Acoustical reflections from aluminum cylindrical shells immersed in water

Acoustical reflections from aluminum cylindrical shells immersed in water a)

William Wade Ryan, Jr. b) Applied Research Laboratories, The University of Texas, Austin, Texas 78712 (Received 7 February 1977; revised 20 December 1977)

The purpose of this study was to examine and compare theoretical and experimental steady-state response functions for two thin aluminum cylindrical shells immersed in a water medium. Theoretical steady-state response functions are presented for air-filled and water-filled cylinders of infinite extent for the nondimensional frequency (ka) ranges from near 0 to 52. Experimental measurements are shown for the air-filled cylinders of finite length. Two sets of circumferential waves which are analytically derived from the steady-state response functions are also described and compared with photographs of the corresponding experimental echoes. Good agreement between calculated and measured values is demonstrated.

PACS numbers: 43.30.Dr, 43.30.Gv

INTRODUCTION

Although the cylinder as an acoustical target for un- derwater applications has been investigated by many authors, there are several areas which have not been thoroughly studied. One of these is the area of steady- state response functions. This paper considers acous- ticat reflections from aluminum cylindrical shells immersed in a water medium. The theoretical steady- state responses for two thin-walled aluminum cylinders, air and water filled, are presented and compared with measured responses for air-filled cylinders over the nondimensional frequency (ka) range from near 0 to 52. Two sets of circumferential waves, analytically derived from the steady-state response functions, are compared with experimental echoes.

Theoretical studies of solid cylinders have been per- formed by Rayleigh • and Morse z for soft and rigid cylinders. Similar studies have been conducted by Bauer, Tamarkin, and Lindsay s and Faran 4 for solid elastic cylinders. Doolittle and 'IJ'bera11 s were the first to present the solution to the boundary-value problem for the scattering of sound by an elastic cylindrical she11. Jun- get ø approached the problem sometime earlier from a Lagrangian formulation. In describing the transient nature of a cylinder, Franz and Depperman • predicted waves that would propagate around the cylinder and be reradiated; they called these waves Kriechwellen, or "creeping waves." Barnard and McKinney ø were the first to observe these waves experimentally. Among others who have contributed to the prediction and under-

' 9 standing of such waves are Horton, King, and Diercks, Diercks, Goldsberry, and Horton, xø and Goldsberryn; and Bunney, Goodman, and Marshall, •z Marshall and Olson, xa Shirley and Diercks, •4 and Neubauer and Dra- gonette xs have shown that there are two sets of creeping waves--one with a low velocity (•-2000 m/s) and one with a faster velocity (•-5500 m/s).

I. THE STEADY-STATE RESPONSE

Doolittle and 'Oberall • derived the general solution for the pressure of the wave scattered from an infinite elas-

a)This work was funded by the Naval Sea Systems Command. b)Present address: U.S. Coast Guard, Washington DC 20590.

tic cylindrical shell when a plane wave is incident upon the shell. Their Eq. (2b) can be used to evaluate the scattered pressure without restriction on the surround- ing fluid, the filling fluid, or the elastic material of the cylinder. The steady-state response R of a cylinder as a function of ka can be calculated from their equation and its high frequency limit as follows'

R = 20 log [Ps/P•f, [, (1) where Ps is the scattered pressure and Phil is defined as

limka-•o Ips I = Po(a/r) x/z , (2) where P0 is the amplitude of the incident pressure Pi wave and (a/r) •/• is the target strength

TS = 20 logp•/pi,

where the high-frequency limit of Pt is P0. The high frequency limit of the scattered pressure is used to nor- malize the back-scattered pressure since the remaining function R becomes a relative response that is indepen- dent of r. The calculated steady state frequency response for two aluminum cylinders is shown in Fig. 1. The cylinders are air-filled in (a) and (b) and water- filled in (c) and (d). The ratios of inner-to-outer radii b/a are 0. 916 and 0.937, respectively. Table I is a listing of the physical parameters of the aluminum and water used in this paper.

A block diagram of the experimental apparatus is shown in Fig. 2. The pulse timing generator (Scientific Atlanta) served as the system clock. It provided proper sequential timing of the transmit'and receive gates at a pulse repetition frequency of 30 pulses/s. A circular piston piezoelectric transducer, with an effective diameter of 6.3 cm and a resonance frequency of 254 kHz, served as the projector and a USRD type H23 standard hydrophone was used as a receiver. The filter in the test setup was a Krohn-Hite Corporation variable bandpass filter, model 3202, set for a passband from 10 kHz to 1 MHz. The low noise broadband preamplifier oper- ated at a 40-rib gain setting. The receive gate selected a 200-•s portion from the steady-state part of the received echo and supplied it as the input to the Scientific Atlanta model 1163 rectilinear chart recorder which

plotted curves proportional to the energy in the gated portion of the echo. The recorder output was manually

1159 d. Acoust Soc. Am. 64(4), Oct 1978 0001-4966/78/6404-1159500.80 ¸ 1978 Acoustical Society of America 1159

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1160 William Wade Ryan, Jr.: Reflection from aluminum shells in water 1160

dB

o

-8

-16

-24

-32

-4o

-48

! ! [ • [ ! ! ! [ ,,

i i i i i i i i i I i i i , i

ALUMINUM AIR FILLED

6.096 cm o.d.

5.588 cm i.d.

(a) b/a ' 0.916 i i i i l, i i I i i i i i i i i i i I i i i i i i

o

-8

dB -16

-24

-32

-40

I i I i i ALUMINUM AIR FILLED

6.096 cm o.d. 5.715 cm i.d.

b/a :0.937 i i i i i

, , i , i i I i' i , i I ! i i i , ,

o

-8

dB -16

5.588 cm i.d.

-32 (c) b/a: 0.916 -40 I I I I I I I I I I I J I I I I I I I I I I

FIG. 1. Calculated steady-state response functions for water-filled and air-filled cylinders having t•vo different wall thicknesses,

o

-8

dB -16

-24

-32

I ! ! ! [ i i I i ! ! ! i i I [ i [ i i ! ! i ! [ ,

5.715 cm ,.d.

(d) b/a ':-0.937 J i i i i i i i i i i i ß i i i i i i i i i i I i

4 8 12 16 20 24 28 32 36 40 44 48 52

ka

digitized (Aka =0.6) and subsequently processed by a Control Data Corporation model 3200 digital computer. A Calcomp plotter was used to present the computer output.

Two cylinders with outside diameters of 6.09 cm and lengths of 30.48 cm, machined from 6061-T6 aluminum to have b/a values of 0.916 and 0.937, were suspended below the surface of the water (in a large fresh water lake) with the major axis vertical. The cylinder was excited with 1-ms tone bursts at the rate of 30/s and with the frequency changing 620 Hz each second. The

TABLE I. Physical parameters for aluminum cylinders in water.

Compressional velocity cl 6420 m/s

Shear velocity c2 3040 m/s

Poisson' s ratio cr 0. 335

Lam• constants X 6.10 x 10 TM N/m 2

• 2.5 x 10 lø N/m 2

Density of aluminum pl 2.7 g/cm 3

Water velocity c 3 1470 m/s

J. Acoust. Soc. Am., Vol. 64, No. 4, October 1978

scattered pressure signal was measured during the steady state part of the echo, normalized to the incident signal, and plotted in dB (target strength) as a function of ka.

A comparison between the experimental frequency re-

J OSCILLATOR 1

ITRANSMIT • J PULSE TIMING J GATE J GENERATOR

REcFIVE J _J STRIP CHART UNIT GlI'E J 'J RECORDER

••,AMPLIFIER 4 PREAMP/IFIER • / POWER

= 3.8 m • = 1.3 m---

4.9 m

PROJECTOR HYDROPHONE TARGET

FIG. 2. Block diagram of experimental apparatus.



, , i ! !

ood.

5.588 cm i.d. b/a = 0.916

•,, . ß

THEORY b/a = 0.937 EXPERIMENTAL MEASUREMENTS

.... ..... ka

-16.3

-16.3 FIG. 3. Experimental backscattered measurement s (response functions) for a aluminum air-filled cylinders.

sponses and the calculated responses, given in Figs. l(a) and l(b), is shown in Fig. 3. The left axis is the relative axis that is normalized to •)hs•- The right axis is the target strength. The agreement between theory and measurement is good for both cylinders. The slight shift in ka between the theoretical and experimental curves can probably be attributed to the difference between the elastic constants of the aluminum used in con-

struction of the cylinders and those assumed for the calculations. The possible discrepancy between the theory for an infinite cylinder and the experiment for a cylinder of finite length is minimized by using normal incidence in the measurements.

II. THE TRANSIENT RESPONSE

Since only small amplitude signals are being considered, the cylindrical shells previously described can be treated as a linear system. Thus, the transient re-

sponse can be computed using the convolution theorem. In the frequency domain, this is stated mathematically as a multiplication, that is

= (3)

where E(co) is the complex frequency spectrum of the tr.ansient echo', T(co) is the complex frequency spectrum of the incident pulse, and H(co) is the frequency response of the target. The magnitude of the frequency response for the aluminum targets under consideration is presented in Fig. 1. In order to obtain a time domain representation of the transient, E(co) is inverse Fourier transformed to yield e(t). The phase of the frequency response was computed and used in the Fo'u- rier transform.

A pulse length of 10 ps for the incident pulse was chosen from prior knowledge of the target size, the

e(t} O

-0.5

+0.07 -

.{,) o

-0.07 - ---T-----T' i T --T---T--T----T- -1' -- -T- ' -•[

TIME 50 /•s/div =

UNEXPANDED. THEORETICAL

EXPANDED, THEORETICAL

EXPERIMENTAL - 50,/•s/div

FIG. 4. Transient signal echoes from an air-filled aluminum cylindrical shell (6.096 cm outer diameter, 5. 588 cm inner diameter, b/a =0.916, ira =39.1).




1.0

0 0

(a) LOW FREQUENCY - 71 kHz, ka = 9.26

'•ß AMPLITUDE - g-6'6887 (t- 0.045) i , i .""""'• ß • ß I

0.2 0.4 0.6

TIME - ms

2.0

u.I

• 1.o

(b) HIGH FREQUENCY - 300 kHz, ka = 39.1

AMPLITUDE = •-4.3944 (t - o, 18)

o0 I I I I I 0.2 0.4 0.6

TIME - ms

FIG. 5. Peak amplitudes of experimental circumferential waves at two frequencies on an aluminum cylinder (6. 096 cm outer diameter, 5.588 cm inner diameter, b/a =0. 916).

known circumferential wave velocities, •2-•e and the ca- pabilities of the experimental apparatus at hand. This choice of pulse length allows each circumferential re- turn to be resolved in time with no overlap.

By using linear system theory, these experimental waves can now be verified analytically. Two frequencies are chosen for computation of the theoretical echoes--300 kHz, ka= 39.1; and 71 kHz, ka=9.26. The spectrum for each incident pulse is calculated from

T(w) = sin(wot)e'•'•tdt , (4)

where t• = 10 •s. In the high frequency pulse there are three cycles, and in the low frequency pulse there are three quarters of a cycle.

For the two frequencies being considered, the echo amplitudes ½(t) are plotted on an unexpanded ordinate scale which shows the relationship between the reflection from the front surface and the circumferential re-

turns, and on an expanded ordinate scale which allows the circumferential waves to be resolved.

Experimental circumferential waves from a finite aluminum cylinder (5/a=0. 916) are shown in Fig. 4. Examination of the unexpanded, high-frequency echo shows that the signal following the initial pulse from the

front surface of the cylinder is approximately 23 dB lower than that of the peak. On the expanded amplitude scale, one will note a series of pulses decreasing in amplitude. As mentioned earlier, these have been termed "creeping waves," but are more often described as circumferential or surface waves. The time bases

between the experimental and theoretical curves are aligned so that the leading edges of the pulses from the front surface of the cylinder correspond. There is good correspondence between the predicted and measured times of arrival of the successive circumferential

waves; the time separation of the waves is 34.6 •s, which yields a group velocity of 5530 m/s. This value is in agreement with previous studies. •e With the ex- ception of the first few values, the peak values of the envelope for both the experimental and theoretical echoes decrease exponentially with time, with a time constant of - 6. 6887 (Fig. 5).

In the low-frequency echo shown in Fig. 6, the first circumferential wave is only 14 dB down from the front reflection of the cylinder, which is an indication that this is a stronger mode, or a mode easier to excite than the high frequency mode. The'time separation between the circumferential waves is about three times that for

the high-frequency circumferential waves. The average time between circumferential wave groups is 100 •s, which yields a group velocity of 1920 m/s. A comparison of •he experimental data with the calculated echoes again yields good agreement. Here also, the envelope of the peaks has an exponential decay, but with a short- er time constant of -4. 3944 (Fig. 5).

A reason for the deviation of the peak amplitudes in the first few circumferential returns of the high-frequency echo is that both the high-frequency and low-frequency circumferential waves are supported by the cylinder simultaneously, and destructive interference oc- curs. This can be seen at time equal to 0.045 ms in Fig. 5.

Neubauer and Dragonette •5 showed by the use of schlieren photographs that two circumferential waves could be generated on the same cylinder. Another ap- proach to demonstrate that two sets of circumferential waves can be supported by a cylinder simultaneously is by theoretically calculating the impulse response for the cylinder. The impulse response for the cylinder whose b/a ratio is 0. 916 is shown in Fig. 7(a). The impulse response is calculated by inverse transforming the frequency response for the cylinder shown in Fig. l(a). From the knowledge of the spacing in time of the circumferential waves presented in Figs. 4 and 6, one can observe wave packets at those times in the impulse response. When the impulse response is appropriately bandpass filtered, each set of waves can be separated [Figs. 7 (b) and 7 (c) ].

III. SUMMARY

The steady-state response functions for two aluminum cylindrical shells whose b/a ratios are 0.916 and 0.937, respectively, are computed and plotted for both water- filled and air-filled shells. The steady-state response




e(t) 0 -

+025 -

e(t) 0 -

-0.25

UNEXPANDED, THEORETICAL

ii !I

TI ME - 100 ,u.s/div =• .

.

EXPERIMENTAL

100 p.s/div

EXPANDED, THEORETICAL

FIG. 6. Transient signal echoes from an air-filled aluminum cylindrical shell (6.096 cm outer diameter, 5. 588 cm inner diameter, 5/a =0. 916, }a =9.26).

+1.0 -

+05 -

o

-0.5-

ß 0133 -

e(,) 0

-0133 -

ß 0.133 -

IMPULSE RESPONSE: UNFILTERED

IMPULSE RESPONSE: FILTERED AT 50-100 kHz

{a)

LOW FREQUENCY MODE FIG. 7. Low- and high- velocity .modes obtained by filtering the impulse response function of an aluminum cylinder with b/a = 0. 916.

elt) 0

-0.133

IMPULSE RESPONSE: FILTERED AT 275-350 kHz

i I i i i I i i I i i i i i i i 1 I I

TIME - 50

It)

HIGH FREQUENCY MODE

i i i i ! i I i ! i i i i i I i I I i i

u.$/div



1164 1164 William Wade Ryan, Jr.: Reflectiqn from aluminum shells in water

functimm for the air-filled cylinder are compared to experimental data and good agreement is observed.

Linear system theory is used to calculate the echo structure from the known frequency response of the cylinder and the incident pulse. Theoretical echoes are then calculated for the aluminum cylindrical shells.

The impulse response function for the aluminum cylinder (b/a=0.916) is calculated•frorn the frequency response (ka= 133). TWO echoes are calculated using a 10-•s incident pulse: one for ka= 9.25 and the other for ka= 39.1. The echoes for each of these pulses show that the cylinder •an be excited in two different modes simultaneously. One mode corresponds' to a low velocity (2000 m/s) and a low ka value (ka = 9.26); the other mode corresponds to a high velocity (5400 m/s) and a

,

high ka value (ka = 39.1). An experimental echo corre- s/)onding to each theoretical echo is presetired, and there is good agreement between theory and experiment.

ACKNOWLEDGMENTS

The author wishes to thank Dr. C. W. Horton, Sr., Dr. C. M,. McKinney, and Mr. K. J. Diercks for their assistance in preparing this paper.

,

1L. Rayleigh, Theory of Sound (Dover, New York, 1945). 2p. M. Morse, Vibration and Sound (McGraw-Hill, New York,

1936). 3L. Bauer, P. Tamarkin, and R. B. Lindsay, J. Acoust. Soc.

Am. 20, 858 (1948). 4j. j. Faran, Jr., J. Acoust. Soc. Am. 23, 405 (1951). 5R. D. Doolittte and H. •Jbei•all, J. Acoust. Soc. Am. 39ø 272

(1966).

6M. C. Junget, J. Acoust. Soc. Am. 24, 366 (1952). 7W. Franz and K. Depperman, Ann. Physik 10, 361 (1952). 8G. R. Barnard and C. M. McKinney, J. Acoust. Soc. Am:

33, 266 (1961). 9C. W. Horton, W. R. King. and K. J. Diercks, J. Acoust.

Soc. Am. 34, 1929 (1962). løK. J. Diercks, T. G. Goldsberry, and C. W. Horton, J.

Acoust. Soc. Am. 35, 59 (1963). lIT. G. Goldsberry, J. Acoust. Soc. Am, 42, 1298 (1967). 12R. E. Bunhey, R. R. Goodman, and S. W. Marshall, J.

Acoust. Soc. Am. 46, 1223 (1969). 13S. W. Marshall and T. G. Olson,•J. Acoust. Soc. Am. '•7,

949 (1970). 14D. J. Shirley and K. J. Diercks, J. Acoust. Soc. Am. 48,

1275 (1970).

15W. G. Neubauer and L. R. Dragonette, J. Acoust. Soc. Am. z•8, 1135 (1970).

16C. W. Horton and M. V. Mechler, J. Acoust. Soc. Am. 51, 295 (1972).



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Acoustical reflections from aluminum cylindrical shells immersed in water