Transverse whispering gallery waves in scattering by elastic cylinders

Transverse whispering gallery waves in scattering by elastic cylinders N. D. Veksler Department Akadeemia Received

of Mechanics, tee 21, 200108 73 June

1989;

Institute Tallinn, revised

of Cybernetics, USSR

27 June

Estonian Academy

of Sciences,

1989

The steady-state problem of a normally incident plane acoustic wave scattered by an infinitely long elastic (aluminium) cylinder embedded in a liquid (water) is considered. The far field form function is analysed as a function of x = ka. The modal resonance components are joined into families. It is shown that in the region 0
transverse

wave;

whispering

There currently exist detailed methods for calculating the acoustic pressure scattered by an elastic cylinder. The majority of the calculated form functions have been obtained by the direct summation of the Rayleigh series on the eigen functions. The Sommerfeld-Watson transformation is often applied to this series to interpret the obtained results. The analysis of the integrand in the complex plane and, in particular, the search of the poles generating the different families of creeping and circumferential waves, permit calculation of the phase and group velocities, as well as the entrance-emanating angles and the damping factors of these waves. Occasionally the eva!uation ofthe residues of the poles is accomplished and, extremely rarely, the form function is calculated by this method, especially in the case when all the contributions of the waves generated in the cylinder are taken into account and superposed. The resonance scattering theory, form function filtration, the method of isolation and the identification of resonances are powerful tools for analysing the form function. Each of these procedures explicitly uses the existence of the exact solution in series form, and allows decomposition of the form function. This isolates the families of modal resonance components and gives, for every peripheral wave, its full description, including the magnitude, phase velocity, damping factor and phase. The results of such an investigation are often presented in the form of Regge trajectories for the waves of different types, dispersion curves and acoustic spectrograms. The last are closely connected with the properties of the elastic scatterer and therefore are often proposed as a means of acoustic labelling. In the present study, the analysis of the influence of the longitudinal, c,, and transverse, c,, waves in the

gallery

wave;

scattering;

elastic

material of the cylinder on the form function and on the properties of the peripheral waves is presented. The form function was calculated in series form. For its interpretation, resonance scattering theory was used as being the most suitable in approach. The computation was accomplished in the narrow frequency band 0 < x < 50 for the case of an aluminium cylinder immersed in water. It turns out that the incident plane wave generates a Rayleigh wave and transverse whispering gallery wa$es in the cylinder. The velocity c, influences these waves weakly, while velocity c, influences them strongly.

Methodology The formulation of the problem and the method of solution are standard’ -4. The two-dimensional steadystate problem of a plane acoustic wave scattered by an elastic cylinder of circular cross-section is considered. The plane wave strikes on the infinite cylinder normally. The pressure in the incident wave, pi, has the form pi=p*

exp[i(kS

- Wt)]

(1)

where p.+ is a constant with the dimensions of pressure, k = w/c is the wave number in the liquid, w is the frequency, c is the Cartesian coordinate in the direction of the initial wave propagation, t is time, c is the sound velocity in the liquid and i = ( - 1)“‘. At a fixed observation point the exact solution of the steady-state problem for the pressure scattered by the cylinder is given by p, = p* exp( -iwt)

i q,i”R,H!,‘)(kr) n=0

cos n4

(2)

0041-624X/90/020067-10 @ 1990

Butterworth

& Co (Publishers)

Ltd

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1990

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67

Transverse

whispering

gallery

waves

in scattering

by elastic

where (3)

R,, = A,,IR,,

and the determinants A,, and B, of the third order are defined in reference 3. The notation is as follows: p, is the pressure in the secondary (scattered) field, r and 4 arc the polar coordinates, HL”(kr) is the cylindrical function of the first kind, c0 = 1, c, = 2(n 3 1). The angle 4 = rc corresponds to back-scattering. In the far field (at kr >> 1 ), instead of Hf,‘)(kr) its asymptotic representation can be used li2mn Wt”(kr)

-

i

&

C

exp( ikr).

cylinders:

(creeping) waves are rather well modelled in scattering by an acoustically rigid cylinder and can be described as waves of the same type. Thus the meaning of the difference introduced in Equation ( 11) is the isolation of the break waves in a ‘pure form’, caused by the peripheral waves revolving around the cylinder. In Equation ( 13) the second term corresponds to the phone component (the background) and the sum to the resonant component. The bigger Y is, the better the modal resonance component is defined. The form function and the modal resonance components were calculated in the case of an aluminium cylinder immersed in water with the following parameters:

1

Then Equation

(2) becomes

aluminium:

asymptotically

exp[i(krE,,

wt)] ,zO 2(inx)-“’

water:

R, cos nd,

(5)

where u is the external radius of the cylinder. If, following references 2 and 5, the partial function S, = 2R, + 1

scattering (‘5)

and the form function .f(d) = & consisting

(7)

L(43) of the partial

.L($) = (i7r.w)) l%,(S, are introduced,

form functions

~ 1) cos nC$

(8)

then (5) may be presented

as

expCi(kr

-

wtJlf(4)

In the limiting case, when liquid density/density of cylinder material, p/pi -to, one can obtain from Equation (2) the Rz’ function, corresponding to the scattering from an acoustically rigid cylinder n

J;(x)

where J,(x) is a Bessel function, and the prime denotes differentiation with respect to argument. In the far field, the difference between the acoustic pressure scattered by a solid elastic cylinder pa and an acoustically rigid cylinder ~6” may be written in the form l/2

(1 u

5

expCi(kr

-

wt)lti”‘(d)

(11)

where

(12) Ic/~)(~)=2(izx)m1!2c,

cos n4

(13)

At fixed 4 = c$~, we define for simplicity (14)

P = If > in = I $!T’ I

We shall call p(x) the form function and i,(x) the modal resonance component. As the results of computations and measurements have shown, the specular reflected wave and the diffracted

68

These parameters were chosen in the experimental investigation of the same problem6 which allows our results to be compared with experiment. In order to investigate the influence of the velocities c, and c, on the properties of peripheral waves, the computations of p(x) and i,(x) were carried out for cylinders from hypothetical materials immersed in water. In one series C, was constant and c, was taken in the form c; = qc,

Ultrasonics

(16)

where for variant numbers II, III, IV, V, q = 0.90, 0.95, 1.05, 1.10, respectively. In the other series c, was constant .and c, was taken in the form

where for variant numbers VI, VII, VIII, IX, 0.95, 1.05, 1.10, respectively. The basic variant, at which c,’ = c,, c; = c,, name the first. The observation point was situated in the backscattering. The computation was carried step

(17) q = 0.90,

we shall far field out with

(10)

,x-ka

H!y’( x)

PS- P?’ = P*

(15)

p = 1 x lo3 kg mm3, c = 1470 m 5-l.

c; = qc,

li2

R(r) = _

0, = 2.79 x lo3 kg mm3, C, = 6380 m s-r, c, = 3100 m SC’;

1:2

x

N. 0. Veksler

1990

Vol 28 March

in the domain 0
Transverse

whispering

gallery

waves

in scattering

by elastic

cylinders:

N.D.

Veksler

Figure 1 The form function of the acoustic pressure scattered by a solid aluminium cylinder immersed in water as a function of x = ka for the parameters defined in Equation (15). Two symbols (n. I) mark the partial mode resonance position

I”“l”“l”“l”“l’“‘l”“r”“l”“I”“t”“l 0 Figure

5 2

70

15

20

25

30

35

40

45

x

50

The acoustic spectrogram

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1990 Vol 28 March

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whispering

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by elastic

20

25

cylinders:

N. D. Veksler

Cl25

0

5

Figure

3

The isolated

zoo

75

70 modal

resonances

of the Rayleigh

30

35

40

45

x

50

wave

T

(1,7)

6” 0.75

0150

0.25

\

! ,

\

,‘lb” Figure

4

The isolated

~

\ / I

75

modal

20

resonances

of the first

25 (/=

1 ) transverse

40

30 whispering

gallery

45

1

R x

:

wave

0.75 <"

(I21

0.50

:, 0.25

/

0.00I( 10 Figure

70

7

35 5

The same as in Figure

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1990

4 but for /=

Vol 28 March

2

40

i

50

: iii 55

X

60

Transverse

Figure

6

Figure

Figure

7

8

The same as in figure

The same as in Figure

whispering

gallery

waves

in scattering

by elastic

cylinders:

N. D. Veksler

4 but for /= 3

4 but for /= 4

The same as in Figure 4 but for /== 5

050 (7X?

(7016)

1

Figure

Figure

9

10

4

The same as in Figure 4 but for I= 40 6

The same as in Figure 4 but for /=

Jy-, 45

,A, 50

A

, ,,(Jq*< 55

A

I\

I\

I //I,

A

II

I\ !&J:

60

65

70

x

75

7

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1990

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whispering

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by elastic

considered, the total number of resonances of the peripheral waves equals 90, but the number of identified extrema of the form function curve, corresponding to these resonances, is only 28. The identified resonances are marked in Figt4w 1. A similar situation arises for all the computed variants of the problem. We cannot indicate a priori how the extrema of the form function curve will shift when C, and C, are changed. General

structure

of the

form

function

The resonances are marked by two symbols (n, I). The first of them defines the ordinal number of the resonance and the second indicates the type of the peripheral wave (the family of resonances); I= R corresponds to the Rayleigh wave and I = 1, 2, 3, to the transverse whispering gallery waves. Here the notations in reference 7 are used, although it should be noted that in reference 6 the Rayleigh wave is not distinguished and the following notations are used: I = 1 corresponds to the Rayleigh wave and I = 2, 3,. . to the transverse whispering gallery waves. From the dispersion curves computed for the first (basic) variant, it is obvious that for every peripheral wave with x increasing, the dispersion curve tends to c,. This means that in the domain considered, 0 < .u < 50, the longitudinal whispering gallery waves are not generated. The acoustic spectrogram computed for the basic variant is presented in Figuw 2. It is concordant with the one given in reference 8. It should be noted that the extrema of the form function curve do not always explicitly correspond to the resonance frequencies (spectral lines in ~+Jc/,-cJ2). As a result of the interference of the waves the extrema positions on the form function curve are slightly shifted with respect to the spectral lines. Even m the case of very narrow, high quality resonances, the extrema on the form function curve which are mainly connected with (n, 1) resonance, other resonance(s) (II’, I’) can have an influence. As a rule, the resonances of peripheral wave with a high quality or magnitude are well observed on the form function curve. When the resonance has both these properties, its chance of being distinguished, on the form function curve is enhanced. One can distinctly see on the curve the extrema caused by the narrow (high quality) resonances. On many occasions one can clearly observe extrema corresponding to the constructive interference of two or more waves. Thus, for example, the extremum on the curve (see Figurr I) situated at Y = 40.39 is connected with resonances ( 10.2). (5, 5) and (3, 6). It should be mentioned that both resonances and anti-resonances of the peripheral waves may produce extrema. This is especially true for waves with a large magnitude, but not very high quality. As is well-known, such a phenomenon takes place in the case of scattering by an elastic sphere (see, for example, the plots of the form function and the contribution of the Rayleigh wave, Figures 3 and 4 in reference 9). It is characteristic that sometimes one can see on the form function curve extrema which have no particular distinction; they have neither high quality nor magnitude and correspond to rather large indices. They were fortunate in that the other waves do not significantly influence the form function curve in the region of the x domain where they are situated. Only one domain of the form function curve is distinguished for all the nine calculated variants. It is situated on the x-axis for values between zero and

72

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cylinders:

N. D. Veksler

approximately five. This is the so called quasi-rigid region”. In terms of peripheral waves it is restricted from above by a value corresponding to the second resonance of the Rayleigh wave. As is well known, the first resonance of this wave is generated on neither a sphere nor a cylinder. For seven of the nine computed variants of the form function, the position on the x-axis and magnitude of the first four maxima and three minima of the form function curve do not change from one variant to another. This can be possible if, and only if, the properties of the waves generating these extrema depend neither on c, nor on c,. This is indeed the case, since these extrema are caused by the slip waves, reradiated by the creeping waves revolving around the cylinder. These waves are often called the Franz waves. They propagate mainly in the liquid on the surface of the cylinder and their properties are defined not by the elastic cylinder but by the liquid. In the sixth and seventh variants the second resonance of the Rayleigh wave is situated so low on the x-axis that it ‘shades’ the resonances of the creeping wave which are small in magnitude. In these two variants the third minimum and the fourth maximum are not seen on the form function curve, while they clearly manifest themselves in the basic variant. So, for these two variants the quasirigid region is rather narrower than for all the other variants. On the plots of the modal resonance components presented below, the resonances of creeping waves are missing. The standard procedure of the resonance isolation used, in the form given by Equations (11) to ( 13) just does not allow separation of these resonances. However, the properties of the creeping waves generated on a solid elastic cylinder are investigated in detail in reference 7. The analysis of the computed form function shows that the variation of c, or ct on 5- 10 % affect the form function, change position, magnitude and number of the form function curve extrema. but so far it is not possible to correlate the variation of the parameters of the elastic cylinder with changing the properties of the form function. Detailed

results

The plots of modal resonance components of the Rayleigh wave and the transverse whispering gallery waves are shown in Figures 3 to IO. They are computed for the first variant. The plots for the other variants have a similar form. The results computed for variants II-V show that the variation of c, by 5- 10% affects the position of modal resonance components only weakly. In Tuhl~ I the resonance positions of the Rayleigh wave are compared for these four variants. It is evident that at small values of II, increasing c,’ has almost no effect on the resonance positions. With y1 increasing, the influence of the c; becomes apparent, but even for the largest value chosen, 17 = 23, it is rather small. The variation of C; by 20%, from 0.90 c, to I .lO c,, shift the 23rd resonance to the right only by 0.66 on the x-axis. At fixed ~1, the resonance magnitudes also change a little. They decrease by 0.003 when c,’ increases from 0.90 c, to 1.10 c,. The resonance magnitudes calculated for the first variant are presented in Tuhl~ 2. The qualities of the modal resonance components are almost unchanged from one variant to another. The modal resonance components of the transverse whispering gallery waves are also little affected by variations in c, for any of the variants. In contrast, the variation of c, has a marked effect on the modal resonance components connected with both

Transverse

whispering

gallery

The positions of the modal resonances of the Rayleigh wave for differentc; values

Table1

waves

in scattering

by elastic

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

N. D. Veksler

Table2 The positions (upper row) andthevalues (bottom row) of the modal resonances of the Rayleigh wavefordifferentc;values Variant

Variant "

cylinders:

II

III

IV

4.88 7.58 9.88 12.07 14.18 16.25 18.32 20.35 22.34 24.34 26.37 28.36 30.31 32.30 34.30 36.29 38.24 40.23 42.19 44.18 46.13 48.13

4.88 7.58 9.92 12.11 14.26 16.33 18.40 20.43 22.46 24.45 26.48 28.48 30.47 32.46 34.45 36.45 38.44 40.43 42.42 44.38 46.37 48.36

4.88 7.58 9.96 12.15 14.30 16.41 18.48 20.55 22.58 24.61 26.64 28.67 30.66 32.70 34.63 36.72 38.71 40.70 42.70 44.69 46.68 48.67

V 4.88 7.58 9.96 12.19 14.34 16.45 18.52 20.59 22.62 24.69 26.72 28.71 30.74 32.77 34.77 36.80 38.79 40.82 42.81 44.80 46.80 48.79

" 2 3 4 5 6 7 8 9 10 11 12 13 14

the Rayleigh wave or the whispering gallery waves. This can be clearly seen from the data presented in Tables 2 to 9 where, for variants VI-IX and the first (basic) variant the position on the x-axis (upper row) and the magnitude (bottom row) of the modal resonance components are given. It can be seen from the data presented, that at fixed n, with increasing cl, the resonance position shifts to the right on the x-axis and its magnitude slightly diminishes. The larger n is, the larger is the effect of c; on both the resonance position and its magnitude. This is particularly clearly seen on the low quality (slanting) resonances of the partial modes. In principle it should also be observed for high quality (narrow) resonances, but at the chosen (rather large) computational step, 1, = 10/256, they are occasionally lost. From the data presented in the tables, it can be seen that the 20% variation of cI (from 0.90 c, to 1.10 c,), strongly shifts to the right the position of the partial mode resonances, for example: resonance (21, R) by 6.75, resonance (15, 1) by 3.95, resonance (12, 2) by 3.40 and resonance (10, 3) by 2.03. It should be noted that for.every peripheral wave the first resonance of the partial mode is practically fixed on the x-axis. The wave is too long to ‘feel’ the c, variation. This fact evokes the following speculation. Although each of the partial mode resonances describes the elastic scatterer, observations of some of them give better information on the object. In order for the influence of the elastic properties of the scatterer to become prominent, it is advantageous to observe some resonance of high order n, if the latter possesses sufficient magnitude and is not of low quality. The resonances with n N lo-20 as a rule fulfil this condition. In contrast to the curve representing the form function (see Figure I), the curves of the modal resonance components permit a relation to be established between the variation of the elastic parameters of the scatterer, for example, its velocities c, and c, and the positions of

15 16 17 18 19 20 21 22 23 24

VI

VII

I

VIII

IX

4.88 1.02 7.58 0.820 9.88 0.718 12.03 0.650 14.10 0.601 16.09 0.562 18.05 0.531 19.92 0.505 21.80 0.483 23.59 0.464 25.39 0.448 27.19 0.433 28.95 0.419 30.70 0.407 32.42 0.396 34.18 0.386 35.90 0.377 37.62 0.368 39.34 0.360 40.98 0.352 42.70 0.345 44.41 0.338 46.17 0.332

4.88 1.02 7.58 0.819 9.92 0.716 12.11 0.649 14.18 0.599 16.25 0.560 18.24 0.528 20.20 0.502 22.15 0.479 24.10 0.460 26.02 0.442 27.89 0.427 29.80 0.413 31.68 0.401 33.55 0.390 35.39 0.379 37.27 0.370 39.14 0.361 40.98 0.352 42.85 0.345 44.73 0.337 45.56 0.331 48.44 0.324

4.88 1.02 7.58 0.819 9.92 0.716 12.15 0.647 14.26 0.597 16.37 0.558 18.44 0.525 20.47 0.498 22.54 0.475 24.53 0.455 26.56 0.438 28.59 0.422 30.59 0.408 32.58 0.395 34.61 0.384 36.60 0.373 38.59 0.363 40.58 0.354 42.58 0.346 44.57 0.338 46.56 0.331 48.55 0.323

4.88 1.02 7.58 0.818 9.96 0.715 12.19 0.646 14.38 0.595 16.48 0.555 18.63 0.523 20.74 0.495 22.85 0.472 25.00 0.451 27.11 0.433 29.22 0.417 31.37 0.403 33.48 0.390 35.59 0.378 37.73 0.367 39.84 0.357 41.95 0.348 44.10 0.340 46.21 0.332 48.32 0.325

4.88 1.02 7.62 0.818 10.00 0.714 12.23 0.645 14.45 0.594 16.64 0.553 18.83 0.521 21.02 0.492 23.20 0.468 25.39 0.448 27.62 0.429 29.84 0.413 32.07 0.398 34.34 0.385 36.56 0.373 38.79 0.362 41.05 0.352 43.28 0.343 45.55 0.334 47.73 0.326

the resonance lines. Thus the information about the elastic scatterer in the ‘raw’ form cannot be effectively used. Decomposition of the total form function on specularly reflected, creeping and peripheral waves and successive analysis of the latter permit some knowledge of the scatterer to be determined. A recently developed method of form function filtering” permits isolation of the families of resonances, and identification of the ordinal number of each resonance. Peripheral

wave

phase

velocity

dispersion

The resonance position of a standing wave (partial mode) corresponds to the resonance of a travelling (peripheral) wave. Using the known position of the partial mode resonance, one can find the phase velocity of the peripheral wave

cp”(x) =

x( n, 1) (’ ~

(19)

n

Ultrasonics

1990

Vol 28 March

73

Transverse

whispering

gallery

waves

in scattering

by elastic

Table 3 The positions (upper row) and the values (bottom row) of the modal resonances of the first (/= 1 ) transverse whispering gallery wave for different c; values

cylinders:

Table 5

VI

VII

I

VIII

1

5.94 0.670 9.14 0.305 12.38 0.635 15.59 0.572 18.67 0.522 21.64 0.485 24.45 0.456 27.19 0.432 29.76 0.413 32.30 0.397 34.77 0.383 37.19 0.370 39.53 0.359 41.88 0.349 44.18 0.339 46.44 0.331 48.71 0.323

5.98 0.621 9.30 0.508 12.66 0.615 15.98 0.564 19.18 0.515 22.23 0.477 25.16 0.450 27.97 0.427 30.66 0.407 33.24 0.391 35.74 0.377 38.24 0.365 40.66 0.354 43.05 0.344 45.39 0.335 47.73 0.327

6.05 0.917 9.45 0.623 12.93 0.620 16.33 0.547 19.65 0.509 22.81 0.471 25.78 0.443 28.63 0.421 31.41 0.402 34.02 0.386 36.60 0.373 39.14 0.361 41.60 0.349 44.06 0.340 46.48 0.331 48.87 0.323

6.13 0.618 9.61 0.0691 13.16 0.622 16.88 0.552 20.08 0.502 23.28 0.467 26.33 0.439 29.22 0.416 32.03 0.398 34.69 0.383 37.30 0.369 39.88 0.357 42.42 0.346 44.92 0.336 47.38 0.328 49.80 0.319

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

IX 6.1 7 0.908

n

VI

VII

I

VIII

IX

1

16.17 0.561 21.68 0.485 25.94 0.441 28.98 0.412 31.84 0.400 34.61 0.372 37.34 0 335 ~

16.17 0.561 21.68 0.485 25.98 0.440 29.18 0.417 32.07 0.385 34.92 0.232 37.70 0.309

42.77 0.0599 45.47 0.169 48.13 0.299

43.24 0.0932 46.02 0.286 48.79 0.311

16.17 0.561 21.68 0.484 26.05 0.441 29.30 0.406 32.27 0.373 35.16 0.373 38.01 0.352 40.86 0.0182 43.71 0.0550 46.56 0.236 49.41 0.320

16.17 0.561 21.64 0.485 26.05 0.442 29.41 0.414 32.46 0.379 35.39 0.326 38.28 0.362 41 .17 0.0238 44.10 0.243 47.03 0.279

16.17 0.561 21.64 0.485 26.05 0.441 29.49 0.414 32.58 0.395 35.55 0.370 38.52 0.348 41.48 0.143 44.49 0.0284 47.50 0.325

2 3

13.40 0.547 16.99 0.546 20.43 0.496 23.71 0.463 26.80 0.436 29.73 0.414 32.54 0.395 35.23 0.380 37.89 0.366 40.51 0.354 43.05 0.343 45.59 0.333 48.13 0.325

4 5 6 7 8 9 10 11

Table

6

n 1 2

4

The same as in Tab/e 3 but for /= 2

5

Vanant n

VI

VII

I

VIII

IX

1

13.75 0.601 16.64 0.460 19.95 0.367 22.23 0.0993

13.83 0.605 16.76 0.531 19.61 0.405 22.46 0.0924

13.87 0.588 16.88 0.459 19.77 0.388 22.62 0.0878 25.55 0.0175 28.48 0.316 31.45 0.380 34.41 0.385 37.38 0.368 40.35 0.353 43.14 0.343 46.05 0.331 48.87 0.323

13.91

13.94

0.571 16.95 0.483 19.88 0.431 22.81 0.296 25.78 0.0207 28.79 0.249 31.88 0.400 34.96 0.380 38.05 0.366 41.09 0.351 44.06 0.339 46.91 0.329 49.73 0.319

0.575 16.99 0.519 19.96 0.403 22.97 0.301

2 3 4 5 6 7 8 9 10 11 12 13 14

74

27.77 0.415 30.55 0.404 33.32 0.375 36.13 0.370 38.91 0.359 41.64 0.350 44.37 0.339 47.07 0.329 49.73 0.320

Ultrasonics

28.13 0.425 30.98 0.355 33.87 0.387 36.76 0.370 39.61 0.358 42.42 0.346 45.23 0.335 47.97 0.326

6 7 8 9

Table 7 29.14 0.250 32.30 0.389 35.51 0.374 38.71 0.362 41.84 0.348 44.84 0.337 47.77 0.326

Vol 28 March

VI

VII

I

VIII

20.70 0.0306 23.83 0.449 27.81 0.428 32.62 0.395 37.27 0.369 41.05 0.351 44.10 0.333 46.91 0.302 49.65 0.310

20.90 0.0291 24.10 0.458 28.01 0.425 32.73 0.394 37.30 0.369 41 .I 7 0.352 44.34 0.338 47.23 0.320

21.05 0.0305 24.34 0.406 28.20 0.423 32.85 0.394 37.34 0.369 41.21 0.351 44.49 0.337 47.50 0.327

-24.53 0.354 28.40 0.423 32.93 0.393 37.34 0.369 41.25 0.351 44.61 0.336 47.70 0.325

IX

24.69 0.365 28.55 0.422 33.01 0.393 37.38 0.369 41 29 0.351 44.69 0.337 47.85 0.326

The same as in Tab/e 3 but for / = 5 Vanant

n 1 2 3 4 5 6 7

1990

The same as in Tab/e 3 but for / = 4 Vanant

3 Table 4

The same as rn Tab/e 3 but for / = 3 Variant

Variant n

N. D. Veksler

VI

VII

I

VIII

IX

27.19 0.181 30.23 0.0531

27.46 0.183 30.55 0.0423

27.73 0.334 30.86 0.406

27.93 0.223 31 .13 0.0641

28.13 0.320 31.33 0.0896

36.25 0.173 39.38 0.356 43.24 0.342 47.73 0.327

36.64 0.204 39.88 0.351 43.67 0.341 48.01 0.326

37.03 0.255 40.31 0.351 44.06 0.339 48.24 0.325

37.38 0.180 40.74 0.352 44.45 0.338 48.52 0.324

37.73 0.253 41.09 0.344 44.80 0.337 48.75 0.323

Transverse

whispering

gallery

waves

(20) The same as in Tab/e 3 but for /= 6 Variant ” 1 2 3 4 5

Table 9

VI

VII

I

VIII

IX

30.20 0.410 36.13 0.375 39.84 0.297 42.85 0.244 45.78 0.175

30.16 0.410 36.13 0.375 40.16 0.356 43.20 0.250 46.17 0.0435

30.16 0.411 36.13 0.375 40.43 0.349 43.55 0.315 46.56 0.0560

30.16 0.411 36.13 0.375 40.63 0.349 43.87 0.337 46.95 0.176

30.16 0.411 36.09 0.376 40.82 0.353 44.18 0.280 47.30 0.173

VII

I

VIII

IX

37.42 0.350 42.27 0.347 47.70 0.326

34.49 0.0363 37.77 0.361 42.34 0.343 47.73 0.327

34.80 0.0293 38.13 0.329 42.46 0.346 47.77 0.326

35.08 0.0415 38.44 0.316 42.62 0.345 47.77 0.326

1 2

37.07 0.352 42.23 0.347 47.70 0.327

3 4

N. D. Veksler

It is shown above that c, weakly affects the phase velocity of each wave generated in the cylinder. In contrast, c, affects it significantly. The greater c, is, the larger this influence is. It is quite difficult to observe this influence in the total form function curve. Using a procedure of form function analysis (for example, the SommerfeldWatson transformation, the method of resonance isolation and identification, or the so called resonance scattering theory) and selecting from the form function each peripheral wave, this influence can be clearly detected. For n 3 10, the velocity c, strongly influences the position on the x-axis of the resonances of the Rayleigh wave and also of the transverse whispering gallery waves. This fact can be used to extract information about the properties of an elastic cylinder from the registered dependencies. In addition to the acoustic spectrogram, dispersion curves of the phase velocities of the Rayleigh wave and some low order (I = 1,2,3) transverse whispering gallery waves

The same as in Tab/e 3 but for I= 7

VI

cylinders:

Conclusion

Variant n

by elastic

The dispersion curves of relative phase velocities of the Rayleigh wave and the first three transverse whispering gallery waves are shown in Figures 11 to 14. The dispersion curves of successive waves (at I> 3) have a similar character. It is obvious from the plots presented that for every peripheral wave the family of dispersion curves for different c,’ are similar to a divergent spray. At first (at small n and x) the spray diverges quickly then its width gradually becomes established. The larger the value of c,‘, the higher the dispersion curve in this family is situated. On the plane x - ( cph/c,), the resonances of the same H are situated on straight lines. The larger n is the less is the angle between this corresponding line and the x-axis, and the more the phase velocities of the peripheral wave, generated in the cylinder, differ for different c;.

where c.ph(x) is the phase velocity of the peripheral wave of the Ith family, x(n, 1) is the position on the x-axis of the nth resonance of the partial mode of the same family. Relative phase velocity is suitable for the purpose of comparison

Table 8

in scattering

1.20

1 5

o.900 Figure

11

The dispersion

I 70 curves

I 15 of the relative

I 20 phase velocity

’ I 25 of the Rayleigh

I 30 wave

’

/

1 35 (I=

R) for different

Ultrasonics

n=25

/yy$b 40 values

45

x

n=26 VI 50

of c;

1990

Vol 28 March

75

Transverse

whispering

gallery

waves

in scattering

by elastic

cylinders:

N. 0.

Veksler

3.0 Ph Cl c

2.5

Figure

12

I

I

5

10

1

15

I

I

I

I

I

I

20

25

30

35

40

45

The same as in Figure

7 1 but for the transverse

The same as in Figure

72 but for I=

whispering

gallery

wave

Figure

14

(/=

J

x

:j0

1)

/

Figure

13

can be used to obtain scatterer.

information

2

on the nature

of the

References I

76

Flax, L., Dragonette, L.R. and tiberall, H. Theory of elastic resonance excitation by sound scattering J Acousr Sot Am (1978) 63 723-731 Flax, L. and Neubauer, W.G. Acoustic reflection from layered elastic absorptive cylinders J Acoust Sot Am (1977) 61 307 -312 Doolittle, R.D., liberalI, H. and Uginfius, P. Sound scattering by elastic cylinders J Acousr Sot Am ( 1968) 43 1~ I4 Faran Jr., J.J. Sound scattering by solid cylinders and spheres J Aumst Sot Am (1951) 23 405-418 Vogt, R.H. and Neubauer, W.G. Relation between acoustic rellection and vibrational modes of elastic spheres J Acousr Sot Am (1976) 60 15-22

Ultrasonics

1990

Vol 28 March

The same as in Figure

72 but for /=

3

Maze, G., Izbicki, J.L. and Ripocbe, J. Resonances of plates and cylinders: Guided waves J Acoust Sot Am ( 1985) 77 1352Zl355 Gaunaurd, G.C. and Brill, D. Acoustic spectrogram and complexfrequency poles of a resonantly excited elastic tube J Acousf SW Am (1984) 75 1680-1693 Ripocbe, J., Maze, G. and Izbicki, J.L. New research in nondestructive testing: Resonance acoustic spectroscopy. in ‘Ultrasonics lntrrnarional 85 Confvrence Proceedings’ Butterworth Scientific Ltd, Guildford (1985) 364-369 Williams, K.L. and Marston, P.L. Synthesis of back-scattering from an elastic sphere using the Sommerfeld-Watson transformation and giving a Fabry-Perot analysis of resonances .I Acousr SW Am (1986) 79 1702&1708 Gaunaurd, G.C. and tiberall, H. RST analysis of monostatic and bistatic acoustic echoes from an elastic sphere J Acoust Sot Am (1983) 73 l-12 Busson, D.E. DilTraction of a plane acoustic wave by a layered elastic sphere Proc Insr Acousr (1985) 7 16OCl68