u = nabla120601 + nabla times H Here 120601 is a scalar field andH is a vector field For the plane strain case the fields P-SV and SH are decoupled Considering the P-SV field the 119911

component of the displacement field and its partial derivative119906119911

= 120597119906119911

= 0 vanish A solution of the form 119906(119909 119910 119905) =

119891(119910) exp(119896119909 minus 120596119905) appears It expresses the propagation of

a shape 119891(119910) (Figure 1) trapped in the thickness of the layeralong the 119909-direction with wave number 119896 and frequency 120596The dispersion relation for the single infinite layer case is [5]

tan120572119887

tan120573119887= minus[

41205721205731198962

(1198962 minus 1205732)2]

plusmn1

(6)

with

1205722=

1205962

11988821

minus 1198962 120573

2=

1205962

11988822

minus 1198962 (7)

where 1198881is the compressional wave (P) velocity and 119888

2the

shear wave (S) velocity Expression equation (6) is known asthe Rayleigh-Lamb equation (R-L) The plus sign in the (R-L) expression corresponds with symmetric modes and theminus with the antisymmetric ones (Figure 1)

31 Calculation of the Dispersion Curves from the AnalyticalRayleigh-Lamb Equation The R-L equation is transcenden-tal so no closed analytical solution is available It can though


be cast into a form amenable to the use of iterative rootfinding local algorithms but these present various difficultiesarising from the nature of the equations First due to thetangent functions the left hand side is discontinuous at cer-tain points where local algorithms for smooth functions willfind difficulties [29] Due to this and to the sampling ratecharacteristics of the searching algorithm some roots mightbe missed

A visual inspection to the pattern followed by the dis-persion curves on a phase velocity versus frequency (V minus 119891)

plot (Figure 1) clearly shows the presence of near vertical andhorizontal stretches with physical relevance For instance thehorizontal line towards which the modes A0 and S0 convergecontains the information of the so-called Rayleighwaves [30]The S1 mode on the other hand owns a point with verticaltangent that is minimumwave number in an otherwise nearvertical portion of the curve with the most important prope-rty of having zero group velocity producing then a useful res-onance [31]

For local root searching methods like the Newton-Raph-sonmethod [32] the strategy consists in using one found rootas a seed to calculate the next point Methods performing a1D search with a fixed value of 119896 (resp 120596) will very poorlycharacterize a near vertical (resp horizontal) portion of thecurve unless the sampling rate is extremely dense Othermethods start from the frequencies at zero wave numberwhere the roots can be calculated analytically and try to foll-ow up each curve with some sort of linear predictions [33]Problems arise at the mode intersections (Figure 1) It hasbeen suggested that the search grid should locally mimicthe behavior of the dispersion curves [34] At any rate themethods are computationally intensive for useful tolerancesand their extension to the case ofmore layers where the curvepattern is much more intricate is inefficient

In this paper the root loci of the R-L equation have beenobtained using the bisection method which always con-verges It is very slow and fails wherevermultiple roots exist inthe proposed interval Apair of time frequency phase velocityvalues are input in the R-L equation and its sign evaluatedKeeping the frequency fixed the velocity has been variedwitha step of 120575V = 200ms from 0 to 10

4ms When the signchanges the bisection method is iteratively applied to obtaina root with an accuracy 120575120576 = 03ms The process is repeatedfor different frequency values and the roots are plotted in(V minus 119891) (Figure 1)

The elastic parameters input to the R-L equation arethickness 119889 = 0045m longitudinal wave (P) velocity 119888

1=

5800ms and shear wave (S) velocity 1198882= 3200ms because

theywill be shown to produce the best fit to the empirical testsperformed on a steel slab (Section 6)

Some of the problems discussed above can be clearly seenin the (V minus 119891) representation (Figure 1(c)) The upper partof the modes (S1 S2) was impossible to calculate becausedue to their almost infinite slope the number of roots theremay be found to be infinite or zero Starting instead froma fixed velocity shifts the difficulty to the near horizontalsegments of the curves Besides a regular searching grid inthe (119891 minus 119896) plane becomes an irregular sampling in the (V minus

119891) domain where some stretches with the same amount of

complexity independent of the number of layers may notbe sufficiently well characterized The proposed F-B basedmethod will allow an arbitrary degree of accuracy in thecalculation of the curves without suffering from these prob-lematic issuesThis might also be an advantage for systematicperformance of sensitivity analysis testing a relevant amountof perturbed models around one found solution

4 Dispersion Curves Calculation Using FEMwith a MASW Type Scheme

The multichannel analysis of surface waves (MASW) meth-odology is a procedure to numerically or empirically calculatedispersion curvesThe strategy is tomeasure (or obtain num-erically) on the surface of the system the wave field in anumber of equally spaced points and take readings at a certaintemporal sampling rate [35] This sampled 2D time-spacefield is then Fourier-transformed into a 2D time frequency-wave number domain A continuous surface of amplitudes isnow obtained in this (V minus 119891) domain using interpolation orfitting methodsThe dispersion curves can be obtained as theloci of local maxima of that surface The MASWmethod hasbeen discussed in the literature [20 36 37]

Given the inverse relationship between the length of theprofile on the surface and the wave number sampling intervaland correspondingly between the range of sensed wavenumbers and the distance between adjacent sensors [38] fieldconstraints (ie of logistic or economic type) on the feasibleprofile length or on the number of sensors available for usedo influence the representation Reciprocity [21] alleviates thedifficulty allowing keeping one sensor fixed while movingthe impactor in the so-called multichannel record with onereceiver (MROR) technique

When the scheme is applied to perform a numericalcalculation on guided waves the simulation domain has tobe finiteThis brings the problem of unwanted reflections andmode conversions at those boundaries coming back into therelevant domain and corrupting the signal The naive optionof extending the domain implies increasing the number ofnodes and computation times and assumes that boundaryreflection events are separated in time from the studied eve-nts which might not be possible The use of the so-calledabsorbing regions where the wave field enters and is compu-tationally absorbed has been treated in the literature [23ndash26]For instance perfectly matched layers (PML) or absorbinglayers using increasing damping (ALID) have been frequentlyemployed PML are regions attached to the boundaries wherethe wave enters and decays exponentially [25 26] In ALIDthe domain is enlarged with layers of the same material butwith increasing damping parameters [39] Although both areimplemented in commercial finite element codes the successof both techniques relies on iteratively finding the optimumdesign parameters This can be time consuming and varieswith the characteristics lack of the system one is interested in

In this case COMSOL Multiphysics has been used forthe simulation The characteristic parameters employed forthe PML are PML scaling factor 119865 = 1 and PML curvatureparameter 119899 = 1 (Figure 2)


Computational domain

Impact point

PML (F n) PML (F n)

120575L 120575LL

120575x

dmiddot middot middot

Figure 2 Typical scheme for MASW method implementation in aFEM software The computational domain has been set to 119871 = 3mthe PML parameters used 119865 = 1 and 119899 = 1 the distance betweenreceivers 120575119909 = 5 cm the PMLrsquos sizes 120575119871 = 1m and the thickness119889 = 10 cm

41 Numerical Implementation In this section the results ofthe numerical simulations using the FEM software COMSOLMultiphysics following a MASW procedure and employingPML to take lateral unboundedness into account are pre-sented The dispersion curves will be discussed and serveas a reference to be compared with those obtained by thesearching root algorithm (Section 3) those calculated usingthe F-B approach (Section 5) and the field curves presentedin Section 6 The elastic parameters are the same as inSection 3 and the thickness now is 119889 = 10 cm The lengthof the simulated profile is 119871

119901= 2m and that of the PML is

120575119871 = 1mA frequency domain study has been performed with a

step of 120575119891 = 100Hz and sweeping frequencies up to 119891 =

50 kHz with uniform energy distribution 40 point acceler-ometers are separated 120575119909 = 5 cmThe results after 2D Fouriertransform and interpolation are shown in Figure 3

Some observations are in order First certain portionsof some modes are not excited [40ndash42] This affects mainlythe lower frequency parts of the zero order symmetric mode(S0) Should a half cycle sinusoidal function have been usedas impact model [43] the frequency energy input at lowerfrequencies would have been relatively higher and the A0low frequency part would be visible whereas the now visiblehigher frequency branch would be absent This argumentdoes not affect the S0 mode

The empty triangular space in the bottom right part ofFigure 3 corresponds to pairs of frequency-wave numbersimpossible to be reached with a sensor separation of 120575119909 =

5 cm For this sampling distance the highest representablewave number is 119896 = 6282mminus1 As Vph = 2120587119891119896 the phasevelocity stays always over Vmin = 21205871198916282 for any givenfrequency

Additional numerical artifacts arise due to the finitelength of the profile As this is mathematically equivalent tomultiplication by a boxcar window it generates in the wavenumber domain a convolution with a sinc function Thiseffect has been zoomed in Figure 4 This spatial convolutionshows up Figure 4(b) as a spurious repetition of some branc-hes For more complex patterns present in more than onelayer system the real position of the dispersion lines becomesmore uncertain

All the described difficulties with the MASW domain areabsent in the Floquet-Bloch technique

5 Floquet-Bloch Theory andGuided Waves in Layers

The equation of movement for the single layer case was pre-sented in Section 3 (5) To solve it it is necessary to try planewave type solutions

u (119909 119910 119905) = 119891 (119910) exp 119894 (119896119909 minus 120596119905) (8)

Equation (8) establishes that the displacement solution ofthe problem is a certain shape 119891(119910) trapped in the thicknessof the system which propagates in 119909-direction with a certainwave number 119896 and frequency 120596 along the infinite lateralextension (Figure 1)

The layer is considered infinite however the solutions canbe computed over a finite computational domain (unit cell)subject to certain boundary conditions Consider an infinitelayer as is shown in Figure 5

According to [43] a layer behavior will appear experi-mentally if both dimensions of the layer are at least ten timesthe thickness dimension This goal will be achieved in theexperimental study taking into account the values presentedin Figure 7

Consider now the spatial part of (8)

u (119909 119910) = 119891 (119910) exp (119894119896119909) (9)

Therefore the displacement field at the left side of the unitcell (Figure 5(c)) will be related to the displacement at theright side such that

u (119909 + 119871 119910) = 119891 (119910) exp (119894119896119909) exp (119894119896119871)

= u (119909 119910) exp (119894119896119871) (10)

Note that the function 119891(119910) does not change(Figure 5(c)) It defines the form of the considered mode andis only the propagative term (the exponential ones) whichdefines the propagation of 119891(119910)

Moreover a lateral infinite layered system is a triviallyperiodic medium in the propagation 119909-direction Due to itthe F-B theory states that the solution of the problem can bewritten as

u119899119896FB

(119909 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119909) (11)

For certain periodic function u119899119901119896FB

(119909 119910) F-B exponent119896FB and where the 119899th-dependency appears due to the rela-tionship between the wave number 119896 and the F-B exponent119896FB

From (9) and (11) the solution of the problem will beequivalent if the wave vector 119896 and the F-B exponent 119896FB arerelated as

119896 =2119899120587

119871+ 119896FB (12)

where 119871 is the length of the unit cell in the propagation 119909-direction (Figures 5(b) and 5(c)) such that substituting (12)into (9) leads to

u119899119896FB

(119909 119910) = 119891 (119910) exp (2119899120587119909

119871) exp (119896FB119909) (13)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1

S0A0

A1

(a)

1 2 3 4 5 6 7 8

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms

)


S1

S0

A0

S2

ph =2120587

kf

(b)

Figure 3 Dispersion curves for a single layer in frequency-wave number representation (a) and in phase velocity-frequency representation(b) Parts that are lacking in the phase velocity plot (b) correspond to wave numbers greater than the measured ones

05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1S0

A0

A1

(a)

26999 27 27001 27002 27003 2700332

34

36

38

40

42

44

46

Frequency (Hz)

Wav

e num

ber (

m1)

times104

Δk

(b)

Figure 4 Dispersion curves in (119896 minus 119891) representation obtained with padded signals (a) Zoom of the box marked in the left plot and thefast Fourier transform of the box function used for padding the signals (b) The distance between zeros of the obtained sinc function isΔ119896 = 2120587119873120575119909 = 31 where 120575119909 is the spatial sampled rate and 119873 is the number of measured points

where now the term u119899119901119896FB

= 119891(119910) exp(2119899120587119909119871) is clearly 119871-periodic However due to the 119899th-dependent term a fixedvalue of the F-B exponent 119896FB and 119899 values of the wave vector119896 appear

Now the relation of the solutions at the left and right sidesof the unit cell can be used to define the proposed F-B boun-dary conditions as

u119899119896119861

(119909 + 119871 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119871) (14)

Due to the relationship between 119896FB and the 119899th-depe-ndent periodic term the dispersion relation 120596(119896) becomes120596119899(119896FB) which can be calculated over a unit cell of arbitrary

length 119871 solving the following eigenvalue problem (Figure 5)

1205881205962

119899(119896FB) u119899119896FB (119909 119910 120596119899) + nablaC (119909 119910)

times nabla120576(u119899119896FB

(119909 119910 120596119899)) = 0 forall (119909 119910) isin Ω

119877


Infinite

Infinite

Unit cell45 cm

(a)

Unit cell

Aspect ratio

L

ΩR

f(y)120575ΩRx

120575ΩRx

y

x

2b = 45 cm

Ra = L2b

(b)

Unit cell

F-B boundary conditionu(x + L y) = u(x y) middot eikFBL

f(y) f(y)

L

(c)

Figure 5 Infinite layer and the unit cell elected (a) Computational domain (unit cell) properties and dimensions (b) F-B boundary conditionsapplied to the laterals of the computational domain (unit cell) for a certain shape 119891(119910)

First zone

10

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100 120 140

Freq

uenc

y (H

z)

Propagative wave number (radm)

1205961 k0

1205960 k0

1205961 k1

times104 120587L

(a)

Good infinite layer dispersion curves

representation zoneFirstzone

45

4

35

3

25

2

15

1

05

00 200 400 600 800 1000 1200 1400

Freq

uenc

y (H

z)


times105 120587L

(b)

Figure 6 Dispersion curves in (119891 minus 119896) representation obtained with the proposed Floquet-Bloch method for different aspect ratios of theunit cell Ra = 1 (a) and Ra = 01 (b)

[nablaC(119909 119910)nabla120576(u119899119896FB

(119909 119910 120596119899))]119910= 0 forall (119909 119910) isin 120575Ω

119877119910

u119899119896119861

((119909 119910) + (119871 2119887) 120596119899)

= u119899119896119861

((119909 119910) 120596119899) 119890119894119896FB119871 forall (119909 119910) isin 120575Ω

119877119909

(15)where u

119899119896FBare the displacement field and [nablaC(119909 119910)nabla

120576(u119899119896FB

(119909 119910 120596119899))]119910any component in the 119910-direction of the stress

tensor (traction free-surface boundaries in the unit cell)The theory presented in this section applied to the lateral

sides of the unit cell can be used for any kind of layeredsystems whatever the nature of the layers (isotropic or aniso-tropic) and the number of them are The reason is that theproposed F-B boundary conditions affect only the propaga-tive part of the solution (119909-direction) which appears in allterms of the equations when the problem is treated in a clas-sical analytical way and is simplified disappearing from theequations

Therefore the complicated part of the problem which isto obtain the function 119891(119910) (given the propagation modes)is solved by the finite element software This kind of softwareallows solving a huge kind of complicated systems which areintractable analytically This is the case for example of lay-ered systems with a big number of layers or when the layersare anisotropic composites

However the perspective for getting analytical solutionsis always the same to obtain a function (complicated in gen-eral) in the thickness direction (the mode) which propagateslaterally In this approach the propagation part is always extr-acted in the equations when infinite layered systems are con-sidered

Because of the above reason the theory applied to definethe F-B boundary conditions can be used for any complicatedsystems while they are laterally infiniteThe boundary condi-tions only affect the propagation part and allow converting


15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)

Figure 7 Outline of the experiment assembly (a) a photograph (b) and the impacts generated in the experiment in the time domain (c) andspectra (d) The average time has been Δ119905 = 143 plusmn 7 120583119904

the infinity analytical problem in a finite problem whichcan be solved with commercial Finite Element softwareTherefore it is not necessary to develop equations (generallycomplicated) for each certain problem and perform complexnumerical codes based on searching roots algorithms [4445]

51 Aspect Ratio Effects Using COMSOL Multiphysics soft-ware the eigenvalue problem equation (15) can be solvedby sweeping different values of 119896FB and obtaining the corre-sponding values 120596

119899(119896FB)

However to transform the problem of the infinite layeredsystem to a finite problem through the F-B boundary con-ditions introduce artificial aspects due to the periodicity ofthe 119899-dependent term For this reason given certain F-Bwavenumber 119896FB from (12) 119899 values of the wave number 119896

119899(and

eigenfrequencies 120596119899) will be obtained

An example is presented in Figure 6(a) where the valueof the F-B wave vector is fixed as 119896FB = 119896

0 For this value

all eigenfrequencies 120596119899corresponding with the propagation

wave vector values derived from (12) are obtained (in theexample only are presented the fundamental one 120596

0and the

first one corresponding to 119899 = 1 as 1205961 which match with the

wave vector 1198961= 120587119871 + 119896

0)

52 Results andDiscussionDerived from the Proposed Floquet-Bloch Method Since the calculation is developed in terms ofthe F-Bwave vector the solution becomes119871-periodic and thedispersion representation obtained has all eigenfrequenciesinside the first periodic zone of the solution (Figure 6) Dueto it the branches of the dispersion curves are reflected backin the limits of the zone being necessary to take into accountthe aspect ratio of the computational domain (unit cell) interms of getting a good dispersion curves representation

Different aspect ratios of the unit cell Ra = 2119887119871 havebeen explored Ra = 1 04 02 01 to obtain the dispersioncurves Results for the cases Ra = 1 and Ra = 01 are shownin Figure 6 The value of 119871 chosen for the unit cell establishes


at 120587119871 the periodicity in the wave number domain Thereflected lines intersect the canonical modes in the unit cellpreventing them from being clearly identified (Figure 6(a))Aspect ratios smaller than 02 are usually enough to obtaina representative number of modes defining clear dispersioncurves (Figure 6(b))

In the case Ra = 1 the limit 120587119871 is situated at low wavenumber values and the reflected branches render the spec-trum unclear However a good representation is obtainedfor Ra = 01 (emphasized in the blue frame of Figure 6(b))because the reflected branches arise at higher wave numbers(Figure 6(b))

A good criterion is to choose the lateral dimension at leastfive times the thickness in the computational domain

6 Real Test on a Steel Slab and Results

A real NDT experiment has been conducted on the surface ofa steel slab shown in Figure 7The profile was set up along thesymmetry axis and included 57 equally spaced measurementpoints separated at a distance of 5 cm An instrumentedhammer and accelerometers have been employed followingthe MROR method described in Section 4 The accelerome-ters were threaded producing a coupling resonance around12KHz An outline of the experiment configuration togetherwith the relevant dimensions and photographs of the testsetup and the impacts generated is shown in Figure 7

The shape of the hammer impacts in the time domainis very consistent (Figure 7) The average impact duration isΔ119905 = 143 plusmn 7 120583119904 Beyond 119891 = 15 kHz the amplitudes of theimpact spectra are very weak and fall mostly under the meas-urement noise level

61 Results and Discussion The resulting empirical disper-sion curves can be seen in Figure 8(c)

The estimated parameters using the proposed F-B bound-ary conditions are presented in Table 1

The estimated errors are obtained with the propagationlaw (16) taking into account the temporal and spatial fre-quency steps 120575119891 = 1667Hz 120575119896 = 01754 based on theNyquist criteria and on the fact that V = 2120587119891119896 The com-putation time in finite element software is under 2 minutescomputing the curves in an i7 PC processor

120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)

A fit has been achieved where only the A0mode is clearlyseen in the measurements There are two main reasons forthat On the one hand the tip of the hammer only signif-icantly inputs frequencies until 15 kHz (Figure 7(d)) so thatevery mode including the A0 above this frequency will notbe seen On the other hand the S0 mode is not there becauseits excitability is very low [42] The excitability is a conceptrelated to what parts of the different modes are detectable atthe surface These commercial receivers measure the out-of-plane (perpendicular to the surface) acceleration componentwhich has a very low excitability value at the low frequency

Table 1 Estimated elastic parameters of a real steel slab using theexperimental signals obtained with MASW method fitted with theproposed F-B method

Density (kgm3) P wave velocity (ms) S wave velocity (ms)7850 5800 plusmn 30 3200 plusmn 23

part of the S0 mode The vertical band of high energy near1ndash12 kHz is the coupling frequency of the accelerometer

62 Comparison of theThreeMethodologies This section pre-sents the comparison of the different methods to obtain dis-persion curves the searching root method the numericalMASW method together with the proposed F-B methodand their ability to match the empirical dispersion curvesThe results have been calculated for a layer with the samethickness as the slab used for the experiment The numericalMASW simulation performed in COMSOLMultiphysics hasused as impacts the experimental ones Figure 8(a) presentsthe dispersion curves obtained with the searching rootalgorithm and with F-B method and the numerical MASWsimulation together with the relevant part of the experiment

From Figure 8 the proposed F-B method matches per-fectly with the analytical solution of the dispersion rela-tion in the layer and with the experimental and simulateddispersion curves obtained with MASW method As wasemphasized in previous sections the F-B method providesbetter results in the vertical zones of the modes than themethod based on searching roots algorithm (Figure 8(a))Another feature is that the F-B method as analytical methodis not affected by the excitability concept as happens withthe MASW method because the F-B method is not based onthemeasurements of the displacement fieldThe results of thesimulated experiment (Figure 8(b)) are again affected by theexcitability and the S0mode is not obtained as happens in thereal experiment which might be taken as a proof that the S0absence is not due to pitfalls in the measuring process Theexperimental results are affected by the resonant frequenciesof the accomplished method used for the accelerometers(Figure 8(c)) highlighting that the measuring process couldbe improved proving different coupling systems

7 Conclusions

Themain conclusion of this study is that the F-B theory can beused to compute the theoretical dispersion curves of layeredsystems with infinite lateral extension over a finite unit cellThe method is applied directly using a finite element com-mercial software and is free of the drawbacks associated withother numerical procedures used It is also a tidy method tocalculate curves with more than one layer avoiding ambi-guities at crossing points or with particular slopes Basedon the obtained results aspect ratios with lower values thanRa le 02 of the computational domain are enough to obtaina good number of modes in a clear plot representation Themethod allows obtaining the associated eigenvectors in (15)


0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)


Searching root methodF-B method

(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms

)Frequency (Hz) times104

(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)

Figure 8 Superposition of the dispersion curves obtained with the numerical searching root method (6) and with the proposed F-Bmethodfor aspect ratio Ra = 01 (a) Overlap of F-B method and the simulated MASW method dispersion curves (b) Overlap of F-B method andthe experimental dispersion curves (c)

so the excitability curves could be obtained too in the samecomputing process and compared with those results obtainedwith the MASW procedure The infinite lateral extension isimplicitly taken into account without the need to use ad hocdomain extensions

Conflict of Interests

The authors Pablo Gomez Garcıa and Jose-Paulino Ferna-ndez-Alvarez declare that there is no conflict of interests rega-rding the publication of this paper


Acknowledgments

Thanks are due to the Wave Propagation Division of Chal-mers University and specially to Professor Anders Bostrombecause of his suggested corrections in the development ofthis work

References

[1] G Floquet ldquoSur les equations differentielles lineaires a coef-ficients periodiquesrdquo Annales scientifiques de lrsquoEcole NormaleSuperieure vol 12 pp 47ndash88 1883

[2] F Bloch ldquoUber die Quantenmechanik der Elektronen in Krist-allgitternrdquo Zeitschrift fur Physik A vol 52 no 7-8 pp 555ndash6001929

[3] C Kittel and H Y Fan ldquoIntroduction to solid state physicsrdquoAmerican Journal of Physics vol 25 no 5 p 330 1957

[4] L Brillouin Wave Propagation in Periodic Structures ElectricFilters and Crystal Lattices 1946

[5] K F GraffWave Motion in Elastic Solids Dover 1991[6] M Lowe ldquoMatrix techniques for modeling ultrasonic waves in

multilayered mediardquo IEEE Transactions on Ultrasonics Ferro-electrics and Frequency Control vol 42 no 4 pp 525ndash542 1995

[7] MColletMOuisseM Ruzzene andMN Ichchou ldquoFloquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic damped mechanical systemsrdquo Interna-tional Journal of Solids and Structures vol 48 no 20 pp 2837ndash2848 2011

[8] J Tausch ldquoComputing Floquet-Blochmodes in biperiodic slabswith boundary elementsrdquo Journal of Computational and AppliedMathematics vol 254 pp 192ndash203 2013

[9] J Butler G Evans L Pang and P Congdon ldquoAnalysis ofgrating-assisted directional couplers using the Floquet-Blochtheoryrdquo Journal of Lightwave Technology vol 15 no 12 pp 2301ndash2315 1997

[10] A Safaeinili D E Chimenti B A Auld and S K Datta ldquoFlo-quet analysis of guided waves propagating in periodically lay-ered compositesrdquo Composites Engineering vol 5 no 12 pp1471ndash1476 1995

[11] S Dahmen M Ben Amor andM H Ben Ghozlen ldquoAn inverseprocedure for determination of material constants of a periodicmultilayer using Floquet wave homogenizationrdquo CompositeStructures vol 92 no 2 pp 430ndash435 2010

[12] C Potel J-F de Belleval and Y Gargouri ldquoFloquet waves andclassical plane waves in an anisotropic periodicallymultilayeredmedium application to the validity domain of homogeniza-tionrdquo The Journal of the Acoustical Society of America vol 97no 5 p 2815 1995

[13] N Ryden ldquoSurface wave testing of pavementsrdquo The Journal ofthe Acoustical Society of America vol 125 no 4 p 2603 2009

[14] N Ryden andM J S Lowe ldquoGuidedwave propagation in three-layer pavement structuresrdquoThe Journal of the Acoustical Societyof America vol 116 no 5 p 2902 2004

[15] R C Thompson ldquoOptical waves in layered mediardquo Journal ofModern Optics vol 37 no 1 pp 147ndash148 1990

[16] P Yeh A Yariv and A Y Cho ldquoOptical surface waves in peri-odic layered mediardquo Applied Physics Letters vol 32 no 2 pp104ndash105 1978

[17] H Chen and C T Chan ldquoElectromagnetic wave manipulationby layered systems using the transformation media conceptrdquo

Physical Review BmdashCondensed Matter and Materials Physicsvol 78 no 5 Article ID 054204 2008

[18] W T Thomson ldquoTransmission of elastic waves through astratified solid mediumrdquo Journal of Applied Physics vol 21 pp89ndash93 1950

[19] NAHaskell ldquoThedispersion of surfacewaves onmulti-layeredmediardquo Bulletin of the Seismological Society of America vol 43pp 17ndash34 1953

[20] R D Miller J Xia C B Park and J M Ivanov ldquoMultichannelanalysis of surface waves tomap bedrockrdquo Leading Edge vol 18no 12 pp 1392ndash1396 1999

[21] N Ryden P Ulriksen C B Park R D Miller J Xia and JIvanov ldquoHigh frequency MASW for non-destructive testingof pavementsmdashaccelerometer approachrdquo in Proceedings of the14th EEGS Symposium on the Application of Geophysics toEngineering and Environmental Problems vol 14 of RoadbedApplications p RBA5 2001

[22] D Penumadu and C B Park ldquoMultichannel Analysis of SurfaceWave (MASW) method for geotechnical site characterizationrdquoin Geotechnical Special Publication pp 957ndash966 2005

[23] D Givoli ldquoNon-reflecting boundary conditionsrdquo Journal ofComputational Physics vol 94 no 1 pp 1ndash29 1991

[24] M Israeli and S AOrszag ldquoApproximation of radiation bound-ary conditionsrdquo Journal of Computational Physics vol 41 no 1pp 115ndash135 1981

[25] J-P Berenger ldquoA perfectly matched layer for the absorption ofelectromagnetic wavesrdquo Journal of Computational Physics vol114 no 2 pp 185ndash200 1994

[26] M B Drozdz Efficient finite element modelling of ultrasoundwaves in elastic media [PhD dissertation] Imperial College2008

[27] M S P Eastham The Spectral Theory of Periodic DifferentialEquations Texts in Mathematics Scottish Academic PressChatto amp Windus London UK 1973 httpbooksgoogleesbooksid=LUHvAAAAMAAJ

[28] J D AchenbachWave Propagation in Elastic Solids vol 16 1973[29] Z Su L Ye and Y Lu ldquoGuided Lamb waves for identification

of damage in composite structures a reviewrdquo Journal of Soundand Vibration vol 295 no 3ndash5 pp 753ndash780 2006

[30] I A Viktorov Rayleigh and Lamb Waves Physical Theory andApplications Plenum Press New York NY USA 1967

[31] A Gibson and J S Popovics ldquoLambwave basis for impact-echomethod analysisrdquo Journal of Engineering Mechanics vol 131 no4 pp 438ndash443 2005

[32] J L Rose Ultrasonic Waves in Solid Media C U Press 1999[33] N Gandhi and J E Michaels ldquoEfficient perturbation analysis of

Lamb wave dispersion curvesrdquo in Review of Progress in Quan-titative Nondestructive Evaluation vol 29 of AIP ConferenceProceedings pp 215ndash222 2010

[34] L de Marchi A Marzani S Caporale and N Speciale ldquoUltra-sonic guided-waves characterization with warped frequencytransformsrdquo IEEE Transactions on Ultrasonics Ferroelectricsand Frequency Control vol 56 no 10 pp 2232ndash2240 2009

[35] D N Alleyne and P Cawley ldquoA 2-dimensional Fourier trans-form method for the quantitative measurement of Lambmodesrdquo in Proceedings of the IEEE 1990 Ultrasonics Symposiumvol 2 pp 1143ndash1146 December 1990

[36] C B Park R D Miller and J Xia ldquoMultichannel analysis ofsurface wavesrdquo Geophysics vol 64 no 3 pp 800ndash808 1999


[37] R D Miller J Xia J Ivanov and C B Park ldquoMultichannelanalysis of surfacewaves (MASW) active and passivemethodsrdquoThe Leading Edge vol 26 pp 60ndash64 2007

[38] R G Lyons Understanding Digital Signal Processing 2004[39] M Drozdz L Moreau M Castaings M J S Lowe and P

Cawley ldquoEfficient numerical modelling of absorbing regions forboundaries of guidedwaves problemsrdquoAIPConference Proceed-ings vol 820 no 1 pp 126ndash133 2006

[40] K Aki and P G Richards Quantitative Seismology UniversityScience Books 2002

[41] P Wilcox ldquoModeling the excitation of lamb and SH waves bypoint and line sourcesrdquo in AIP Conference Proceedings vol700 pp 206ndash213 AIP 2004 httplinkaiporglinkAPC7002061ampAgg=doi

[42] K Luangvilai Attenuation of ultrasonic Lamb waves with appli-cations to material characterization and condition monitoring[PhD dissertation] 2007

[43] M Sansalone ldquoImpact-echo the complete storyrdquoACI StructuralJournal vol 94 no 6 pp 777ndash786 1997

[44] A Maghsoodi A Ohadi and M Sadighi ldquoCalculation ofwave dispersion curves inmultilayered composite-metal platesrdquoShock and Vibration vol 2014 Article ID 410514 6 pages 2014

[45] A Kamal and V Giurgiutiu ldquoStiffness transfer matrix method(STMM) for stable dispersion curves solution in anisotropiccompositesrdquo in Health Monitoring of Structural and BiologicalSystems 906410 vol 9064 of Proceedings of SPIE San DiegoCalif USA March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Many relevant structures can be assumed to be layeredsystems of infinite extent for example [13 14] in civil engi-neering constructions [15 16] in optics or [17] in electroma-gnetics Therefore theoretical methods and experimentaltechniques to obtain their dispersion curves have beendevised From the theoretical side different matrix tech-niques have been developed to address the calculation Theyinvolve numerical computational methods whose complexityincreases with the number of layers in the system [18 19] ormore recently [6]

In laboratory experiments or field work the dispersioncurves can be obtained using for example the multichannelanalysis of surface waves (MASW) methodTheMASW pro-cedure involves collecting equally spaced measures of vibra-tion along a profile on the system surface using for exampleaccelerometers The resulting 2D space-time discrete imageis Fourier-transformed to the frequency-wave number (120596 119896)

domain and then processed to build the dispersion curves[20 21] The method has some drawbacks inherent to theFourier transform limitations which will be discussed laterThe MASW has been applied successfully in the characteri-zation of pavement systems [13] as a seismic data acquisitiontechnique [20] or for geotechnical characterization [22] TheMASW strategy is here also used to perform a computernumerical simulation of the system closely mimicking thefield setupThe issue of infinite lateral extent is usually tackledby using perfectly matched layers (PML) [23ndash26] as has beendone here or absorbing regions Both techniques presentdrawbacks [26]

In this paper an alternative way to calculate the disper-sion curves of layered systems with infinite lateral extentusing the F-B theory is presentedThemethod has never beenapplied to the dispersion curves calculations of nonperiodiclayered systems Here it is used to obtain the dispersioncurves of a single layer case and to estimate the elastic param-eters of a real steel slab for showing themethodHowever thenovelty in this work is that it can be applied to an arbitrarynumber of layers even if the layers are anisotropic or ortho-tropic with the same complexity levelThe power of the met-hod is that the equations are solved by the finite elementsoftware because the F-B theory only affects the propagationterm which is the same whatever the nature of the layersIt is not necessary to develop the equations for each specificproblem and to generate complex codes to get the dispersionrelations

The F-B theory reduces the problem to calculationsperformed in the so-called unit cell subject to certain specificboundary conditions derived from the F-B theory and elasto-dynamics The influence of the size of the unit cell is ascert-ained The results are first compared with the dispersion cur-ves derived from the Rayleigh equations [5] solved by asearching root numerical method Comparison is also madewith the curves resulting from a FEM computer simulationfollowed by a 2D Fourier transformation Finally a realexperiment was performed on a steel slab employing theMASW method and the empirical dispersion curves werecompared with the analytical and numerical ones

The results show that the F-Bmethod compares favorablywith other methods and fits accurately the empirical data

providing a good alternative to obtain dispersion curves inlayered systemsThe F-B technique can be run on a finite ele-ment package like COMSOL Multiphysics can be applied toan arbitrary number of layers in the system with the samecomplexity level and eliminates issues of infinite lateral ext-ent

2 Floquet-Bloch Theory Explanation

The Floquet-Bloch theory provides a strategy to obtain a setof solutions of a linear ordinary equations system of the form

f1rfloor (119909) = M (119909) f (119909) (1)

where f(119909) is the solution vector and thematrixM is periodicsuch thatM(119909+119871) = M(119909) for a certain period 119871 At first sightit might seem that the solution of such a problem would haveto be also 119871-periodic But Floquet showed that this need notbe soThere exists however a simple relationship between thesolutionrsquos behaviors inside one period and outside it If F is afundamental matrix of solutions then another matrix B canbe found such that

F (119909 + 119871) = BF (119909) (2)

B can be constructed by setting 119909 = 0 in (2) such thatB = Fminus1(0)F(119871) A simplest case is obtained using F(0) = Iso that B = F(119871) As there is not a unique choice for thefundamental matrix F and how it is exactly chosen dependson the problem B is also not unique But its eigenvalues areintrinsic of the problem and under the right transformationcan be used as a propagator or evolution factor 120588 relating thevalue of the solution at a point inside the period with its valueat a point outside of it Only the solution inside a period istherefore needed verifying that

f (119909 + 119871) = 120588f (119909) (3)

Following the classical nomenclature 120588 = exp(119896119865119871) is known

as Floquetmultiplier 119896119865being the complex Floquet exponent

Moreover Floquet found that the solution at any point canalso be factored in two terms

f (119909) = p (119909) exp (119896119865119909) (4)

Here p(119909) is a periodic function playing the role of theeigenvectors if M was a constant matrix and carrying theperiodicity 119871 of the coefficients of the problem The complexexponential distorts the strict periodicity of p(119909) incorporat-ing damping or ever growing effects in the amplitudes of thesolution depending on the value of |120588| This is why solutionsare in general not periodic and also why the Floquetperspective is usually employed to study their stability Asolution will be stable if the Floquet multipliers verify |120588| le 1

[27]

3 Guided Waves in LayersAnalytical Dispersion Curves

The guided wave propagation problem in a homogeneousisotropic and infinite single layer has been widely treated in



2b

rarrt

Layer


infinite extent

x


(a)

Symmetric mode

Antisymmetric mode

(b)

0 5 10 150

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000Ph

ase v

eloci

ty (m

s)


A0

S0

A1

S1

A2

S2

c1

c2cR

(c)


1= 5800ms and 119888





120588u (119909 119910 119905) + nabla [C (119909 119910) nabla120576(u (119909 119910 119905))] = 0 (5)





= 120597119906119911




tan120572119887

tan120573119887= minus[

41205721205731198962

(1198962 minus 1205732)2]

plusmn1

(6)

with

1205722=

1205962

11988821

minus 1198962 120573

2=

1205962

11988822

minus 1198962 (7)


2the











1=













Impact point

PML (F n) PML (F n)

120575L 120575LL

120575x















u (119909 119910 119905) = 119891 (119910) exp 119894 (119896119909 minus 120596119905) (8)





u (119909 119910) = 119891 (119910) exp (119894119896119909) (9)


u (119909 + 119871 119910) = 119891 (119910) exp (119894119896119909) exp (119894119896119871)

= u (119909 119910) exp (119894119896119871) (10)



u119899119896FB

(119909 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119909) (11)




119896 =2119899120587

119871+ 119896FB (12)


u119899119896FB

(119909 119910) = 119891 (119910) exp (2119899120587119909

119871) exp (119896FB119909) (13)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1

S0A0

A1

(a)

1 2 3 4 5 6 7 8

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms

)


S1

S0

A0

S2

ph =2120587

kf

(b)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1S0

A0

A1

(a)

26999 27 27001 27002 27003 2700332

34

36

38

40

42

44

46

Frequency (Hz)

Wav

e num

ber (

m1)

times104

Δk

(b)





u119899119896119861

(119909 + 119871 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119871) (14)



1205881205962

119899(119896FB) u119899119896FB (119909 119910 120596119899) + nablaC (119909 119910)

times nabla120576(u119899119896FB

(119909 119910 120596119899)) = 0 forall (119909 119910) isin Ω

119877


Infinite

Infinite

Unit cell45 cm

(a)

Unit cell

Aspect ratio

L

ΩR

f(y)120575ΩRx

120575ΩRx

y

x

2b = 45 cm

Ra = L2b

(b)

Unit cell


f(y) f(y)

L

(c)


First zone

10

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100 120 140

Freq

uenc

y (H

z)


1205961 k0

1205960 k0

1205961 k1

times104 120587L

(a)



45

4

35

3

25

2

15

1

05

00 200 400 600 800 1000 1200 1400

Freq

uenc

y (H

z)


times105 120587L

(b)


[nablaC(119909 119910)nabla120576(u119899119896FB

(119909 119910 120596119899))]119910= 0 forall (119909 119910) isin 120575Ω

119877119910

u119899119896119861

((119909 119910) + (119871 2119887) 120596119899)

= u119899119896119861

((119909 119910) 120596119899) 119890119894119896FB119871 forall (119909 119910) isin 120575Ω

119877119909

(15)where u


120576(u119899119896FB








15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)




119899(119896FB)


119899(and



0 For this value



0and the


wave vector 1198961= 120587119871 + 119896

0)













120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2b

rarrt

Layer


infinite extent

x


(a)

Symmetric mode

Antisymmetric mode

(b)

0 5 10 150

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000Ph

ase v

eloci

ty (m

s)


A0

S0

A1

S1

A2

S2

c1

c2cR

(c)


1= 5800ms and 119888





120588u (119909 119910 119905) + nabla [C (119909 119910) nabla120576(u (119909 119910 119905))] = 0 (5)





= 120597119906119911




tan120572119887

tan120573119887= minus[

41205721205731198962

(1198962 minus 1205732)2]

plusmn1

(6)

with

1205722=

1205962

11988821

minus 1198962 120573

2=

1205962

11988822

minus 1198962 (7)


2the











1=













Impact point

PML (F n) PML (F n)

120575L 120575LL

120575x















u (119909 119910 119905) = 119891 (119910) exp 119894 (119896119909 minus 120596119905) (8)





u (119909 119910) = 119891 (119910) exp (119894119896119909) (9)


u (119909 + 119871 119910) = 119891 (119910) exp (119894119896119909) exp (119894119896119871)

= u (119909 119910) exp (119894119896119871) (10)



u119899119896FB

(119909 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119909) (11)




119896 =2119899120587

119871+ 119896FB (12)


u119899119896FB

(119909 119910) = 119891 (119910) exp (2119899120587119909

119871) exp (119896FB119909) (13)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1

S0A0

A1

(a)

1 2 3 4 5 6 7 8

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms

)


S1

S0

A0

S2

ph =2120587

kf

(b)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1S0

A0

A1

(a)

26999 27 27001 27002 27003 2700332

34

36

38

40

42

44

46

Frequency (Hz)

Wav

e num

ber (

m1)

times104

Δk

(b)





u119899119896119861

(119909 + 119871 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119871) (14)



1205881205962

119899(119896FB) u119899119896FB (119909 119910 120596119899) + nablaC (119909 119910)

times nabla120576(u119899119896FB

(119909 119910 120596119899)) = 0 forall (119909 119910) isin Ω

119877


Infinite

Infinite

Unit cell45 cm

(a)

Unit cell

Aspect ratio

L

ΩR

f(y)120575ΩRx

120575ΩRx

y

x

2b = 45 cm

Ra = L2b

(b)

Unit cell


f(y) f(y)

L

(c)


First zone

10

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100 120 140

Freq

uenc

y (H

z)


1205961 k0

1205960 k0

1205961 k1

times104 120587L

(a)



45

4

35

3

25

2

15

1

05

00 200 400 600 800 1000 1200 1400

Freq

uenc

y (H

z)


times105 120587L

(b)


[nablaC(119909 119910)nabla120576(u119899119896FB

(119909 119910 120596119899))]119910= 0 forall (119909 119910) isin 120575Ω

119877119910

u119899119896119861

((119909 119910) + (119871 2119887) 120596119899)

= u119899119896119861

((119909 119910) 120596119899) 119890119894119896FB119871 forall (119909 119910) isin 120575Ω

119877119909

(15)where u


120576(u119899119896FB








15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)




119899(119896FB)


119899(and



0 For this value



0and the


wave vector 1198961= 120587119871 + 119896

0)













120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















1=













Impact point

PML (F n) PML (F n)

120575L 120575LL

120575x















u (119909 119910 119905) = 119891 (119910) exp 119894 (119896119909 minus 120596119905) (8)





u (119909 119910) = 119891 (119910) exp (119894119896119909) (9)


u (119909 + 119871 119910) = 119891 (119910) exp (119894119896119909) exp (119894119896119871)

= u (119909 119910) exp (119894119896119871) (10)



u119899119896FB

(119909 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119909) (11)




119896 =2119899120587

119871+ 119896FB (12)


u119899119896FB

(119909 119910) = 119891 (119910) exp (2119899120587119909

119871) exp (119896FB119909) (13)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1

S0A0

A1

(a)

1 2 3 4 5 6 7 8

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms

)


S1

S0

A0

S2

ph =2120587

kf

(b)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1S0

A0

A1

(a)

26999 27 27001 27002 27003 2700332

34

36

38

40

42

44

46

Frequency (Hz)

Wav

e num

ber (

m1)

times104

Δk

(b)





u119899119896119861

(119909 + 119871 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119871) (14)



1205881205962

119899(119896FB) u119899119896FB (119909 119910 120596119899) + nablaC (119909 119910)

times nabla120576(u119899119896FB

(119909 119910 120596119899)) = 0 forall (119909 119910) isin Ω

119877


Infinite

Infinite

Unit cell45 cm

(a)

Unit cell

Aspect ratio

L

ΩR

f(y)120575ΩRx

120575ΩRx

y

x

2b = 45 cm

Ra = L2b

(b)

Unit cell


f(y) f(y)

L

(c)


First zone

10

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100 120 140

Freq

uenc

y (H

z)


1205961 k0

1205960 k0

1205961 k1

times104 120587L

(a)



45

4

35

3

25

2

15

1

05

00 200 400 600 800 1000 1200 1400

Freq

uenc

y (H

z)


times105 120587L

(b)


[nablaC(119909 119910)nabla120576(u119899119896FB

(119909 119910 120596119899))]119910= 0 forall (119909 119910) isin 120575Ω

119877119910

u119899119896119861

((119909 119910) + (119871 2119887) 120596119899)

= u119899119896119861

((119909 119910) 120596119899) 119890119894119896FB119871 forall (119909 119910) isin 120575Ω

119877119909

(15)where u


120576(u119899119896FB








15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)




119899(119896FB)


119899(and



0 For this value



0and the


wave vector 1198961= 120587119871 + 119896

0)













120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Impact point

PML (F n) PML (F n)

120575L 120575LL

120575x















u (119909 119910 119905) = 119891 (119910) exp 119894 (119896119909 minus 120596119905) (8)





u (119909 119910) = 119891 (119910) exp (119894119896119909) (9)


u (119909 + 119871 119910) = 119891 (119910) exp (119894119896119909) exp (119894119896119871)

= u (119909 119910) exp (119894119896119871) (10)



u119899119896FB

(119909 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119909) (11)




119896 =2119899120587

119871+ 119896FB (12)


u119899119896FB

(119909 119910) = 119891 (119910) exp (2119899120587119909

119871) exp (119896FB119909) (13)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1

S0A0

A1

(a)

1 2 3 4 5 6 7 8

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms

)


S1

S0

A0

S2

ph =2120587

kf

(b)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1S0

A0

A1

(a)

26999 27 27001 27002 27003 2700332

34

36

38

40

42

44

46

Frequency (Hz)

Wav

e num

ber (

m1)

times104

Δk

(b)





u119899119896119861

(119909 + 119871 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119871) (14)



1205881205962

119899(119896FB) u119899119896FB (119909 119910 120596119899) + nablaC (119909 119910)

times nabla120576(u119899119896FB

(119909 119910 120596119899)) = 0 forall (119909 119910) isin Ω

119877


Infinite

Infinite

Unit cell45 cm

(a)

Unit cell

Aspect ratio

L

ΩR

f(y)120575ΩRx

120575ΩRx

y

x

2b = 45 cm

Ra = L2b

(b)

Unit cell


f(y) f(y)

L

(c)


First zone

10

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100 120 140

Freq

uenc

y (H

z)


1205961 k0

1205960 k0

1205961 k1

times104 120587L

(a)



45

4

35

3

25

2

15

1

05

00 200 400 600 800 1000 1200 1400

Freq

uenc

y (H

z)


times105 120587L

(b)


[nablaC(119909 119910)nabla120576(u119899119896FB

(119909 119910 120596119899))]119910= 0 forall (119909 119910) isin 120575Ω

119877119910

u119899119896119861

((119909 119910) + (119871 2119887) 120596119899)

= u119899119896119861

((119909 119910) 120596119899) 119890119894119896FB119871 forall (119909 119910) isin 120575Ω

119877119909

(15)where u


120576(u119899119896FB








15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)




119899(119896FB)


119899(and



0 For this value



0and the


wave vector 1198961= 120587119871 + 119896

0)













120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1

S0A0

A1

(a)

1 2 3 4 5 6 7 8

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms

)


S1

S0

A0

S2

ph =2120587

kf

(b)


05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

Frequency (Hz)

Wav

e num

ber (

m1)

times104

S1S0

A0

A1

(a)

26999 27 27001 27002 27003 2700332

34

36

38

40

42

44

46

Frequency (Hz)

Wav

e num

ber (

m1)

times104

Δk

(b)





u119899119896119861

(119909 + 119871 119910) = u119899119901119896FB

(119909 119910) exp (119894119896FB119871) (14)



1205881205962

119899(119896FB) u119899119896FB (119909 119910 120596119899) + nablaC (119909 119910)

times nabla120576(u119899119896FB

(119909 119910 120596119899)) = 0 forall (119909 119910) isin Ω

119877


Infinite

Infinite

Unit cell45 cm

(a)

Unit cell

Aspect ratio

L

ΩR

f(y)120575ΩRx

120575ΩRx

y

x

2b = 45 cm

Ra = L2b

(b)

Unit cell


f(y) f(y)

L

(c)


First zone

10

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100 120 140

Freq

uenc

y (H

z)


1205961 k0

1205960 k0

1205961 k1

times104 120587L

(a)



45

4

35

3

25

2

15

1

05

00 200 400 600 800 1000 1200 1400

Freq

uenc

y (H

z)


times105 120587L

(b)


[nablaC(119909 119910)nabla120576(u119899119896FB

(119909 119910 120596119899))]119910= 0 forall (119909 119910) isin 120575Ω

119877119910

u119899119896119861

((119909 119910) + (119871 2119887) 120596119899)

= u119899119896119861

((119909 119910) 120596119899) 119890119894119896FB119871 forall (119909 119910) isin 120575Ω

119877119909

(15)where u


120576(u119899119896FB








15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)




119899(119896FB)


119899(and



0 For this value



0and the


wave vector 1198961= 120587119871 + 119896

0)













120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Infinite

Infinite

Unit cell45 cm

(a)

Unit cell

Aspect ratio

L

ΩR

f(y)120575ΩRx

120575ΩRx

y

x

2b = 45 cm

Ra = L2b

(b)

Unit cell


f(y) f(y)

L

(c)


First zone

10

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100 120 140

Freq

uenc

y (H

z)


1205961 k0

1205960 k0

1205961 k1

times104 120587L

(a)



45

4

35

3

25

2

15

1

05

00 200 400 600 800 1000 1200 1400

Freq

uenc

y (H

z)


times105 120587L

(b)


[nablaC(119909 119910)nabla120576(u119899119896FB

(119909 119910 120596119899))]119910= 0 forall (119909 119910) isin 120575Ω

119877119910

u119899119896119861

((119909 119910) + (119871 2119887) 120596119899)

= u119899119896119861

((119909 119910) 120596119899) 119890119894119896FB119871 forall (119909 119910) isin 120575Ω

119877119909

(15)where u


120576(u119899119896FB








15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)




119899(119896FB)


119899(and



0 For this value



0and the


wave vector 1198961= 120587119871 + 119896

0)













120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15 cm5 cm

5 cm 075 cm

45 cm

3 4 5

G5556 57

15m

L

075m

View ldquoArdquo

View ldquoArdquo

(a) (b)

000

98

000

98

000

99

001

001

001

01

001

01

001

01

001

02

001

03

001

03

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time (s)

Forc

e (N

)

Δt = 143 plusmn 7120583s

(c)

05 1 15 2 250

005

01

015

02

025

03

035

Frequency (Hz)

Mag

nitu

de

times104

(d)




119899(119896FB)


119899(and



0 For this value



0and the


wave vector 1198961= 120587119871 + 119896

0)













120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















120575V =10038161003816100381610038161003816100381610038161003816

120597V120597119891

10038161003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816

120597V120597119896

10038161003816100381610038161003816100381610038161003816120575119896 =

1003816100381610038161003816100381610038161003816

2120587

119896

1003816100381610038161003816100381610038161003816120575119891 +

10038161003816100381610038161003816100381610038161003816minus2120587119891

1198962

10038161003816100381610038161003816100381610038161003816120575119896 (16)







7 Conclusions



0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 20

2000

4000

6000

8000

10000

12000

14000

Phas

e velo

city

(ms

)



(a)

05 1 15 2 25

1000

2000

3000

4000

5000

6000

7000

8000

9000

Phas

e velo

city

(ms


(b)

02 04 06 08 1 12 14 16 18

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Phas

e velo

city

(ms

)


(c)






Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Floquet-Bloch Theory and Its Application to the ...downloads.hindawi.com/journals/mpe/2015/475364.pdf · Research Article Floquet-Bloch Theory and Its Application