Experimental determination of resonant forward scattering of solid inclusions in a fluid

Experimental determination of resonant forward scattering of solid inclusions in a fluid J.M.

PerdigZio, A. Ferreira,

J.E. Lefebre*

and C. Bruneel*

Centro de Electrotecnia, Universidade de Coimbra, Largo Marques de Pombal, 3000 Coimbra, Portugal * Laboratoire d’Opto-Acousto-Electronique, Universite de Valenciennes, 59326 Valenciennes, France Received

30 July

1987;

revised

9 October

1987

Continuous growth over recent years of the utilization of heterogeneous materials has introduced the problem of characterization. Variations of attenuation with frequency and the physical characteristics of the media are major problems. A first approach is obtained using the classic models based on the scattering cross-section. Theoretical develop’ments by Watermann et al., however, have not yet been fully demonstrated by practical forward scattering experimentation. In this Paper we present results obtained for the scattering cross-section of a polystyrene obstacle immersed in water. We have concentrated on the design and realization of an experimental set-up and the results show good agreement with theory.

Keywords:

underwater

acoustics;

wave

Ultrasonic propagation in heterogeneous media has been studied by many researchers all over the world. Historically, two distinct approaches have been developed to treat the problem of wave propagation through heterogeneous media. The first of these approaches, the transport theory, establishes the energy transfer in a medium which is heterogeneouslm3, and the starting equation (transfer equation) is equivalent to the Boltzmann equation so that by an energy balance we may achieve a flexible formulation leading to the treatment of a large number of physical phenomena. The second approach, the analytic theory, is obtained by equating components of stress4-6. Following an amplitude balance and having introduced the action of several statistically weighted heterogeneities, namely those of Watermann7.8 and Twersky’, one may arrive at differential equations which characterize certain physical quantities related to the ultrasonic propagation. If we analyse the development of these two theories, a common point is found: both make use of the scattering cross-section of single obstacles (usually spherical). Taking this into account, we have developed a theoretical formulation for the scattering cross-section and an experimental apparatus suitable for its determination in the case of polystyrene obstacles immersed in water. Experimental observations are in good agreement with theory, which confirms and validates our theoretical formulation and experimental set-up. Classic

model

of scattering

cross-section

For plane waves propagating in heterogeneous with wavelengths larger than the dimensions 004-624X/88/020102-05 $03.00 0 1988 Butterworth & Co (Publishers)

102

Ltd

Ultrasonics 1988 Vol 26 March

media of the

scattering;

heterogeneous

media

heterogeneities, the characteristic attenuation of the medium may be determined by the scattering crosssection. This has been calculated by Ying and Truelll’ for a longitudinal wave, and by Einspruch, Witterholt and Truell” for a transverse wave. The work of Truell, Elbaum and Chick” has been further investigated and reformulated by Lefebvre’ 3. (This last work includes full details to put an end to the confusion created by some misprints in the expressions in References 14 and 15.) Theory

It is well known (see any of References 10-15) that once the scattering cross-section of an embedded matrixmedium obstacle is determined, the medium characteristic attenuation, CI,is given by c1= n,y

(1)

where no represents the number of heterogeneities per unit volume in the matrix medium and y is the scattering cross-section. As the scattering cross-section is, by definition, the ratio between scattered energy per unit time and incident energy density per unit area and unit time, it may be obtained by an energy balance of the different wave types, taking into account the wave mode conversion: 1 2 3

incident longitudinal wave associated with scalar potential cpi; scattered wave associated with scalar potential cpS; and longitudinal and transverse waves transmitted to the obstacle, described by potentials cpt and II/,.

The potentials

may be expanded

as finite summations

Resonant

forward

scattering

of solid inclusions

in a fluid: J. M. PerdigSo

1

of orthogonal functions (using appropriate symmetry), such as the Hankel, Bessel or Neuman functions or the Legendre polynomials. Following Watermann and applying the boundary conditions one may determine the scattering cross-section of a scatterer by using r=(47clk:)(llcp;) CI‘%12

+dm+

2

(2m+

et al.

1)

l)(~Il~I)I&I121

(2)

where K, and k, represent transverse and longitudinal wave numbers in the matrix medium, respectively, and A and B are the scattering coefficients corresponding to m iterations of cpSand II/,. As the matrix medium is a fluid (water) it does not support shear waves and, in working with Equation (2), we should note that k,/K, goes to zero because there is a negligible contribution from the transverse wave in water.

Theoretical

results

In writing Equation (2), we evaluate the summation of an infinite series and use the above-mentioned functions. In practice, this evaluation is carried out on a digital computer programmed to obtain an accuracy on the order of 10m3 by using special routines appropriate to the evaluation involved. In the present case, evaluation of scattering cross-section has been carried out for a spherical polystyrene scatterer in water, following Watermann’s method using the classic decomposition. For aluminium spheres in water, Bostrom16 uses a different decomposition based on Green’s functions and the integral representation developed by Varatharajulu and Pao17. We applied our decomposition to this particular case. Comparison between our results and those of Bostrom show good agreement (see Figure I). Any discrepancies are probably due to different precisions used in the evaluation. The particular case we were interested in, polystyrene scatterer in water, has been carried out, and the scattering

0 1

I

I

I

I

3

5

7

9

ka Figure 1 Normalized scattering cross-section, ynor, of aluminium spheres in water versus ka (ynor = y/s, k = 2xV/rZ). P-PI This work; -. Bostrom’6

I

I

I

1.0

1.5

2.0

Frequency,

F (MHz)

Figure 2 Theoretical normalized scattering cross-section, versus frequency for polystyrene spheres in water. Radius: 0.8 mm; -, 0.4 mm

cross-section

determined,

Scatterer u,2 =2470 V

for the following Matrix

m s-r

ynor, ---,

situation:

medium

ai1 = 1500 m s-i

tz=1150ms-’

-0

V 11 -

p2 = 1052 kg rnp3

p1 = 1000 kg me3

a = 0.440.8 mm where: ui and u, are the ultrasonic longitudinal and transverse wave speeds; p is the density; a is the sphere radius. Theoretical calculations of the normalized scattering cross-section are shown in Figure 2, which clearly indicates the presence of resonance phenomena for which the scattering cross-section values may be greater than the geometric cross-section of the scatterer. However, these resonance phenomena are attenuated as frequency increases, as was also shown by Ishimaru18 and Morse and Feshbach”, and the scattering crosssection tends towards twice the cross-section for high frequencies. Large scattering cross-section values may be explained by scatterer effects due to resonance deformations. In fact, a resonant scatterer may introduce some deformations to the surrounding medium. As shown in Figure 3, the deformations cause an increase in scattered energy which, in turn, leads to a remarkable increase in the scattering cross-section. Thus the scatterer shows an effective crosssection which may be considerably larger than the geometric cross-section. In a real scattering medium these resonance phenomena usually have a smaller effect. Inverse problems in acoustics are exceedingly difficult and often not well founded. A

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1988

Vol 26 March

103

Resonant forward scattering of solid inclusions in a fluid: J. M. Perdigso et al. an average scattering cross-section. This is shown in Figure 4 where the normalized scattering cross-section for two scatterers (a = 0.3 and 0.6 mm) is super-imposed. When modelling ultrasonic propagation in heterogeneous media this effect must be taken into account.

Scattered wave 11

Transmitted

wave

Experiments

Obstacle in resonance Incident wave

Deformations Figure

3

Surrounding

in surrounding medium medium

deformations

due to obstacle

In the literature many studies may be found relating to scattering cross-section evaluation’0-‘3~‘6-‘8. The basic work and experimental techniques have already been developed for the determination of the scattering crosssection of an obstacle20m25. Much of the literature, especially that directed to the NDE community, deals with fluid inclusion or voids in solid materials. There is, however, a vast body of literature that deals with the scattering of pressure waves from solid inclusions in a fluid matrix. This is the work of people dealing primarily with underwater acoustics and a number of classic papers detail the scattering pattern of solid spheres of various materials submerged in water. A possible source of difficulty is that workers in the field of underwater acoustics express their results in terms of far-field form functions, rather than scattering cross-section. However, a simple conversion factor can be used to turn the form function into a scattering cross-section: the relationship dy/dQ

= a2/41U,(Q12

with f( 0) = C,f,( 6) = form function

0

.f ....

I

I

1.0

1.5

2.0

0.5

Frequency,

F (MHz)

Figure4 Theoretical weighted normalized scattering cross-section, ynor, versus frequency for polystyrene spheres in water. Radius: ---, 0.6 mm; ....., 0.3 mm. -, Average scattering cross-section

scattered field does not point uniquely to a particular form of scatterer, but it is possible to make inferences and indeed this should be done. In our opinion, the disappearance of the sharp peak is due to the presence of spherical inclusions of various size, therefore the equivalent scattering cross-section of a scatterer dispersion may be comparable to that of the geometric cross-section scatterer. In the situation described in this Paper, a distribution of spherical scatterers of various size (radius 0.3-0.6 mm) with a radius probability distribution of p. at the extremes and 2p, in the middle, the resonance phenomena can be attenuated by the interaction between different obstacles. Taking this distribution into account and inserting the proper values into Equation (2), we are able to introduce

104

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1988

Vol 26 March

is cited in Gaunard et ~1.~~. Once the correspondance between the form function and the scattering crosssection is known, the whole field of scattering data from submerged spheres becomes useful and relevant. The theoretical results indicate the existence of resonance phenomena ~ a prime subject for experimental observation. These phenomena correspond essentially to the back-scattering cross-section or are expressed in terms of far-field form functions. Our original contribution to this field is made by using forward scattering geometry to measure the scattering cross-section. We believe that a direct measurement has not yet been carried out (in the literature) for spherical inclusions, thus we have tried to measure forward scattering directly with reasonable precision. Experimental measurements were taken under identical conditions to those assumed by the theory. One of these conditions is that the emitted wave is equivalent to the incident plane wave, i.e., amplitude and phase uniformity. In view of the ultrasonic field distribution created by a circular transducer, this uniformity is valid only in the far-field. The far-field distance, using a -3 dB contour criterion, is given by Dmin= 4b2/1

(3)

where b is the radius of the transducer and I is the wavelength in water. We must, therefore, have a transducerobstacle distance greater than Dmin. However, measurement of the scattering cross-section will only be possible if we determine the perturbations introduced by the obstacle. When using gated pulses, usually all interfering signals can be gated out. However, if the ultrasonic beam dimension over the scatterer is limited by a screen, a better ratio is obtained between the captured energies (with and without scatterer). The area, S, of the limiting

Resonant forward scattering of solid inclusions in a fluid: J. M. PerdigZo et al.

screen hole, however, must satisfy two conflicting requirements in relation to the geometric cross-section, s, of the obstacle: the edge effects and a reasonable precision. A good compromise is obtained for a S/s ratio in the range 20- 100. In addition to the above, pulses must be used to avoid multiple reflections. Therefore, a narrow pulse width is needed to take into account the conflicting requirements between the time and frequency domain resolution. Experimental conditions

Experiments have been performed under the following conditions: 1 piezoelectric transducer mechanically and electrically matched. These transducers have a wide pass-band and their dimensions have been chosen to ensure that the ultrasonic field covers the scatterer, for a transducer-obstacle distance of the order of 10 cm (F = 3 MHz, b = 3.5 mm); 2 pulse width in the range 100-200 ps; 3 wood screen with holes, with S/s = 50 in all cases; and 4 spherical polystyrene scatterer with dimensions in the range 0.8-1.6 mm diameter. The acoustic scattering cross-section may be easily determined by transmitted energy measurements carried out in the absence and presence of an obstacle placed in the centre of the limiting screen hole. Deconvolution techniques are usually employed for the flaw impulse response, however when one waveform is merely superimposed on another a subtraction in the time domain can be used. Thus, by simple subtraction one may deduct the scattered energy and scattering cross-section. To ensure the significance of our measurements, it is necessary that the field conditions near the hole are uniform for both source and receiver. In practice, the source and receiver were chosen to have similar characteristics, being at least a distance of 20,i” apart. Under these conditions the received energy in the absence of an obstacle is E,,=cA2=dZS

(4)

where c and d are constants, I is the field intensity and A is the amplitude of the recorded signal. The received energy in the presence of an obstacle is E,, = b(A - AA)’ = cl(S - y)

(5)

The expression for the scattering cross-section, obtained from previous equations, is y = S[A2 -(A

- AA)‘]/A

(6)

If we assume weak amplitude variations in relation to the signal amplitude, AA <
(7) set-up

Values AA and A in Equation (6) were determined using the apparatus shown in Figure 5. Detection and sampling of the recorded signal enabled us to regulate the system which creates reference signal A and measure AA with an adequate accuracy. Special attention was given to the signal system reference generator. We used a regulated

I

Figure 5

0

,

Schematic diagram of the experimental set-up

20

40

60

Section ratio, S/s Figure 6 Experimental scatterer perturbation section ratio

evolution

versus

power supply (with regulation better than 0.01%) and compensated for the temperature drifts of components integrated in this system. With these precautions we maintained A and AA fluctuations under 0.1%. Experimental results

Having solved the stability and regulation problems of the measurement chain, we determined y experimentally using screens with holes of 8-16 mm diameter, placing the polystyrene sphere at the hole centre. The positioning of this sphere was achieved by a suitable mechanical system with a thin suspension cord of 15 pm. The effect of the glue fixing the cord to the sphere is negligible in this situation. As can be seen in Figure 6, the edge and other mechanical effects are quite negligible for section ratio S/s > 40. In this situation, the recorded experimental perturbation, A’A, represents essentially the scatterer influence, AA. Results obtained for different holes and obstacles uersus frequency allow us to confirm the validity of field uniformity as well as negligible edge effects. Experimental measurements have been undertaken using obstacles with diameters in the range 0.8-1.6 mm. A reasonable fit was found between theory and experiments. A comparison between theory and experiments versus frequency (using steps of a few kHz) is shown in Figures 7 and 8, for obstacles of 0.8 and 1.6 mm diameter,

Ultrasonics 1988 Vol 26 March

105

Resonant forward scattering of solid inclusions in a fluid: J. M. PerdigBo et al.

10

8

6

.;1 ,... 8Al l

l

l

0

1.0

0.5

1.5

/

2.0

2

0 Frequency,

0.5

1.0 Frequency,

Conclusions

10

When studying and applying the criterion of scattering cross-section in heterogeneous media we must deal with certain difficulties attached to the experimental confirmation of theoretical results. Resonance effects due to ultrasonic excitation of solid inclusions in water were confirmed using experimental apparatus. The original contribution to this field made in this Paper is mainly the use of a forward scattering geometry measuring technique. Most of the bistatic measurements in this area have been carried out by researchers in France, but nearly all of these measurements were performed on cylinders and cylindrical shells. In this Paper, experimental forward scattering results are presented for spherical solid inclusions in water and reasons are proposed for the eventual disappearance of the sharp peak of resonant scattering in real materials. Good agreement between theory and experiments is found; this confirms the validity of the proposed experimental set-up. References Reynolds, L., Johnson, C.C. and Ishimaru, A. Appl Opt (1976) 15

2

Reichmean, J. Appl Opt (1973) 12 1811-1815 Kattawar, G.W., Plass, G.N. and Catching, F.E. Appl Opt (1973)

12 13 14

15 16

17 18 19 20 21 22 23

25

2059-2067

106

11

24

1

1071L1084

Ultrasonics

1988 Vol 26 March

F (MHz)

Figure 8 Comparison between experimental and theoretical results of the normalized scattering cross-section, ynor, versus frequency (a = 0.8 mm). -, Theory; 0, experiments

respectively, and a hole area 2 cm’. These results clearly show the qualitative and quantitative agreement as well as the presence of resonance phenomena with scattering cross-section values 10 times the cross-section. (For small obstacles the fixation cord and glue seem to introduce any disturbances in the peak resonance amplitude and in the off-peak region.)

12 314-319,

2.0

F (MHz)

Figure 7 Comparison between experimental and theoretical results of the normalized scattering cross-section, ynor, versus frequency (a = 0.4 mm). -, Theory; 0, experiments

3

1.5

26

Lin, J.C. and Ishimaru, A. J Acoust Sot Am (1974) 56 1965- 1700 Ishimaru, A. Proc IEEE ( 1977) 65( 7) 1030- 1061 Hemenger, R.P. J Opt Sot Am (1974) 64 503-509 Watermann, P.C. and Truell, R. J Math Phys (1961) 24 512-537 Fikioris, J.G. and Watermann, P.C. J Math Phys (1964) 5( 10) 1413-1420 Twersky, V. Proceedings of the American Mathematical Society Symposium on Stochastic Processes in Mathematical Physical Engineering ( 1964) 16 84- 116 Ying, C.F. and Truell, R. J Appl Phys (1956) 27 1086&1097 Einspruch, N., Witterholt, E.J. and Truell, R. J Appl Phys (1960) 31 108 Truell, R., Elhaum, C. and Chick, B. Ultrasonic Methods in Solid State Physics Academic Press, New York, USA (1969) Lefebvre, J.E., Frohly, J., Torguet, R. and Bruneel, C. Ultrasonics (1980) 22 170-174 Truell, R. and Elbaum, C. Encyclopaedia of Physics: High Frequency Ultrasonic Stress Waues Vol XI/2 (Ed Fliigge, S.) Sections 11 and 12 (1969) Latiff, R.H. and Fiore, N.F. J Acoust Sot Am (1975) 57 144l- 1447 Bostrom, A. Scattering of stationary acoustic waves by an elastic obstacle immersed in a fluid, internal report from Institute of Theoretical Physics, Fack, Goteborg, Sweden (1978) Varatharajulu, V. and Pao, Y.-H. J Acoust Sot Am (1976) 60 556-566 Ishimaru, A. Wave Propagation and Scattering in Random Media Academic Press, New York, USA (1977) Morse, P. and Feshbach, H. Methods of Theoretical Physics Vol I, McGraw Hill (1953) 1485 Harbold, M.L. and Steinberg, B.N. J Acoust Sot Am (1968) 45 592-602 Hickling, R. J Acoust Sot Am (1962) 34 1582-1592 Vogt, R.H. and Neubauer, W.G. J Acoust Sot Am (1976) 60 15-22 Flax, L., Dragonette, L.R. and Uberall, H. J Acoust Sot Am (1978) 63 723-73 1 Dardy, H.D., Bucaro, J.A., Schuetz, L.S. and Dragonette, L.R. J Acoust Sot Am (1977) 55 1373-1376 Dragonette, L.R., Numrich, S.K. and Frank, L.J. J Acoust Sot Am (1981) 69 1186-1189 Gaunaurd, G.C., Tanglis, E., Uberall, H. and Drill, D. II Nuouo Cimento (1983) 76B(2) 153