Time-frequency processing of underwater echoes generated by explosive sources

Time-frequency processing of underwater echoes generated by explosive sources Guiller~o

C. Gaunaurd*

and Hans 6. Striforsj-

*Naval Surface Warfare Center, Carderock Division, White Oak (Code 684), Silver Spring, MD 20903-5640, USA TNational Defense Research Establishment (FOA 6) S-172 90 Stockholm, Sweden Revised 3 October 1994 We study the advantages and performance of an ‘impulse sonar’ that uses the short, energetic pulses emerging from underwater explosions as its sources. This transient analysis is carried out for various simple targets of interest, in the time-domain, in the frequency domain and in the combined time-frequency domain. The first two, more traditional, approaches are contrasted with a third and more recent processing scheme in the combined tim~frequency domain, which utilizes the windowed or pseudo-Wigner distribution (PWD). This third processing approach exhibits the time-evolution of the target resonances in a more informative way and it is seen to offer additional advantages for target identification purposes. We display calculations showing the quite different patterns generated by the echo of each target, which can then be used unambiguously and remotely to characterize them. The procedure, implemented here for the PWD, could be repeated using any of the many distribution members of the general bilinear class. Keywords: acoustic processing; Wigner

echoes; submerged elastic shells; scattering distribution; underwater explosions

The processing of echoes returned by various underwater structures of interest has target identification as one of its main goals, second only to detection. The conventional way to process sonar echoes has been in the frequency domain, followed later by time-domain approaches. A third, more recent and seemingly more advantageous way is to use schemes in the co~~~~e~ time-frequency domain. These can be carried out by using the many members of the general bilinear class of distributions. We show how the use of the pseudo-Wigner distribution seems to simplify the classification goal provided we use the short pulses that an ‘impulse sonar’ is usually designed to use. Since the high-amplitude pressure levels required by such a sonar are hard to generate in short times, we will use here the strong short pulses produced by underwater explosive sources. Theoretical

of waves; signal

amplitude of the incident wave of circular frequency CO. The form-function, f,, in the backscattering direction, here defined to be 9 = n, is given by the Rayleigh series

f&h -4 = ; n

z.(- 1)“m + 1Mxf

(2)

If the body is a spherical shell, of outer and inner radii a and b, possibly containing an inner filler, the coefficients T,(x) can be written as the ratios of two 6 x 6 determinants: &,(x)/D,(x) of the form

background

(i) From steady-state to transient interrogations If steady-state, continuous plane sound waves are incident on a spherical elastic body of radius a, submerged in water, the backscattered pressure far-field is (see Figure I) po* iW - ktr)f,(e ~~~(r,@= a,G = 2re

= 7c, x.

(3)

(1)

where x E k,a = (w/c& is the non-dimensional frequency variable in the outer fluid of sound speed cl, and p,-,is the ~1-624X~95/$~.50

Q 1995 - Elsevier

SSDI 004 t-624X(94)00020-4

Science B.V. All rights reserved

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Time-frequency

processing

of underwater

echoes: G. C. Gaunaurd

and H. C. Strifors

(6) is the travel time between the observation point and the back of the sphere. This fact implies that values of r/a > 2 correspond to a location of the observer outside the scatterer, which accordingly will give a positive value to the arrival time of the backscattered pulse. (ii)

Figure 1 Scattering geometry plane-wave incidence

for an elastic spherical shell with

where the 28 non-vanishing elements dij can all be determined from the boundary conditions of the problem, and they have all been listed elsewhere’,*. If the sphere is solid, the T,(x) coefficients are given as the ratios of two 3 x 3 determinants ~ obtained from the upper left corner of the 6 x 6 determinants above - which have also been determined and are listed elsewhere3. Let T = (c,t)/a be the non-dimensional time variable, and consider the Fourier transform4 pair +7. + 3t g(t) = L G(x)eiX7dx G(x) = s(r)e -‘-” dt, 271s _-3j s -3; (4) In numerical calculations, we use the corresponding discrete-time Fourier transform4 (DFT) pair G(k) =

N-l c g(n)epi(2"lN)k",

g(n) = $:$I

G(k)&2nlN)kn

It=0

(5) where the sequences g(n) nd G(k) both contain N elements. It can be shown’ that the backscattered pressure in the z-domain is given by

T)? =L 2~ a

m f

t ,(x, x)G(x)eiz(’ - ‘la)dx (6) s 3cI whenever a pulse g(z) is used as the incident interrogating waveform. This result takes us from the steady-state situation in Equation (2), in the frequency domain, to the transient situation in the time domain in Equation (5). The spectrum G(x) of the incident pulse acts as a window or ‘filter function’, and it allows the extraction of whatever portion of the form-function, f,, falls within irs band of definition. Narrow pulses will have broad spectra that will consequently permit the extraction of broad spectral portions of f,. This is the essence of the short-pulse/ broad-band techniques used in impulse radar6 or impulse sona?. The broader the portion of the (steady-state) If,1 that the incident short pulse extracts, the easier it will be to identify (resonance) features in the returned echo that will later permit the unambiguous (active) classification of the scatterer. The exponential exp( - ixr/a) in Equation

psc(ll,

148

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Signalprocessing domain

in the combined

time-frequency

The analysis of the returned echoes has traditionally7 been done in the frequency domain (cf. Equation (2)), and later”.’ in the time domain (cf. Equation (5)). A recent method of processing that seems to be gaining acceptance is to work in the combined time-frequency domain. This approach seems to give the most information since it can show the evolution of the identifying resonance features of the scatterer and their amplitudes as surfaces in a general time-frequency-amplitude three-dimensional space. Usually, projections of these three-dimensional surfaces are shown in the two-dimensional time-frequency plane. This evolution can be extracted from the echoes and displayed in as much detail as is feasible, limited however by the influence of the noise contained in the echoes. The extraction is carried out by any of the many distributions that are membrs of the general bilinear’0,” class

xf(.

+ $*(u

-f)dudTd{

(7)

where the asterisk denotes complex conjugation, and O(t, T)is the kernel” function. This bilinear class includes the distributions attributed to Wigner, Ville, MargenauHill, Kirkwood-Rihaczek, Choi-Williams, etc, each with its own characteristics, although sharing the essential properties of time-frequency distributions. Using the kernel a,((, z) = 1 in Equation (7) and integrating the exponential exp[i(t - u)5] with respect to 5 gives the result 2&(t - u) in terms of the Dirac delta function, and it follows immediately that the bilinear form C,(w, t; 0 s 1) is equal to the (auto-) Wigner distribution (WD) of the function F (t)

+m =2s

f(t + t)f*(t - T)e-‘*‘” dz (8) -cc The WD shares with some other time-frequency distributions the property of preserving the time and frequency energy marginals of a signal, i.e. the integration of the WD over the frequency variable at a generic time (or over the time variable at a generic frequency) yields the signal’s instantaneous power at that time (or energy density spectrum at that frequency)“. Another property of the WD, which is desirable for target recognition purposes, is its ability of concentrating the features of a function in the combined time-frequency domain. Digital evaluation of the WD of continuous-time functions requires a re-formulation of Equation (8) to its analogue for discrete-time functions. Existing algorithms for FFT can then be adapted to the discrete Wigner distribution. Analogous to Equation (5), the discrete-time version of the second part of Equation (8), for a sequence

Time-frequency

f(n) containing

N elements

processing

of underwater

and the corresponding DPWD) is

is

N-l

W,(k, l) = 2 c f(l n=O

+ n)f*(l

- ,)e-i(4x’N)k”

(9)

frequency and where k,E=0,1,2 ,... N - 1 represent time, respectively, and f(l + n - N) is substituted for f(l + n) whenever I + n > N. Comparing Equation (9) with the first of Equations (5) shows that the WD is periodic with period X, rather than 2~, as is the case for the DFT. Thus, aliasing is, in general, present in the WD even when the sampling rate satisfies the Nyquist criterion. To avoid aliasing, the ‘analytic function’ could be used when computing the WD. This function is defined by L(n) =f(n)

i3(4

+

(10)

where f(n) is a given real-valued function and 3(n) is the discrete Hilbert transform4 of f(n), defined by

(11) The spectrum

F,(k) =

of the analytic

function

2F(k)

0 < k < N/2

F(k)

k = 0, N/2

i 0

f,(n) is then given by

(12)

+ = f(t + r)f*(t s -30

-

discrete pseudo-Wigner

distribution

N-l

$.(k,

r) = 2 c f(l + n)f*(I - n)w, (n)w)i( -n)e-i(4=‘N)kn n=O (14)

One of the most commonly used window Gaussian in nature, and we use the form

functions

7)~/(7)~/*(-7r)e~'*"'~

ds (13)

is (15)

wJt) = exp (-at*)

where a is a positive real number that controls the width of the window. This function can be applied to the backscattered pressure field returned by any scatterer, such as the spherical body considered earlier, provided the spectrum G(s) of the used incident pulse is known. The choice of window size u used in the calculations depends on the type of feature one wishes to emphasize.

Selection pulse

and

description

of

the

shock

An initial and somewhat simplistic approximation for the shape of a shock-wave pulse emerging from the explosion of an underwater charge is given by an exponential, namely Pint = pmaxe-f’H

N/2 < k < N

where F(k) is the DFT of the sequence f(n) according to the first of Equations (5). The Hilbert transform can be obtained by using Equation (12) jointly with a good FFT algorithm, which makes the analytic function approach to a non-aliasing WD very convenient in numerical implementations. While, in practice, the length N of the sequence f(n) is often selected to be an integer power of 2, it could be any integer. We note that when N is an odd number, F,(k) = F(k) only for k = 0. When analytic signals are used, the distribution in Equations (8) or (9) is often referred to as the Wigner-Ville distribution. Practical applications of the WD are limited by the presence of ‘cross-terms.’ The cross-terms attributed to the bilinear nature of the distribution, generate features that lie between two auto-components and can have peak values larger than those of the auto-components. However, using the analytic function f,(n) eliminates cross-terms between positive and negative frequency components. It is possible to suppress the remaining cross-terms by weighting the function before evaluating the WD using a window function. This window function can be made to slide along the time axis with the instant t at which the WD is being evaluated. Different window functions will place different weights on the time segments of the time-varying function 1; and the resulting pseudo-Wigner distributions (PWD) will have different physical interpretations. Another important property of the window function is that, if narrow enough, it suppresses the influence of noise on the distribution function. If w,(t) is the window function, the PWD of f(t) is R&0, t) = 2

echoes: G. C. Gaunaurd and H. C. Strifors

(16)

In reality, the shock wave pulse has a more gradual decay, proportional to CC’.* for the later times, as has been proposed by Kirkwood and Bethe”,13. The transition region between t = 0 ~ the initial time constant ~ and the later-time region characterized by a decay of the form t-O.*, can be approximated in various ways. One way to describe this middle region is by means of a parabolic decay law: p = p(to) + A(t - to)*, where the parameters to, A,. . are numerically determined for the particular explosive and ranges of interest. These quantities have been tabulated for many explosives, distances from the explosion, size of charges, etc in a useful handbook14. There is also a ‘hump’ in the transition regionI ~ not always observableI that is often ignored, because its

F,,, \, 0

10

,,, ,, 20

Frequency

30

40

,I 50

[kHz]

Idealized incident pressure pulse emerging from Figure 2 TNT-charge (insert plot) and the modulus of the spectrum pulse (main plot)

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1995

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a small of the

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Time-frequency

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echoes: G. C. Gaunaurd and H. C. Strifors

effect is small. The pulse, to be used in our calculations, is shown in Figure 2 (insert plot). This pulse is normalized

8 = 0.05 ms. It would to Pmax and has the time-constant: correspond14 to a charge of 4 g of TNT (trinitrotoluene) at about 50m (- 115Oft) away. The modulus of the spectrum of the pulse is shown in Figure 2 (main plot) in the frequency interval &50 kHz.

Steady-state bodies

responses of various spherical

The steady-state response (i.e. the response of the target to incident CW plane waves) of four submerged spherical bodies is displayed in Figures 34, main plots. The four spherical bodies, which all have an outer diameter of 1 m, are an air-filled spherical steel shell of thickness 10 mm (relative shell thickness of h/a = 2.0%) (Figure 3, main plot), a TNT-filled spherical steel shell of thickness 5 mm (relative thickness h/a = 1.0%) (Figure 4, main plot), a TNT-filled spherical steel shell of thickness 2 mm (Figure 5, main plot), and a solid granite sphere (Figure 6, main plot). The four form-functions are displayed in the usual

Air-filled Diameter.

“0 2 1

2

g

1

n 0

10

20

30

40

50

Frequency [kHz] Figure 5 Backscattering response (insert plot) (h/a = 0.4%)

form-function of a TNT-filled

(main plot) spherical

and impulse steel shell

spherical steel shell thickness 10 mm

I m,Shell

-0 6 ”

I

2

3

4

Time

0

10

20

30

Frequency

40

50

0

[kHz]

IO

20

Frequency

Figure 3 Backscattering form-function (main plot) and impulse response (insert plot) of an air-filled spherical steel shell (h/a = 2%)

30

6

7

8

[ms]

40

50

[kHz]

Figure 6 Backscattering form-function response (insert plot) of a solid granite

(main sphere

plot)

and

impulse

frequency domain for: 0 < f 6 50 kHz. The values of the form-functions are computed in the non-dimensional frequency interval: 0 < k,a d 120, where the upper limit for bodies of 1 m diameter corresponds to the frequency .f= 56.9 kHz, in 8192 equidistant frequency points. To avoid aliasing error when computing inverse discretetime Fourier transformations according to the second equation in Equation (5) the spectrum is lowpass filtered using a third-order Butterworth filter4 with nondimensional passband cut-off frequency k,a = 0.95 x 120 = 114 (outside the frequency band shown in Figures 3-6). There are obvious differences between the four responses. The physical properties of water, air, steel, TNT and granite are given in Table 1. 10

20

Frequency

30

40

50

[kHz]

Figure 4 Backscattering form-function (main plot) and impulse response (insert plot) of a TNT-filled spherical steel shell (h/a = 1%)

150

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Transient

responses

of targets

If the targets in the previous section (Figures 3-6) are hit by the pressure wave in Equation (16) (cf. Figure 2, insert

Time-frequency processing of underwater echoes: G. C. Gaunaurd and H. C. Strifors Table 1

Material properties of targets and ambient fluid

Material

Density p (kg mm3)

Dilatational wavespeed (m s-l)

Air Water Steel TNT Granite

1.2 1000 7800 1650 2700

340 1490 5880 2100 4770

cd

plot), their responses can be determined in the time domain, in the frequency domain, or in the combined timeefrequency domain. Because the incident pulse has short duration, its spectrum is broad, as can be seen in Figure 2, main plot. The time-domain response is obtained from Equation (6) using the form-function of the steady-state case (cf. Figures 34, main plots), and the spectrum of the incident waveform (cf. Figure 2, main plot). When the incident waveform is not the pulse given by Equation (16) but an ideal impulse, i.e. given by a Dirac delta function (namely g(r) = 6(r)), the frequency response is obtained by a discrete-time form of Equation (6) with G(x) = 1. The result in all four cases is displayed in the insert plot of each one of Figures 34 in the time interval: 0 6 t < 8 ms. The ordinates have been normalized to unity. When the incident waveform is given by Equation (16) the response in the time domain is similarly obtained using the spectrum displayed in Figure 2 (main plot) for G(x). The result in all four cases is displayed in Figures 7-10, on the left grid plane. The corresponding right grid plane displays the pulse-extracted spectrum, i.e. the spectrum of the waveform on the left grid plane. The dotted line on each right grid plane is the spectrum of the incident waveform (cf. Figure 2). The ordinates of the plots on the grid planes have been normalized to unity. Moreover, the initial high-amplitude features of the pulse-extracted spectrum in Figure 7 have been cut to make the rest of the spectrum more easily discernible. It is clear that since the incident pulse is so broad-banded, the frequency responses extracted from the backscattered echoes are good replicas of the form-functions in Figures 34 for the steady-state cases or, alternatively, the delta-function extracted spectra. We have seen5 that short energetic pulses are the best ones to obtain such good replicas in quite broad frequency bands. We now consider the response in the combined time-frequency domain. The combined time-frequency displays are generated by means of the discrete-time version of the pseudo-Wigner distribution (PWD) in Equation (14). The Gaussian window used (cf. Equation (15)) has parameter x = 5 (ms))*. When computing the PWD we use the Hilbert transform (cf. Equation (11)) of the spectrum of the backscattered echo that is extracted by the waveform given by Equation (16). The output of this evaluation can be exhibited as a three-dimensional surface that displays the absolute value of @f at each point (f, t). For the four pertinent targets, these threedimensional surfaces are shown in Figures 7-10, together with projected two-dimensional contour plots of 20 equidistant levels of the respective three-dimensional surface. These three-dimensional surface plots and two-dimensional contour plots together provide signature

Shear wavespeed (m s-l)

c,

3140 1120 3180

Young’s modulus E (GPa)

Poisson’s ratio Y

200 5.4 60

0.3 0.3 0.1

Waveform L

C-+C--Yp

;

4

Figure 7 Surface plot and its plane projection contour plot of the PWD of the transient response when an air-filled spherical steel shell (1 m diameter, 10 mm thickness, h/a = 2%) is insonified by a shock-wave pulse. The grid planesshow the waveform and spectrum of the transient response

Waveform mm

Figure 8 Surface plot and its plane projection contour plot of the PWD of the transient response when a TNT-filled spherical steel shell (1 m diameter, 5 mm thickness, h/a = 1%) is insonified by a shock-wave pulse. The grid planes show the waveform and spectrum of the transient response

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representations of the targets that can be conveniently interpreted. The frequency-domain plots (right grid plane) and time-domain plots (left grid plane) help in the interpretation of the central PWD-plots. The evolution of the resonance features is easily traced in the central plots. For example, in Figure 7 the low-frequency multipole feature16, near f~ 2 kHz, shows a timedevelopment that may appear to be more prominent for t > 2 ms, but which is present in the entire time-interval shown (0 < t 6 5 ms). The ‘hump’ at the coincidence frequencyi7, present for: 20 < f < 35 kHz, does not seem to develop until t E 1.5 ms. It can also clearly be seen that features related to the coincidence hump occur repeatedly at multiple times of 1.5 ms, at about 3 and 4.5 ms, times which are caused by the ringing of anti-symmetrical Lamb waves below coincidence5. In Figure 8, the low-frequency (multipole) feature of Figure 7 is washed-out and is absent, since now the filler is denser16. Also, features related to the coincidence hump now appear smeared-out over a broader band (namely, 5
Figure 10 Surface plot and its plane projection contour plot of the PWD of the transient response when a solid granite sphere (1 m diameter) is insonified by a shock-wave pulse. The grid planes show the waveform and spectrum of the transient response

is that the four objects show quite different ‘signatures’, which can be used to identify them from their remotely sensed echoes. If a neural network or some other automatic target recognizer were given the opportunity to learn these patterns, it would be simple to distinguish these four bodies unambiguously. The active classification capability of the process is clearly enhanced by the broadband nature of the incident pulse. Shorter pulses produce responses in broader spectral bands, which contain more identifying features, and this is the main advantage of an ultra-wideband projector.

Conclusions

Figure 9 Surface plot and its plane projection contour plot of the PWD of the transient response when a TNT-filled spherical steel shell (1 m diameter, 2 mm thickness, h/a = 0.4%) is insonified by a shock-wave pulse. The grid planes show the waveform and spectrum of the transient response

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We have studied the response of various underwater scatterers to incident energetic pulses emerging from (transient) explosive sources. The analysis was illustrated with four simple targets (i.e. an air-filled steel shell, two TNT-filled steel shells of different thickness, and a granite sphere) which are simplistic models for certain man-made scatterers and rocks. The processing of the echoes has been accomplished in the frequency domain, in the time domain, and in the combined time-frequency domain, using incident short pulses in all cases. The first two approaches are the more conventional and traditional ones. The time-frequency domain approach is more recent and it seems to offer more advantages because both the spectral content of the target response and its time-evolution can be utilized for target recognition purposes. In the present work we have used the windowed Wigner distribution or pseudo-Wigner distribution (PWD) with a narrow window. The resulting plots (cf. central plots of Figures 7-10) exhibit easily discernible differences between the four considered targets by emphasizing the time-evolution of the resonance features generated by each of the objects in clearer and different ways. Those patterns could be

Time-frequency

processing

used later in conjunction with neural networks or other pattern recognizers to achieve the unambiguous (active) classification of non-cooperative submerged structures. It may be argued that identifying features are also present in the waveform or the spectrum plots of Figures 7-10, and that these could also be used for classification. However, it is apparent that those ‘features’are not always easy to distinguish or to analyse. For example, the coincidence ‘hump’ is smeared-out and hardly noticeable in the spectrum plots of Figures 8-10. Such plots contain rapid successions of noise-like peaks and dips that are difficult to see or interpret. However, the (two-dimensional projections of the) PWDs show clear and interpretable features as they successively develop in time, and this is the main advantage of these two-dimensional plots. In brief, we have described the advantages and performance of an ultra-wideband (or quasi ‘impulse’) sonar that uses the short pulses emerging from explosions or explosion-like sources, to achieve broad-band active classification of non-cooperative targets, by means of processing schemes in the combined time-frequency domain. The physical interpretations and origins of all the identifying features present in the separate frequency or time domains have been given earlier in considerable detail for sonars,16 and radar18. Any arbitrary filler substance and shell material or thickness (or material composition of the solid sphere) is possible under the present formulation. These material properties must be assumed known in order to generate plots such as the ones shown here. Then, these plots would be the ones to be used as ‘templates’ to classify an object under investigation as one of the (four) possibilities presented here. Additional possible alternatives could be also generated at will in the same fashion.

of underwater

References 1

2

3

4 5

9 IO

11 12 13

Gaunaurd, G.C. and Kalnins, A. Resonances in the sonar cross sections of coated spherical shells Int J Solid Structures (1982) 18 1083-l 102 Ayres, V.M., Gaunaurd, G.C., Tsui, C.Y. and Werby, M.F. The effects of Lamb waves on the sonar cross section-sections ofelastic spherical shells Inr J So/id Structures (1987) 23 937-946 Gaunaurd, G.C. and Wertman, W.H. Asymptotic and numerical determinations of the complex eigenfrequencies and the real resonances of an acoustically insonified sphere J Acoust Sot Am (1992) 91 248992501 Oppenheim, A.V. and Schafer, R.W. Digital Signa/ Processing Prentice-Hall, Englewood Cliffs, NJ (1975) Gaunaurd, G.C. and Strifors, H.C. Frequency- and time-domain analysis of the transient resonance scattering resulting from the interaction of a sound pulse with submerged elastic shells IEEE Tram Ulirason Ferroelec Frey Control (1993) 40 3 13-324 Noel, B. (Ed) Ultra-widehand Radar: Proceedings of the First Los Alamos Swnposium CRC Press, Boca Raton, FL (1991) Faran, J.J. Sound scattering by solid cylinders and spheres J Acousr Sot Am (1951) 23 405-418 Hampton, L.D. and McKinney, C. Experimental study of the scattering of acoustic energy from solid metal spheres in water J Acoust Sot Am (1961) 33 664-673 Diercks, K.J. and Hickling, R. Echoes from aluminium spheres in water J Acoust Sot Am (1967) 41 38@393 Claasen, T.A.C.M. and Mechlenbriiuker, W.F.G. The Wigner distribution - A tool for time-frequency signal analysis Philips J Res(1980)35, Part I, 217-250; Part II, 276-300; Part III, 372-389 Cohen, L. Time-frequency distribution A review Proc IEEE (1989) 77 941-981 Kirkwood, J.G. and Bethe, H.A. Basic Propagation Theory OSRD Report No 588 (1942) Cole, R.H. .!Jnder~~~a/er E.~plosions Dover Publications, New York

(1965) 14

15 16

17

Acknowledgements The authors wish to thank the Independent Research Programs of their respective Institutions for support.

echoes: G. C. Gaunaurd and H. C. Strifors

18

Swisdak, Jr., M.M. (Ed) E.uplosion .Efects and Properlies: Part II Esplosion .Eficts in Water Naval Surface Warfare Center White Oak Laboratory, Tech. Report TR-76-116 (1978) (110 pp.) [Supercedes NOL-TR-65-2181. Also: NTIS AD A056694 Personal communication: Staff members of the Damage Assessment Branch, NSWCjWOL (R-14) Strifors, H.C. and Gaunaurd, G.C. Multipole character of the large-amplitude, low frequency resonances in the sonar echoes of submerged spherical shells Inr J So/ids Structures (1992) 29 121-130 Junger, M.C. and Feit, D. Sound, Structures, tmd Their Interaction MIT, Cambridge, MA, 2nd edn (1986) Strifors, H.C., Gaunaurd, G.C., Brusmark, B. and Abrahamson, S. Transient interactions of an EM pulse with a dielectric spherical shell IEEE Tram Antennas Propagut (1994) AP-42 453-462

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