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Backscattering of sound pulses by elastic bodies underwater
Applied Acoustics 33 (1991) 87-107
B a c k s c a t t e r i n g of S o u n d Pulses by Elastic Bodies Underwater
Donald Brill, a Guillermo Gaunaurd, b Hans Strifors c & William Wertman b ° US Naval Academy, Physics Department, Annapolis, Maryland 21402-5026, USA b N.S.W.C., White Oak Lab, (R-42), Silver Spring, Maryland 20903-5000, USA c Sv~edish Defense Research Establishment (FOA2), S10254 Stockholm, Sweden {Received 8 May 1990; revised version received 24 August 1990: accepted 24 September 1990)
ABSTRACT This paper studies the'scattering interaction of a n incident acoustic pulse and an elastic target. The pulse emerges from a distant transducer and can have any arbitrary shape and finite duration. The target is an elastic body. here assumed to be spherical and homogeneous. The Fourier integral representation of the incident pulse is combined with the resonance-scattering representation of the scattered pressurefield to yield afilter-type integral that can be viewed as filtering the form function of the scatterer through the spectral window of the incident transient pulse (ping). We anal.roe the backscattered pulses when the incident pulses are of three simple types. The analysis is carried out in the frequency and time domains, and results are illustrated with numerical predictions for a variety of instances of increasing complexity and interest. The nature of the backscattered echo is explained in two instances for either short or long pulses. These instances correspond to cases in which the incident pulse has a carrier frequency that either coincides with an)' o.f the natural resonances of the submerged sphere or not. The main advantage of short pulses is that the)" can be used to replicate the (stead)" state) sonar cross-section of scatterers. Ultimately, if the incident pulse were a delta function in r, the spectrum of the backscatteredpulse would exactly be the form function f~(n, x) divided by r. The backscattering sonar cross87 © 1991 US Government.
88
Donald Brill, Guillerrno Gaunaurd, Hans Strifors, William Wertman section ( B S C S ) would then he represented by !/~ (n. x)!'-. For a narrow incident continuous wat'e ( c.w. ) pulse in ~ (but not infinitely narrow as is the delta fimetion ), hating a carrier frequency x o, the replication o/" the B S C S would be accurate up to about that ralue o f x o. Long pulses excite single target resonances and produce echoes that are quite similar to those produced b t' c.w. incidences. This analysis can he analogously carried out f o r arbitrary pulses incident on targets o f more general shapes and lossv (i.e. t'iscoelastic) composition.
1 INTRODUCTION The last generation has seen a lot of study devoted to the resonant scattering of acoustic waves from elastic bodies.~ Most of this work has been for steadystate monochromatic plane waves incident upon an elastic target which scatters this energy into outgoing spherical (or cylindrical) waves in the far field. Targets have usually been either solids or shells which are spherical, cylindrical or more recently spheroidal in shape. ~ A form function is developed which describes the distribution of the scattered pressure field in a given direction in the far field as a function of frequency. If, for example, one examines the backscattered form function, certain characteristics will emerge. 2 As the frequency becomes small and we pass into the Rayleigh region where diffraction predominates, the form function vanishes, while at the other extreme where the frequency becomes very large, the form function varies about some constant geometrical acoustics value. For frequencies above the Rayleigh region the p h e n o m e n o n of resonance is observed and as a consequence sudden large variations in the form function occur.-" Here the incident waves pass both into and around the elastic body exhibiting resonances at a set of frequencies determined by the characteristics of the body and its environment, a These resonance frequencies carry the most explicit information about the nature of the target and therefore offer significant hope that a scheme will emerge whereby the target's identity can be made out when this set of characteristic frequencies is recognized. "~For the present discussion we consider the active insonification of a target at some great distance with a pulse of our own making. Hence, we know the spectral content of the illuminating plane-wave pulse and wish to determine how the structure of the scattered spherical wave is modified by the resonant response of the target. For the present discussion we shall assume the target to be a solid sphere immersed in water, realizing that the technique which emerges extends to the general case of an arbitrary elastic target in any liquid. This same approach can be extended to sound-absorbing (i.e. lossy or viscoelastic) bodies ~ in a straightforward manner.
Backscattering of sound pulses by elastic bodies underwater
89
2 THEORY
A. Scattering of steady-state incident waveforms A monochromatic plane pressure wave moving with the speed c in the + direction is incident on the south pole of an elastic sphere of radius a. This incident wave with the angular frequency co and propagation constant k has the representation: t'2"3 pine(r, O, t, 09) = Po ei(krc°s°-a'° __ Po e-i,ot ) ' , i"(2n + 1)/.(kr)P,(cos O) (1) n=O
where j.(kr) is the spherical Bessel function of order n and P.(cos 0) is the Legendre polynomial of order n. The outgoing scattered pressure wave can be given the form: 3 p~c(r, O, t, co) = Poe - i,o,
2
i"(2n + i) T.(x)h~. 1~(kr)P.(cos O)
(2)
n=O
where x =- ka = coa/c and h~.t~(kr) is the spherical Hankel function of order n. The total pressure is the superposition of these two pressures Pt = Pi.¢ + Ps¢ = P0 e - ~ " ) ' i"(2n + 1)[.L(kr) + T.( x)h~.l )(kr) ] P.(cos O) (3) /,,,,,,,,,,,,,d n=O
where the three boundary conditions at r = a (continuity of elastic normal displacement and of normal and shear stress) determine the coefficients T.(x) in the form R e d 11 dl2 T,(x)=-Red21 d22 0
dt3l-/ldlt d23[/I d21
d32
d33
0
dt2 d13 d22 d23 d32 d33
(4)
where the elements, di~, are combinations of the spherical Bessel functions evaluated at the sphere's surface and are given elsewhere.3 In the far field (r>>a) the asymptotic expansion 6 h~l~(kr) ~ ( _ i ) . ÷ i
eikr
kr
(5)
places the scattered pressure given by eqn (2) in the form psc(r, O, t, co) ~ P° ei(k'-'°'~:~ (0, x) r
(6)
90
Donald Brill. Guillermo Gaunaurd, Hans StriJors, William Wertman
The quantity f~(O,x) is referred to as the 'scattering pattern' or "angular distribution' at fixed frequencies, and is given by:
--f~.(O, x) a
(2n + 1)T~(x)P~(cos 0)
tx
(7)
n=O
Plots can be made of the modulus o f this function versus the angle 0 at some fixed non-dimensional frequency x or versus frequency x at some fixed observation direction 0. The acoustic backscattering (0 = 7t) cross-section (BSCS) is proportional to the square of the normalized 'form function', _f~(~z,x), viz.' ~rR = 4rtlJ~:(rc, x)[ 2 = 4~z lim r Ps¢ z
(8a)
Now substitute eqn (2) with 0 = rc forp~ c and eqn (1) with 0 = 0 for p.,¢, and finally use the expansion in eqn (5) as r---, vc to show that ( - l)"(2n + 1)T,(x) "
a• = 4re ~ n=0
which leads to the normalized (BSCS): a_._~B rca 2 =
2
a f~(r~'x)
2
2
=
~x~
( - 1)"(2n +
1)T.(x)
2
(8b)
n=O
This result gives the cross-section and form function for a m o n o c h r o m a t i c incident plane wave backscattered by an elastic sphere in a liquid. There are m a n y discussions o f this result 3 which give calculations and plots; however, the character of the scattering process is essentially described by the scattering coefficients, T.(x).
B. Scattering of pulsed incident waveforms When the incident waveform is a pulse o f finite duration, it must contain a superposition of waves with a spectrum of frequencies. We have already seen the far-field backscattering pressure for a unit amplitude m o n o c h r o m a t i c incident wave of frequency x, viz. O{3
ei(kr-~t)
p~c(r, re, t, co) - - - r
cl
f ~ (re, x) = ~r e.k~_o, ,_2 tx V ( - 1)"(2n + 1)T~(x) (9) L.a n=O
Backscattering o f sound pulses by elastic bodies underwater
91
For the far-field scattered pressure this can be viewed as a frequency response to the above incident unit amplitude pressure wave. If we are to synthesize a scattered pressure due to an incident superposition of waves, a Fourier transform is necessary for both the incident and scattered waveforms. To that end we choose the following conjugate variables for each of these waveforms: Incident waveform:
x - ka ~
Scattered waveform:
x =- ka e=~ r = (ct - r)/a
r z - (ct - r
cos O)/a
This choice makes - ixr correspond to i ( k r - cot) and - ixrz correspond to i(kr cos 0 - o~t). The associated Fourier transform pairs are (where we show for a scattered wave and replace it with r z when representing an incident wave): G(x)=
g(r)e"dr
and
g(r)=~-_
G(x)e-i"dx
(10a, b)
with Dirac delta function representation: =
2n6(x-xo)
e"X-x°~dr
(11)
We now use concepts from the T h e o r y of Linear Systems 7 and assume there is a transfer function,-H(x), which converts the Fourier transform o f the incident waveform, Pi.c(x), into the Fourier transform o f the scattered waveform, Psc(x), as shown by the following 'black box' illustration:
Pinc(X)~
Ps¢{x) )
A representation for H ( x ) is achieved using the case of a unit amplitude m o n o c h r o m a t i c incident wave with the non-dimensional frequency x o as follows: pi,¢(Xo, rz) = e-ixo~z= Pine(Xo. X) = 2n 6(x - Xo)
(12)
This produces the scattered wave: Ps¢(x o, x) = H(x)Pi.¢(Xo, x) = 2nil(x) 6(x - Xo) =p~¢(x o, 3) = H ( x o ) e -i~o~
(13) and comparison with eqn (9) shows that H ( x ) = f ~- ( n-, x ) r
= ra --
(-
lxr n=O
1)"(2n + 1 ) T . ( x )
(14)
92
DonaM BrilL Guillermo Gaunaurd, Hans Strifors, William Wertman
Now we can write the backscattered pressure in terms of the Fourier transform, P.~¢(x), of a general incident pulse of the form: P~'(~) = 2 x J _ ~
" = 2 ~ j _ ,:
which in our specific case expands to a Ps¢(r) = i2rrr
dx
( - 1)"(2n + 1)
-
x::
Tn(x)Pi.¢(x)e - ~ ~ - ) (
(16)
n=0
Thus, the far-field backscattered pressure can be evaluated from knowledge of: (a) the spectrum Pi.c(x) of the incident pulse, and (b) the transfer function of the scatterer, which is proportional to the form function f..(x, x) of the target. In what follows we shall replace P~.¢(x) with G(x) to remove subindices as we study the effect of incident pulses, g(r), with various spectra, and go on to write: rp.,¢(r) = 1__[ 2To j _ +~ ~ [f~(n,x)G(x)] e-,X~d~.
{15a)
This is an alternative form o f e q n (15), which leads to the same result in eqn (16) for an elastic sphere in water, specifically showing its inverse Fourier transform structure. It is possible to apply i n p u t / o u t p u t concepts from linear theory s to scattering problems. Define the Fourier transform pair: J~: 0i, x) =
_ ~ f.. 0i, r) e i~ dr
i ÷'-~. f~0i, x) e -'x* dx j~(fi, r) = ~1 .j_
(17a)
(17b)
valid in any direction r~; then eqn (15a) can be expressed alternatively in the equivalent convolution form: rpse(r ) =
g(r'-- v)f~ (re, r') dr'
(18)
which states that the scattered pulse in the r - d o m a i n is the convolution of the incident pulse g(r) and the scattered waveform for an impulsive source, also called the impulse response in the r-domain. These linear-systems concepts leading to the general eqn (15) were introduced into the (radar) scattering literature 8'9 in 1958, and these works presented useful approximations to generate estimates of the scattered pulses. In what follows, we have c o m p u t e d our results by exact Fourier synthesis, and we have not used the
Backscattering of sound pulses by elastic bodies underwater
93
mentioned approximations. For the case ofpi.¢ given by a monochromatic c.w. of non-dimensional frequency Xo, it is clear that substitution of P~n¢(x) = G(x) = 2npo 6(x - x o) into the transient solution, eqn(16), produces a steady state c.w. solution of the form given by eqns (6) and (7). Also, the case pi,c(r)=6(0 leads with the help of eqns (8a), (10) and (14) to the conclusion that the BSCS is proportional to the square of the scattered spectrum, IP~¢(x)[2. The form of eqn (15a) most suitable for numerical evaluations is ~E
rpso(~) =- )~ Q.(r) n= 0
(19a) ~:
where Q.('r) = ~-n _ [G(x)f.(n, x)] e-ix, dx
(19b)
and a
f.(n, x) = _ ( -
tx
1)"(2n + 1)T.(x)
(19c)
The T.(x) are the determinant ratios given in eqn (4), and we have further defined the expansion of the form function in partial waves,f~(rc, x), given by
f~(rc, x) =- ~ f~(r~,x)
(19d)
n=0
C. Some possible interrogating incident pulses Pulses of different shapes and durations will accomplish different goals. What pulse to use depends upon the application in question. The literature in high resolution radar ~0 has established the uses of hyperbolic frequencymodulated waveforms and linear FM pulses (i.e., chirps). Recent advances in active target-identification TM t2 schemes have shown the importance of the 'K-pulse'. Although all conceivable pulses can be used in the above formulation, the examples that follow will treat very simple instances as illustrations. We do this with the understanding that these pulses have very limited physical applications, even in idealized conditions. It is well known that long sinusoidal pulses (i.e. sinusoidal wave-trains of long but finite duration) have narrow spectra and will excite only a few resonances on any elastic target.~3 On the other hand, short sinusoidal c.w. pulses have broad spectra and, provided they have enough energy, they will excite many target
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
94
resonances. These are the ones of greatest interest in this study, since long pulses have spectra approaching the type typical of m o n o c h r o m a t i c (c.w.l steady state incidences, as shown in eqns (12). Consider now a selection of three often-used pulses and their Fourier transforms which shall be represented by: pi,c(rz) - g(r z) (i)
Pi,c(x) - G(x) =
g(rz)e ix'' dr z
(20)
A finite constant pulse of duration 3*: gl(rz)=
A, 0,
0<3z<3" elsewhere
(21)
has the spectrum: Gdx) = At* sin (x3"/2) e,X,./2 ~ (x3"/2)
(ii)
(22)
The main lobe of G~(x) is at the origin. The only way to move this lobe away from the origin is to give some frequency content to the incident pulse, i.e. make it sinusoidal rather than constant during the time interval r*. We can then place this lobe at selected frequencies. A cosinusoid with a small a m o u n t o f d a m p i n g (b): g,(rz) -
=)'Ae-b'zc°s(x°zz)' (
0,
rz > 0 rz<0
(23)
has the spectrum G2(x) = ,4 b(b2 + x2 + x2) + ix(b2 + x2 - x°2)
(24)
(b 2 + x g - - x 2 ) 2 -4- 4 b 2 x 2
This waveform starts at 3, = 0 and has infinite duration. The small d a m p i n g present makes the spectral peaks be o f finite rather than infinite amplitudes. Here b (the damping) is small relative to Xo, say 5% of x o. This makes the peaks at + x o high (but not of infinite magnitude) and narrow, and no other side-lobes are present. (iii) A sinusoid of finite duration r*; g3(3,)
J'sin (Xo3z), 0,
l
0 < 3z < 3" elsewhere
(25)
has N cycles, i.e. 3 * = 2 n N / x o. Its spectrum is 1 - e i(x+x°)~* G3(x) = 2(x + Xo)
1 - - e I{x-x°)t*
2(x - Xo)
(26)
Backscattering of sound pulses by elastic bodies underwater
95
The rectified part of this spectrum for positive frequencies is 3" sin ( ~ - ( x - x°))
(27)
(X- Xo) The above spectra can be found by direct evaluation of eqn (20), or with the help of tables. ~4 We henceforth drop the sub-index z in the variable r z in the expressions for the incident pulses, viz. eqns (21), (23) and (25).
3 N U M E R I C A L RESULTS The normalized backscattering cross-section (BSCS) as given by eqns (8) and (4) is plotted versus x in Fig. l(a). The plot is for a solid stainless-steel sphere ofmaterial properties: P2 = 7.9 g/cm 3, ca. = 5-78 x 105 cm/s and % = 3.09 x 105 cm/s, immersed in water of properties: p~ = 1.0g/cm 3, and c~ = 1-5 x 105 cm/s. The displayed band is wide enough (viz. 0 < x < 30) to contain many resonances. The (normalized) BSCSs of a rigid and a soft sphere are shown in the same band in Figs l(c) and l(d), respectively. These are found from eqn (8) with the coefficients T, now given by: 3
T.(x)=
L(x) h r(x)
T.(x)=
L(x)
)
(28)
respectively. The resonances are isolated in this elastic case by the subtraction of the rigid background, and they are shown in Fig. l(b). The resonances manifest themselves as peaks in this type of backgroundsuppressed plot, and 'valleys' denote regions away from resonances. We have marked the particular resonance Xo=8.1, and the valley or 'offresonance' position, .,Co= 9.0, by arrows. The spectrum of the rectangular pulse in eqn (21) is the familiar sinefunction in eqn (22), which need not be plotted. The spectrum of the Type-(ii) pulse in eqn (23) is given in Fig. 2. The peaks occur at x = + Xo, where Xo is the value of the carrier frequency of the damped cosinusoid. The peaks' amplitudes are inversely proportional to b, the damping constant of the pulse, here assumed to be small (viz. b = Xo/20). The spectrum of a Type-(iii) pulse is shown in schematic form in Fig. 3, in absolute value and for positive frequencies. The main lobe occurs at Xomthe carrier frequency of the c.w. pulse. The amplitude of the main lobe is r*/2, where 3" is the pulse duration (c.f. eqn (25)). The separation between
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
96
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Backscattering of sound pulses by elastic bodies underwater
97
IIG~.)I
•
Jlk
l J k.._
-~o
'
~o
'
X
Fig. 2. Spectrum of an incident Type-(ii} pulse with carrier frequency x o and damping constant b = Xo/20. This spectrum is complex and it is delayed here in modulus verus x. The peak occurs at x = .%. nulls is uniform and equal to xo/N, where N is the numbers ofcycles in the sinusoidal pulse. The following relation holds: r*x o = 2rcN. The effect o f using short (i.e. r* = 1) incident pulses is illustrated in Fig. 4 in the time and frequency domains. These results will later be contrasted with the ones for long pulses in Fig. 5. Short c.w. pulses with N ~ 1.3 cycles for the carrier frequencies Xo=8.10 (at resonance, Fig. 4(a)) and Xo=9-0 (off resonance, Fig. 4(b)) have spectra shown in Figs 4(c) and 4(d), respectively. The main lobes occur at the carrier frequency Xo in each case (cf. eqn (26)). Long incident c.w. pulses having N = 20 cycles are illustrated in Fig. 5. The carrier frequency is x o = 8-1 (at resonance, Fig. 5(a)) and x o = 9"0 (off resonance, Fig. 5(b}). The corresponding spectra (cf. eqn (26)) are illustrated in Figs 5(c) and 5(d). The main peaks occur at + x o with 2 ( N - 1) minor lobes between these two main peaks, and infinitely m a n y more for Ixl > Xo. The pulse duration, which can be read off the top graphs, is r* = 15.5 and 13-7, respectively. The study of the backscattered pulses is considerably more important and complicated. It can be accomplished, however, by means o f e q n (15), or more specifically, for this particular case, by eqn (19). I G~(x)l
' . . . . . .
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'. . . . . . .
. . . . . . ~. . . . . . r
' . . . . . .
7-~o
L . . . . . .
.
. . . . . . ::
t
XO
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Backscattering of sound pulses by elastic bodies underwater
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Figure 6(a) exhibits the pulse backscattered from the sphere when the incident pulse is of Type (i). The spectrum of each echo pulse is illustrated in Fig. 6(b). The pulse backscattered from the sphere when the incident pulse is a Type-(ii) damped cosinusoid of carrier frequency x o = 8.1 (at resonance), is shown in Fig. 6(c), and its corresponding spectrum in Fig. 6(d). Two features are noticeable in Fig. 6(c). There is a narrowing of the returned echo signals which occur between observable transients both before and after ringing begins. This narrow section has been explained elsewhere, ~3 and will be further discussed in Section 4. In addition, the amplitudes of the returned pulses decay in discrete steps. This indicates that a target resonance has been excited, t3 which is indeed the case for .,co = 8'1 (cf. Fig. l(b)). Note that the spectrum in Fig. 6(d) has peaks at these frequencies, + Xo, with a splitting of the peaks due to the double transient nature of the time-response at resonance illustrated in Fig. 6(c). The backscattered response of the sphere to a long Type-(iii) pulse such as the one in Fig. 5(a) is shown in Fig. 7(a). The narrowing of the echo in the region in between the two transients is clearly observable. The second transient '3 begins to appear at approximately z - r* (--- 15.5), and then the 'ringing' starts, again with an amplitude decay that decreases in discrete steps. The spectrum in Fig. 7(c) exhibits the split of the main lobe as in Fig. 6(d), and the lobe structure is quite non-uniform in contrast to that in Fig. 5(c) for the long incident pulse. At an off-resonance frequency (viz. -'co= 9"0), Fig. 7(b) shows the backscattered pulse without any narrowing of the middle portion, and with a decay after r = r* that is pretty much exponential in nature. Its spectrum in Fig. 7(d) is substantially more uniform than in Fig. 7(c), and it resembles the spectrum of the incident pulse in Fig. 5(d) much more closely. The main lobes in Fig. 7(d) do not exhibit any splitting. Prior to r = r* the backscattered pressure pulse in Fig. 7(b) is pretty much a replica of the incident pulse. All these are precisely the features one should expect when the incident pulse has a carrier frequency that does n o t coincide with any target resonance. As we mentioned earlier, and according to Ref. 13, the behavior of the long pulse is quite similar to that of a steady-state monochromatic plane c.w. excitation. We finally arrive at the case of the backscattered pressure pulse when the incident pulse was of Type (iii) as shown in Fig. 4(a). The echo is displayed in Fig. 8(a) for -'co= 8"1 (at resonance). This short incident pulse generates a quite long backscattered pulse that goes on far past r = r* ( ~ 1). This time response has the spectrum shown in Fig. 8(b). The main advantage of this spectrum is that its square replicates the backscattering cross-section of the target as shown in Fig. l(a). This is to be expected, because any short pulse will behave as a delta function and will imprint the cross-sectional features of Fig. l(a) into the spectral response squared (see remarks just before eqn
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(19)). This is indeed the case, and the various features are indicated in Fig. 8(b) up to x --. 14. It follows that an easy way to obtain the BSCS o f any target is to study its response to a short incident c.w. pulse. When the carrier frequency, x 0, o f the incident pulse does not coincide with the resonance of the elastic target, then the process by which the spectrum squared o f the backscattered pressure pulse replicates the BSCS, which is optimum at resonance, starts to degrade giving a distorted cross-section. The farther Xo is from a resonance, the greater the distortion will be. There is a limit to this degradation because as Xo starts to separate from any given resonance it also begins to get close to the next one.
104
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
Although the use of pulses as interrogating waveforms in sonar systems is very c o m m o n in practice, the detailed theoretical analysis of echo responses from elastic targets in water insonified by transients is very limited. Some studies of this interaction which have used eqn (15) to analyze the returns in various ways have appeared in the radar literature, 9t5"~6 as well as that of acoustics.t" Most of the calculations shown here requiring a Fast Fourier Transform (FFT) package have been performed using the listings given in a recent monograph. ~s 4 DISCUSSION, I N T E R P R E T A T I O N A N D C O N C L U S I O N S We have shown that the elastic form-function of a scatterer (c.f. eqn (7)) divided by the range to the observer acts as a transfer function which transforms the spectrum of the incident pressure pulse into that of the farfield scattered pressure (c.f. eqn (14)). Short sinusoidal c.w. pulses have wide spectra, and provided that they are energetic enough, they will excite many resonances in the elastic target. This becomes manifest by comparing the spectrum of the incident pulse (c.f. Fig. 4(c)) with that of the backscattered pressure pulse (c.f. Fig. 8(bt). The main advantage of using short pulses as interrogating waveforms is that the spectra of the backscattered returns they generate replicate the BSCS, or sonar cross-section, of the target in the steady-state case. The use of short pulses is thus an easy way to obtain BSCSs. Long sinusoidal pulses have narrow spectra and hardly excite the target resonances which do not lie near their carrier frequency. When the carrier frequency of the incident pulse coincides with a target resonance, large changes become evident in the central maximum of the scattered pulse's spectrum. Furthermore, there is an observable narrowing of the initial portion of the backscattered echo in the (non-dimensional) time-domain. This narrowing occurs between the t w o transients 13 that develop in this case (c.f. Fig. 7(a)). Furthermore, the decrease in pressure amplitude that follows the second transient, once the ringing starts, occurs in discrete steps or jumps, as we have repeatedly indicated (c.f. Fig. 7(a): more on this below). On the other hand, when the carrier frequency of the pulse lies away from any of the target's natural resonances, the central maximum in the incident pulse's spectrum is communicated to the spectrum of the backscattered pressure pulse with little or no change in shape. In this case, no narrowing of the response in the time domain is then produced or observed (c.f. Fig. 7(b)). Furthermore, the amplitude decay in the final part of the response does not exhibit the stepwise decrease we discussed before at resonance. Whether at resonance or offresonance, the smaller side-lobes present at either side of the carrier frequency Xo in the backscattered spectrum (c.f. Figs 7(c) or 7(d)) are
Backscattering of sound pulses by elastic bodies underwater
105
distorted by the presence of target resonances. This discrepancy with the incident pulse spectra (c.f. Figs 5(a) and 5(b)) is more marked when x 0 coincides with a target resonance than when it does not. This difference-however small--between the incident and backscattered spectra (c.f. Figs 5(c) and 7(c)) is what accounts for the 'ringing' of the scattered pressure pulse in the time (3) domain. It should be pointed out that, since the sphere is elastic, there are portions of the incident wave that penetrate into its interior and become reflected from its north pole (the antipode of the point of incidence in the present arrangement). Other portions of the internally transmitted wave circumnavigate the sphere at selected frequencies arising from the excitation of the elastic Rayleigh (R) and whispering gallery (WG) resonances. Concurrently, there are circumnavigating waves in the outer fluid that are called 'creeping waves'. These (external) creeping waves produce relatively small effects in the bands of the BSCS that we have displayed in our plots. We have stated, and shown in the computed plots, that the backscattered pulse usually decreases its amplitude in discrete steps when the incident c.w. pulse has carrier frequency coinciding with any of the target resonances. An explanation for this behavior can be given in terms of a phase-matching argument 1. t q-21 that has been repeatedly given for the single surface wave that circumnavigates the target and adds in-phase after each circumnavigation, producing the resulting ringing. This phase-matching idea has been explained in various other works 2°'2t dealing with impenetrable and penetrable scatters of various geometries. It follows from Ref. 21 and the above discussion that the group velocities cl~ of the induced surface waves-normalized to the sound speed c~ in the outer fluid--can be obtained from the form-function plot in Fig. l(a) by just reading the difference in abcissa Ax between consecutive resonance features (dips) caused in this case by the Rayleigh wave, viz. 2t
Clg) --
= Ax
(29)
Ct
This value is read to be Ax ~ 2,4. Since the basic sonar/radar uncertainty relation is: t° ArAx--27r it is possible to estimate A~. The value is A t = 27:/2.4 ~ 2-6. This is the value observed in Fig. 7(a) for the (horizontal) duration of the discontinuous step, in the case of a long c.w. pulse. As we see in Fig. 7(b), this behavior is absent away from resonance. It is also absent for short pulses as can be confirmed in Fig. 8. We conclude by stating that a transient resonance scattering methodology has emerged from this analysis that can be used to study the scattering of any pulse of known spectrum from a general elastic target of known form function in the steady-state case.
106
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
REFERENCES 1. Gaunaurd, G., Elastic and acoustic resonance wave scattering. Appl. Mech. Rev., 42 (1989) 143-92, ASME Book No. AMR056. 2. Brill, D., Gaunaurd, G. & Ayres, V., The influence of natural resonances on -scattering and radiation processes. J. Wash. Acad. Sci., 77 (1987) 55-65. 3. Brill, D. & Gaunaurd, G. Resonance theory of elastic waves ultrasonically scattered from an elastic sphere. J. Acoust. Soc. Am., 81 (1987) 1-21 (Invited) (and references therein). 4. Gaunaurd, G., Techniques for sonar target-identification. IEEEJ. Ocean. Engr., OE-I 2 (1987) 419-22 (Special issue on Scattering); also ibid, Inverse scattering techniques for material characterization. Proc. Ultrasonics International '87. Butterworth Scientific Ltd, Surrey, UK, 1987, pp. 520-5. 5. Strifors, H. & Gaunaurd, G., Wave propagation in isotropic linear viscoelastic media. J. Acoust. Soc. Am., 85 (1989) 995-1004. 6. Jackson, J., Classical Electrodynamics, Wiley, New York, 1966, p. 540. 7. Gardner, M. F. & Barnes, J. L., Transients in Linear Systems (Vols. 1 and 2). John Wiley & Sons, New York, 1942. 8. Kennaugh, E. M. & Cosgriff, R. L., The use of impulse response in electromagnetic scattering problems. 1958 IRE National Conv. Rec., Part 1, pp. 72-7. 9. Kennaugh, E. M. & Moffatt, D. L., Transient and impulse response approximations. Proc. IEEE, 53 (1965) 893-901, and references therein. 10. Rihaczek, A. W., Principles of High-Resolution Radar. McGraw-Hill Book Co., New York, 1969. 11. Kennaugh, E. M., The K-pulse concept. IEEE Trans. Antennas & Prop., AP-29 (1981) 327-31. 12. Turhan-Sayan, G. & Moffatt, D. L., K-pulse estimation and target identification of low-Q radar targets. Wave Motion, 11 (1989) 453-61, and references therein. 13. Gaunaurd, G. & Tsui, C., Transient and steady-state target resonance excitation by sound scattering. Appl. Acoust., 23 (1988) 121--40. 14. Champency, D. C., Fourier Transforms and Their Physical Applications. Academic Press, New York, 1973. 15. Ikuno, H. & Felsen, L. B., Complex rays in transient scattering from smooth targets with inflection points. IEEE Trans., AP-36 (1988) 1272-80. 16. Moffatt, D. L., Young, J. D., Ksienski, A A., Lin, H. C. & Rhoads, C. M., Transient response characteristics in identification and imaging. IEEE Trans., AP-29 (1981) 192-205. 17. See, for example, Veksler, N., Information Analysis in Hydroelasticity. Academy of Sciences of the Estonian Soviet Socialist Republic, Valgus Publishing House, Tallinn, 1982 (in Russian); also, Rudgers, A., Acoustic pulses scattered by a rigid sphere immersed in a fluid. J. Aeoust. Soc. Am., 45 (1969) 900-10; also, Howell, W., Numrich, S. & Ueberall, H., Complex frequency poles of the acoustic scattering amplitude and their ringing. IEEE Trans., UFFC-34 (1987) 22-7. 18. Press, W., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T., Numerical Recipes, Ch. 12. Cambridge University Press, Cambridge, UK, 1986, pp. 390--429. 19. Werby, M. & Tango, G., Numerical study of material properties of submerged elastic objects using resonance response. J. Acoust. Soe. Am., 79 (1986) 1260-8.
Backscattering of sound pulses by elastic bodies underwater
107
20. Gaunaurd, G., Resonance acoustic scattering from underwater elastic bodies, in Elastic War'e Propagation, ed. M. F. McCarthy & M. A. Hayes. Elsevier Science Publishers, North Holland, 1989, pp. 335-46. 21. Gaunaurd, G. & Werby, M. F., Resonance response of submerged, acoustically excited thick and thin shells. J. Acoust. Soc. Am., 77 (1985) 2081-93, eqn (27).
B a c k s c a t t e r i n g of S o u n d Pulses by Elastic Bodies Underwater
Donald Brill, a Guillermo Gaunaurd, b Hans Strifors c & William Wertman b ° US Naval Academy, Physics Department, Annapolis, Maryland 21402-5026, USA b N.S.W.C., White Oak Lab, (R-42), Silver Spring, Maryland 20903-5000, USA c Sv~edish Defense Research Establishment (FOA2), S10254 Stockholm, Sweden {Received 8 May 1990; revised version received 24 August 1990: accepted 24 September 1990)
ABSTRACT This paper studies the'scattering interaction of a n incident acoustic pulse and an elastic target. The pulse emerges from a distant transducer and can have any arbitrary shape and finite duration. The target is an elastic body. here assumed to be spherical and homogeneous. The Fourier integral representation of the incident pulse is combined with the resonance-scattering representation of the scattered pressurefield to yield afilter-type integral that can be viewed as filtering the form function of the scatterer through the spectral window of the incident transient pulse (ping). We anal.roe the backscattered pulses when the incident pulses are of three simple types. The analysis is carried out in the frequency and time domains, and results are illustrated with numerical predictions for a variety of instances of increasing complexity and interest. The nature of the backscattered echo is explained in two instances for either short or long pulses. These instances correspond to cases in which the incident pulse has a carrier frequency that either coincides with an)' o.f the natural resonances of the submerged sphere or not. The main advantage of short pulses is that the)" can be used to replicate the (stead)" state) sonar cross-section of scatterers. Ultimately, if the incident pulse were a delta function in r, the spectrum of the backscatteredpulse would exactly be the form function f~(n, x) divided by r. The backscattering sonar cross87 © 1991 US Government.
88
Donald Brill, Guillerrno Gaunaurd, Hans Strifors, William Wertman section ( B S C S ) would then he represented by !/~ (n. x)!'-. For a narrow incident continuous wat'e ( c.w. ) pulse in ~ (but not infinitely narrow as is the delta fimetion ), hating a carrier frequency x o, the replication o/" the B S C S would be accurate up to about that ralue o f x o. Long pulses excite single target resonances and produce echoes that are quite similar to those produced b t' c.w. incidences. This analysis can he analogously carried out f o r arbitrary pulses incident on targets o f more general shapes and lossv (i.e. t'iscoelastic) composition.
1 INTRODUCTION The last generation has seen a lot of study devoted to the resonant scattering of acoustic waves from elastic bodies.~ Most of this work has been for steadystate monochromatic plane waves incident upon an elastic target which scatters this energy into outgoing spherical (or cylindrical) waves in the far field. Targets have usually been either solids or shells which are spherical, cylindrical or more recently spheroidal in shape. ~ A form function is developed which describes the distribution of the scattered pressure field in a given direction in the far field as a function of frequency. If, for example, one examines the backscattered form function, certain characteristics will emerge. 2 As the frequency becomes small and we pass into the Rayleigh region where diffraction predominates, the form function vanishes, while at the other extreme where the frequency becomes very large, the form function varies about some constant geometrical acoustics value. For frequencies above the Rayleigh region the p h e n o m e n o n of resonance is observed and as a consequence sudden large variations in the form function occur.-" Here the incident waves pass both into and around the elastic body exhibiting resonances at a set of frequencies determined by the characteristics of the body and its environment, a These resonance frequencies carry the most explicit information about the nature of the target and therefore offer significant hope that a scheme will emerge whereby the target's identity can be made out when this set of characteristic frequencies is recognized. "~For the present discussion we consider the active insonification of a target at some great distance with a pulse of our own making. Hence, we know the spectral content of the illuminating plane-wave pulse and wish to determine how the structure of the scattered spherical wave is modified by the resonant response of the target. For the present discussion we shall assume the target to be a solid sphere immersed in water, realizing that the technique which emerges extends to the general case of an arbitrary elastic target in any liquid. This same approach can be extended to sound-absorbing (i.e. lossy or viscoelastic) bodies ~ in a straightforward manner.
Backscattering of sound pulses by elastic bodies underwater
89
2 THEORY
A. Scattering of steady-state incident waveforms A monochromatic plane pressure wave moving with the speed c in the + direction is incident on the south pole of an elastic sphere of radius a. This incident wave with the angular frequency co and propagation constant k has the representation: t'2"3 pine(r, O, t, 09) = Po ei(krc°s°-a'° __ Po e-i,ot ) ' , i"(2n + 1)/.(kr)P,(cos O) (1) n=O
where j.(kr) is the spherical Bessel function of order n and P.(cos 0) is the Legendre polynomial of order n. The outgoing scattered pressure wave can be given the form: 3 p~c(r, O, t, co) = Poe - i,o,
2
i"(2n + i) T.(x)h~. 1~(kr)P.(cos O)
(2)
n=O
where x =- ka = coa/c and h~.t~(kr) is the spherical Hankel function of order n. The total pressure is the superposition of these two pressures Pt = Pi.¢ + Ps¢ = P0 e - ~ " ) ' i"(2n + 1)[.L(kr) + T.( x)h~.l )(kr) ] P.(cos O) (3) /,,,,,,,,,,,,,d n=O
where the three boundary conditions at r = a (continuity of elastic normal displacement and of normal and shear stress) determine the coefficients T.(x) in the form R e d 11 dl2 T,(x)=-Red21 d22 0
dt3l-/ldlt d23[/I d21
d32
d33
0
dt2 d13 d22 d23 d32 d33
(4)
where the elements, di~, are combinations of the spherical Bessel functions evaluated at the sphere's surface and are given elsewhere.3 In the far field (r>>a) the asymptotic expansion 6 h~l~(kr) ~ ( _ i ) . ÷ i
eikr
kr
(5)
places the scattered pressure given by eqn (2) in the form psc(r, O, t, co) ~ P° ei(k'-'°'~:~ (0, x) r
(6)
90
Donald Brill. Guillermo Gaunaurd, Hans StriJors, William Wertman
The quantity f~(O,x) is referred to as the 'scattering pattern' or "angular distribution' at fixed frequencies, and is given by:
--f~.(O, x) a
(2n + 1)T~(x)P~(cos 0)
tx
(7)
n=O
Plots can be made of the modulus o f this function versus the angle 0 at some fixed non-dimensional frequency x or versus frequency x at some fixed observation direction 0. The acoustic backscattering (0 = 7t) cross-section (BSCS) is proportional to the square of the normalized 'form function', _f~(~z,x), viz.' ~rR = 4rtlJ~:(rc, x)[ 2 = 4~z lim r Ps¢ z
(8a)
Now substitute eqn (2) with 0 = rc forp~ c and eqn (1) with 0 = 0 for p.,¢, and finally use the expansion in eqn (5) as r---, vc to show that ( - l)"(2n + 1)T,(x) "
a• = 4re ~ n=0
which leads to the normalized (BSCS): a_._~B rca 2 =
2
a f~(r~'x)
2
2
=
~x~
( - 1)"(2n +
1)T.(x)
2
(8b)
n=O
This result gives the cross-section and form function for a m o n o c h r o m a t i c incident plane wave backscattered by an elastic sphere in a liquid. There are m a n y discussions o f this result 3 which give calculations and plots; however, the character of the scattering process is essentially described by the scattering coefficients, T.(x).
B. Scattering of pulsed incident waveforms When the incident waveform is a pulse o f finite duration, it must contain a superposition of waves with a spectrum of frequencies. We have already seen the far-field backscattering pressure for a unit amplitude m o n o c h r o m a t i c incident wave of frequency x, viz. O{3
ei(kr-~t)
p~c(r, re, t, co) - - - r
cl
f ~ (re, x) = ~r e.k~_o, ,_2 tx V ( - 1)"(2n + 1)T~(x) (9) L.a n=O
Backscattering o f sound pulses by elastic bodies underwater
91
For the far-field scattered pressure this can be viewed as a frequency response to the above incident unit amplitude pressure wave. If we are to synthesize a scattered pressure due to an incident superposition of waves, a Fourier transform is necessary for both the incident and scattered waveforms. To that end we choose the following conjugate variables for each of these waveforms: Incident waveform:
x - ka ~
Scattered waveform:
x =- ka e=~ r = (ct - r)/a
r z - (ct - r
cos O)/a
This choice makes - ixr correspond to i ( k r - cot) and - ixrz correspond to i(kr cos 0 - o~t). The associated Fourier transform pairs are (where we show for a scattered wave and replace it with r z when representing an incident wave): G(x)=
g(r)e"dr
and
g(r)=~-_
G(x)e-i"dx
(10a, b)
with Dirac delta function representation: =
2n6(x-xo)
e"X-x°~dr
(11)
We now use concepts from the T h e o r y of Linear Systems 7 and assume there is a transfer function,-H(x), which converts the Fourier transform o f the incident waveform, Pi.c(x), into the Fourier transform o f the scattered waveform, Psc(x), as shown by the following 'black box' illustration:
Pinc(X)~
Ps¢{x) )
A representation for H ( x ) is achieved using the case of a unit amplitude m o n o c h r o m a t i c incident wave with the non-dimensional frequency x o as follows: pi,¢(Xo, rz) = e-ixo~z= Pine(Xo. X) = 2n 6(x - Xo)
(12)
This produces the scattered wave: Ps¢(x o, x) = H(x)Pi.¢(Xo, x) = 2nil(x) 6(x - Xo) =p~¢(x o, 3) = H ( x o ) e -i~o~
(13) and comparison with eqn (9) shows that H ( x ) = f ~- ( n-, x ) r
= ra --
(-
lxr n=O
1)"(2n + 1 ) T . ( x )
(14)
92
DonaM BrilL Guillermo Gaunaurd, Hans Strifors, William Wertman
Now we can write the backscattered pressure in terms of the Fourier transform, P.~¢(x), of a general incident pulse of the form: P~'(~) = 2 x J _ ~
" = 2 ~ j _ ,:
which in our specific case expands to a Ps¢(r) = i2rrr
dx
( - 1)"(2n + 1)
-
x::
Tn(x)Pi.¢(x)e - ~ ~ - ) (
(16)
n=0
Thus, the far-field backscattered pressure can be evaluated from knowledge of: (a) the spectrum Pi.c(x) of the incident pulse, and (b) the transfer function of the scatterer, which is proportional to the form function f..(x, x) of the target. In what follows we shall replace P~.¢(x) with G(x) to remove subindices as we study the effect of incident pulses, g(r), with various spectra, and go on to write: rp.,¢(r) = 1__[ 2To j _ +~ ~ [f~(n,x)G(x)] e-,X~d~.
{15a)
This is an alternative form o f e q n (15), which leads to the same result in eqn (16) for an elastic sphere in water, specifically showing its inverse Fourier transform structure. It is possible to apply i n p u t / o u t p u t concepts from linear theory s to scattering problems. Define the Fourier transform pair: J~: 0i, x) =
_ ~ f.. 0i, r) e i~ dr
i ÷'-~. f~0i, x) e -'x* dx j~(fi, r) = ~1 .j_
(17a)
(17b)
valid in any direction r~; then eqn (15a) can be expressed alternatively in the equivalent convolution form: rpse(r ) =
g(r'-- v)f~ (re, r') dr'
(18)
which states that the scattered pulse in the r - d o m a i n is the convolution of the incident pulse g(r) and the scattered waveform for an impulsive source, also called the impulse response in the r-domain. These linear-systems concepts leading to the general eqn (15) were introduced into the (radar) scattering literature 8'9 in 1958, and these works presented useful approximations to generate estimates of the scattered pulses. In what follows, we have c o m p u t e d our results by exact Fourier synthesis, and we have not used the
Backscattering of sound pulses by elastic bodies underwater
93
mentioned approximations. For the case ofpi.¢ given by a monochromatic c.w. of non-dimensional frequency Xo, it is clear that substitution of P~n¢(x) = G(x) = 2npo 6(x - x o) into the transient solution, eqn(16), produces a steady state c.w. solution of the form given by eqns (6) and (7). Also, the case pi,c(r)=6(0 leads with the help of eqns (8a), (10) and (14) to the conclusion that the BSCS is proportional to the square of the scattered spectrum, IP~¢(x)[2. The form of eqn (15a) most suitable for numerical evaluations is ~E
rpso(~) =- )~ Q.(r) n= 0
(19a) ~:
where Q.('r) = ~-n _ [G(x)f.(n, x)] e-ix, dx
(19b)
and a
f.(n, x) = _ ( -
tx
1)"(2n + 1)T.(x)
(19c)
The T.(x) are the determinant ratios given in eqn (4), and we have further defined the expansion of the form function in partial waves,f~(rc, x), given by
f~(rc, x) =- ~ f~(r~,x)
(19d)
n=0
C. Some possible interrogating incident pulses Pulses of different shapes and durations will accomplish different goals. What pulse to use depends upon the application in question. The literature in high resolution radar ~0 has established the uses of hyperbolic frequencymodulated waveforms and linear FM pulses (i.e., chirps). Recent advances in active target-identification TM t2 schemes have shown the importance of the 'K-pulse'. Although all conceivable pulses can be used in the above formulation, the examples that follow will treat very simple instances as illustrations. We do this with the understanding that these pulses have very limited physical applications, even in idealized conditions. It is well known that long sinusoidal pulses (i.e. sinusoidal wave-trains of long but finite duration) have narrow spectra and will excite only a few resonances on any elastic target.~3 On the other hand, short sinusoidal c.w. pulses have broad spectra and, provided they have enough energy, they will excite many target
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
94
resonances. These are the ones of greatest interest in this study, since long pulses have spectra approaching the type typical of m o n o c h r o m a t i c (c.w.l steady state incidences, as shown in eqns (12). Consider now a selection of three often-used pulses and their Fourier transforms which shall be represented by: pi,c(rz) - g(r z) (i)
Pi,c(x) - G(x) =
g(rz)e ix'' dr z
(20)
A finite constant pulse of duration 3*: gl(rz)=
A, 0,
0<3z<3" elsewhere
(21)
has the spectrum: Gdx) = At* sin (x3"/2) e,X,./2 ~ (x3"/2)
(ii)
(22)
The main lobe of G~(x) is at the origin. The only way to move this lobe away from the origin is to give some frequency content to the incident pulse, i.e. make it sinusoidal rather than constant during the time interval r*. We can then place this lobe at selected frequencies. A cosinusoid with a small a m o u n t o f d a m p i n g (b): g,(rz) -
=)'Ae-b'zc°s(x°zz)' (
0,
rz > 0 rz<0
(23)
has the spectrum G2(x) = ,4 b(b2 + x2 + x2) + ix(b2 + x2 - x°2)
(24)
(b 2 + x g - - x 2 ) 2 -4- 4 b 2 x 2
This waveform starts at 3, = 0 and has infinite duration. The small d a m p i n g present makes the spectral peaks be o f finite rather than infinite amplitudes. Here b (the damping) is small relative to Xo, say 5% of x o. This makes the peaks at + x o high (but not of infinite magnitude) and narrow, and no other side-lobes are present. (iii) A sinusoid of finite duration r*; g3(3,)
J'sin (Xo3z), 0,
l
0 < 3z < 3" elsewhere
(25)
has N cycles, i.e. 3 * = 2 n N / x o. Its spectrum is 1 - e i(x+x°)~* G3(x) = 2(x + Xo)
1 - - e I{x-x°)t*
2(x - Xo)
(26)
Backscattering of sound pulses by elastic bodies underwater
95
The rectified part of this spectrum for positive frequencies is 3" sin ( ~ - ( x - x°))
(27)
(X- Xo) The above spectra can be found by direct evaluation of eqn (20), or with the help of tables. ~4 We henceforth drop the sub-index z in the variable r z in the expressions for the incident pulses, viz. eqns (21), (23) and (25).
3 N U M E R I C A L RESULTS The normalized backscattering cross-section (BSCS) as given by eqns (8) and (4) is plotted versus x in Fig. l(a). The plot is for a solid stainless-steel sphere ofmaterial properties: P2 = 7.9 g/cm 3, ca. = 5-78 x 105 cm/s and % = 3.09 x 105 cm/s, immersed in water of properties: p~ = 1.0g/cm 3, and c~ = 1-5 x 105 cm/s. The displayed band is wide enough (viz. 0 < x < 30) to contain many resonances. The (normalized) BSCSs of a rigid and a soft sphere are shown in the same band in Figs l(c) and l(d), respectively. These are found from eqn (8) with the coefficients T, now given by: 3
T.(x)=
L(x) h r(x)
T.(x)=
L(x)
)
(28)
respectively. The resonances are isolated in this elastic case by the subtraction of the rigid background, and they are shown in Fig. l(b). The resonances manifest themselves as peaks in this type of backgroundsuppressed plot, and 'valleys' denote regions away from resonances. We have marked the particular resonance Xo=8.1, and the valley or 'offresonance' position, .,Co= 9.0, by arrows. The spectrum of the rectangular pulse in eqn (21) is the familiar sinefunction in eqn (22), which need not be plotted. The spectrum of the Type-(ii) pulse in eqn (23) is given in Fig. 2. The peaks occur at x = + Xo, where Xo is the value of the carrier frequency of the damped cosinusoid. The peaks' amplitudes are inversely proportional to b, the damping constant of the pulse, here assumed to be small (viz. b = Xo/20). The spectrum of a Type-(iii) pulse is shown in schematic form in Fig. 3, in absolute value and for positive frequencies. The main lobe occurs at Xomthe carrier frequency of the c.w. pulse. The amplitude of the main lobe is r*/2, where 3" is the pulse duration (c.f. eqn (25)). The separation between
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
96
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Backscattering of sound pulses by elastic bodies underwater
97
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Backscattering of sound pulses by elastic bodies underwater
I01
Figure 6(a) exhibits the pulse backscattered from the sphere when the incident pulse is of Type (i). The spectrum of each echo pulse is illustrated in Fig. 6(b). The pulse backscattered from the sphere when the incident pulse is a Type-(ii) damped cosinusoid of carrier frequency x o = 8.1 (at resonance), is shown in Fig. 6(c), and its corresponding spectrum in Fig. 6(d). Two features are noticeable in Fig. 6(c). There is a narrowing of the returned echo signals which occur between observable transients both before and after ringing begins. This narrow section has been explained elsewhere, ~3 and will be further discussed in Section 4. In addition, the amplitudes of the returned pulses decay in discrete steps. This indicates that a target resonance has been excited, t3 which is indeed the case for .,co = 8'1 (cf. Fig. l(b)). Note that the spectrum in Fig. 6(d) has peaks at these frequencies, + Xo, with a splitting of the peaks due to the double transient nature of the time-response at resonance illustrated in Fig. 6(c). The backscattered response of the sphere to a long Type-(iii) pulse such as the one in Fig. 5(a) is shown in Fig. 7(a). The narrowing of the echo in the region in between the two transients is clearly observable. The second transient '3 begins to appear at approximately z - r* (--- 15.5), and then the 'ringing' starts, again with an amplitude decay that decreases in discrete steps. The spectrum in Fig. 7(c) exhibits the split of the main lobe as in Fig. 6(d), and the lobe structure is quite non-uniform in contrast to that in Fig. 5(c) for the long incident pulse. At an off-resonance frequency (viz. -'co= 9"0), Fig. 7(b) shows the backscattered pulse without any narrowing of the middle portion, and with a decay after r = r* that is pretty much exponential in nature. Its spectrum in Fig. 7(d) is substantially more uniform than in Fig. 7(c), and it resembles the spectrum of the incident pulse in Fig. 5(d) much more closely. The main lobes in Fig. 7(d) do not exhibit any splitting. Prior to r = r* the backscattered pressure pulse in Fig. 7(b) is pretty much a replica of the incident pulse. All these are precisely the features one should expect when the incident pulse has a carrier frequency that does n o t coincide with any target resonance. As we mentioned earlier, and according to Ref. 13, the behavior of the long pulse is quite similar to that of a steady-state monochromatic plane c.w. excitation. We finally arrive at the case of the backscattered pressure pulse when the incident pulse was of Type (iii) as shown in Fig. 4(a). The echo is displayed in Fig. 8(a) for -'co= 8"1 (at resonance). This short incident pulse generates a quite long backscattered pulse that goes on far past r = r* ( ~ 1). This time response has the spectrum shown in Fig. 8(b). The main advantage of this spectrum is that its square replicates the backscattering cross-section of the target as shown in Fig. l(a). This is to be expected, because any short pulse will behave as a delta function and will imprint the cross-sectional features of Fig. l(a) into the spectral response squared (see remarks just before eqn
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Fig. Backscattered pressure pulses resulting when a short incident pulse of carrier frequency Xo = 8.1---coinciding with a target resonance--hits the sphere. The tail-bursts in the z-domain (a) do not decrease in noticeable discrete steps. The spectrum (b), shows the replication of the BSCS shown in Fig. l(a), up to x~-x~.
(19)). This is indeed the case, and the various features are indicated in Fig. 8(b) up to x --. 14. It follows that an easy way to obtain the BSCS o f any target is to study its response to a short incident c.w. pulse. When the carrier frequency, x 0, o f the incident pulse does not coincide with the resonance of the elastic target, then the process by which the spectrum squared o f the backscattered pressure pulse replicates the BSCS, which is optimum at resonance, starts to degrade giving a distorted cross-section. The farther Xo is from a resonance, the greater the distortion will be. There is a limit to this degradation because as Xo starts to separate from any given resonance it also begins to get close to the next one.
104
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
Although the use of pulses as interrogating waveforms in sonar systems is very c o m m o n in practice, the detailed theoretical analysis of echo responses from elastic targets in water insonified by transients is very limited. Some studies of this interaction which have used eqn (15) to analyze the returns in various ways have appeared in the radar literature, 9t5"~6 as well as that of acoustics.t" Most of the calculations shown here requiring a Fast Fourier Transform (FFT) package have been performed using the listings given in a recent monograph. ~s 4 DISCUSSION, I N T E R P R E T A T I O N A N D C O N C L U S I O N S We have shown that the elastic form-function of a scatterer (c.f. eqn (7)) divided by the range to the observer acts as a transfer function which transforms the spectrum of the incident pressure pulse into that of the farfield scattered pressure (c.f. eqn (14)). Short sinusoidal c.w. pulses have wide spectra, and provided that they are energetic enough, they will excite many resonances in the elastic target. This becomes manifest by comparing the spectrum of the incident pulse (c.f. Fig. 4(c)) with that of the backscattered pressure pulse (c.f. Fig. 8(bt). The main advantage of using short pulses as interrogating waveforms is that the spectra of the backscattered returns they generate replicate the BSCS, or sonar cross-section, of the target in the steady-state case. The use of short pulses is thus an easy way to obtain BSCSs. Long sinusoidal pulses have narrow spectra and hardly excite the target resonances which do not lie near their carrier frequency. When the carrier frequency of the incident pulse coincides with a target resonance, large changes become evident in the central maximum of the scattered pulse's spectrum. Furthermore, there is an observable narrowing of the initial portion of the backscattered echo in the (non-dimensional) time-domain. This narrowing occurs between the t w o transients 13 that develop in this case (c.f. Fig. 7(a)). Furthermore, the decrease in pressure amplitude that follows the second transient, once the ringing starts, occurs in discrete steps or jumps, as we have repeatedly indicated (c.f. Fig. 7(a): more on this below). On the other hand, when the carrier frequency of the pulse lies away from any of the target's natural resonances, the central maximum in the incident pulse's spectrum is communicated to the spectrum of the backscattered pressure pulse with little or no change in shape. In this case, no narrowing of the response in the time domain is then produced or observed (c.f. Fig. 7(b)). Furthermore, the amplitude decay in the final part of the response does not exhibit the stepwise decrease we discussed before at resonance. Whether at resonance or offresonance, the smaller side-lobes present at either side of the carrier frequency Xo in the backscattered spectrum (c.f. Figs 7(c) or 7(d)) are
Backscattering of sound pulses by elastic bodies underwater
105
distorted by the presence of target resonances. This discrepancy with the incident pulse spectra (c.f. Figs 5(a) and 5(b)) is more marked when x 0 coincides with a target resonance than when it does not. This difference-however small--between the incident and backscattered spectra (c.f. Figs 5(c) and 7(c)) is what accounts for the 'ringing' of the scattered pressure pulse in the time (3) domain. It should be pointed out that, since the sphere is elastic, there are portions of the incident wave that penetrate into its interior and become reflected from its north pole (the antipode of the point of incidence in the present arrangement). Other portions of the internally transmitted wave circumnavigate the sphere at selected frequencies arising from the excitation of the elastic Rayleigh (R) and whispering gallery (WG) resonances. Concurrently, there are circumnavigating waves in the outer fluid that are called 'creeping waves'. These (external) creeping waves produce relatively small effects in the bands of the BSCS that we have displayed in our plots. We have stated, and shown in the computed plots, that the backscattered pulse usually decreases its amplitude in discrete steps when the incident c.w. pulse has carrier frequency coinciding with any of the target resonances. An explanation for this behavior can be given in terms of a phase-matching argument 1. t q-21 that has been repeatedly given for the single surface wave that circumnavigates the target and adds in-phase after each circumnavigation, producing the resulting ringing. This phase-matching idea has been explained in various other works 2°'2t dealing with impenetrable and penetrable scatters of various geometries. It follows from Ref. 21 and the above discussion that the group velocities cl~ of the induced surface waves-normalized to the sound speed c~ in the outer fluid--can be obtained from the form-function plot in Fig. l(a) by just reading the difference in abcissa Ax between consecutive resonance features (dips) caused in this case by the Rayleigh wave, viz. 2t
Clg) --
= Ax
(29)
Ct
This value is read to be Ax ~ 2,4. Since the basic sonar/radar uncertainty relation is: t° ArAx--27r it is possible to estimate A~. The value is A t = 27:/2.4 ~ 2-6. This is the value observed in Fig. 7(a) for the (horizontal) duration of the discontinuous step, in the case of a long c.w. pulse. As we see in Fig. 7(b), this behavior is absent away from resonance. It is also absent for short pulses as can be confirmed in Fig. 8. We conclude by stating that a transient resonance scattering methodology has emerged from this analysis that can be used to study the scattering of any pulse of known spectrum from a general elastic target of known form function in the steady-state case.
106
Donald Brill, Guillermo Gaunaurd, Hans Strifors, William Wertman
REFERENCES 1. Gaunaurd, G., Elastic and acoustic resonance wave scattering. Appl. Mech. Rev., 42 (1989) 143-92, ASME Book No. AMR056. 2. Brill, D., Gaunaurd, G. & Ayres, V., The influence of natural resonances on -scattering and radiation processes. J. Wash. Acad. Sci., 77 (1987) 55-65. 3. Brill, D. & Gaunaurd, G. Resonance theory of elastic waves ultrasonically scattered from an elastic sphere. J. Acoust. Soc. Am., 81 (1987) 1-21 (Invited) (and references therein). 4. Gaunaurd, G., Techniques for sonar target-identification. IEEEJ. Ocean. Engr., OE-I 2 (1987) 419-22 (Special issue on Scattering); also ibid, Inverse scattering techniques for material characterization. Proc. Ultrasonics International '87. Butterworth Scientific Ltd, Surrey, UK, 1987, pp. 520-5. 5. Strifors, H. & Gaunaurd, G., Wave propagation in isotropic linear viscoelastic media. J. Acoust. Soc. Am., 85 (1989) 995-1004. 6. Jackson, J., Classical Electrodynamics, Wiley, New York, 1966, p. 540. 7. Gardner, M. F. & Barnes, J. L., Transients in Linear Systems (Vols. 1 and 2). John Wiley & Sons, New York, 1942. 8. Kennaugh, E. M. & Cosgriff, R. L., The use of impulse response in electromagnetic scattering problems. 1958 IRE National Conv. Rec., Part 1, pp. 72-7. 9. Kennaugh, E. M. & Moffatt, D. L., Transient and impulse response approximations. Proc. IEEE, 53 (1965) 893-901, and references therein. 10. Rihaczek, A. W., Principles of High-Resolution Radar. McGraw-Hill Book Co., New York, 1969. 11. Kennaugh, E. M., The K-pulse concept. IEEE Trans. Antennas & Prop., AP-29 (1981) 327-31. 12. Turhan-Sayan, G. & Moffatt, D. L., K-pulse estimation and target identification of low-Q radar targets. Wave Motion, 11 (1989) 453-61, and references therein. 13. Gaunaurd, G. & Tsui, C., Transient and steady-state target resonance excitation by sound scattering. Appl. Acoust., 23 (1988) 121--40. 14. Champency, D. C., Fourier Transforms and Their Physical Applications. Academic Press, New York, 1973. 15. Ikuno, H. & Felsen, L. B., Complex rays in transient scattering from smooth targets with inflection points. IEEE Trans., AP-36 (1988) 1272-80. 16. Moffatt, D. L., Young, J. D., Ksienski, A A., Lin, H. C. & Rhoads, C. M., Transient response characteristics in identification and imaging. IEEE Trans., AP-29 (1981) 192-205. 17. See, for example, Veksler, N., Information Analysis in Hydroelasticity. Academy of Sciences of the Estonian Soviet Socialist Republic, Valgus Publishing House, Tallinn, 1982 (in Russian); also, Rudgers, A., Acoustic pulses scattered by a rigid sphere immersed in a fluid. J. Aeoust. Soc. Am., 45 (1969) 900-10; also, Howell, W., Numrich, S. & Ueberall, H., Complex frequency poles of the acoustic scattering amplitude and their ringing. IEEE Trans., UFFC-34 (1987) 22-7. 18. Press, W., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T., Numerical Recipes, Ch. 12. Cambridge University Press, Cambridge, UK, 1986, pp. 390--429. 19. Werby, M. & Tango, G., Numerical study of material properties of submerged elastic objects using resonance response. J. Acoust. Soe. Am., 79 (1986) 1260-8.
Backscattering of sound pulses by elastic bodies underwater
107
20. Gaunaurd, G., Resonance acoustic scattering from underwater elastic bodies, in Elastic War'e Propagation, ed. M. F. McCarthy & M. A. Hayes. Elsevier Science Publishers, North Holland, 1989, pp. 335-46. 21. Gaunaurd, G. & Werby, M. F., Resonance response of submerged, acoustically excited thick and thin shells. J. Acoust. Soc. Am., 77 (1985) 2081-93, eqn (27).