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Scattering of elastic waves from symmetric inhomogeneities at low frequencies
WAVE MOTION 6 (1984) 325-336 NORTH-HOLLAND
SCA'I'TERING OF ELASTIC AT LOW FREQUENCIES
325
WAVES
FROM
SYMMETRIC
INHOMOGENEITIES
J.M. R I C H A R D S O N Rockwell International Science Center, Thousand Oaks, California 91360, USA
Received 25 March, 1983, Revised 6 January 1984
The problem of the scattering of elastic waves is treated in the low-frequency regime by a systematic expansion in powers of the frequency in the spatial domain of the scatterer. The zeroth and first degree scattering amplitudes vanish if the scatterer is localized in all directions. The second degree scattering amplitude corresponds to the so called Rayleigh regime in which the quasi-static result of Gubernatis et al. is valid. In general, the third and higher degree scattering amplitudes are nonvanishing. However, in the case where the scatterer has inversion symmetry about the origin, it is shown that the third degree scattering amplitude vanishes identicallly for all incident and scattered directions and for all polarizations. This result implies that the frequency derivative of the phase shift approaches zero at least quadratically as the frequency goes to zero. In other words, at sufficiently low frequencies the effective scattering center of a scatterer with inversion symmetry is its geometrical center. The use of this result in the processing of experimental scattering data is discussed.
I. Introduction The m a i n p u r p o s e o f this p a p e r is the d e r i v a t i o n o f certain c o n s e q u e n c e s o f inversion s y m m e t r y o f the scatterer as m a n i f e s t e d in the low f r e q u e n c y b e h a v i o r o f the p h a s e shift o f the scattered wave. In m o r e explicit terms it is p r o v e d that if all o f the p r o p e r t y d e v i a t i o n s (i.e., d e v i a t i o n s o f d e n s i t y a n d the elastic c o n s t a n t m a t r i x from their values in the host m a t e r i a l ) are i n v a r i a n t to the o p e r a t i o n o f inversion t h r o u g h the origin o f the c o o r d i n a t e system, then in the f r e q u e n c y e x p a n s i o n o f the scattering a m p l i t u d e the t h i r d degree term vanishes i d e n t i c a l l y (i.e., i n d e p e n d e n t l y o f the directions a n d p o l a r i z a t i o n s o f the i n c i d e n t a n d scattered waves). It is also p r o v e d that the h i g h e r o r d e r terms o f o d d d e g r e e d o not vanish i d e n t i c a l l y ; at least this is p r o v e d e x p l i c i t l y for the fifth d e g r e e term. In this p a p e r we do n o t a t t e m p t to derive all o f the c o n s e q u e n c e s o f i n v e r s i o n s y m m e t r y ; these will be d i s c u s s e d in a l a t e r p a p e r . The d e r i v a t i o n o f the a b o v e results requires the setting u p o f a relatively c o m p l e x f o r m a l i s m for c a r r y i n g o u t the f r e q u e n c y e x p a n s i o n o f the scat-
tering a m p l i t u d e . The s e c o n d degree term (the lowest o r d e r n o n v a n i s h i n g term in the case o f scatterers that are s p a t i a l l y l o c a l i z e d in all directions) r e p r e s e n t s the scattering process in the Rayleigh regime (low f r e q u e n c y or long wavelength), a case that has b e e n i n v e s t i g a t e d in slightly less generality b y G u b e r n a t i s et al. [1]. In Section 2, we i n t r o d u c e essential definitions a n d c o n s t r u c t a c o m p a c t n o t a t i o n that will significantly facilitate the d e r i v a t i o n o f results in subsequent sections. In Section 3, we set u p the f o r m a l ism d e s c r i b i n g the scattering o f elastic waves from a g e n e r a l lossless i n h o m o g e n e i t y a n d i n t r o d u c e an a p p r o p r i a t e definition o f phase-shift. In Section 4, we a d d r e s s the p r o b l e m o f e x p a n d i n g the scattering a m p l i t u d e in a p o w e r series in the t e m p o r a l f r e q u e n c y to a n d e l i m i n a t e terms that vanish for a general l o c a l i z e d scatterer. F i n a l l y , in S e c t i o n 5 we derive the c o n s e q u e n c e s o f inversion s y m m e t r y - n a m e l y , that the c u b i c term in the f r e q u e n c y e x p a n s i o n vanishes i d e n t i c a l l y if the origin o f the c o o r d i n a t e system is p l a c e d at the center o f symm e t r y - a n d we discuss its i m p l i c a t i o n s for signal processing.
0165-2125/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
326 2. N o t a t i o n
and definitions
The present section is devoted to miscellaneous preliminary matters. Because of the complexity of the later analysis, it is necessary to introduce a c o m p a c t notation in order to simplify the manipulations. This notation does not represent a higher level o f abstraction, only a shorthand version of the conventional notation e m p l o y e d in elastodynamic problems. To accomplish this, we first introduce a general purpose o p e r a n d , i.e., an arbitrary complex vector function o f position q~(r). The position vector r is defined by
r = elxl +e2x 2 + e3x3,
(2.1)
where x~, x2 and x3 are the usual Cartesian coordinates and where e,, e2 and ¢3 are the unit vectors directed along the coordinate axes. In this section we will take the d o m a i n of r to be all o f physical space, i.e., D ~ ; later, we will introduce various subdomains. We first introduce the identity operator I defined by l~b = ~.
(2.2)
We next introduce the o p e r a t o r L involved in the description of the p r o p a g a t i o n o f waves in a h o m o g e n e o u s , isotropic host m e d i u m at the frequency (temporal) w, namely
L ~ = pw2d~ +(A +p~)V~7 • q~ +/~V2~,
where here each of the Greek indices takes the values 1, 2, 3 and repeated Greek indices imply summation. In the above expression 6p = 6p(r) is the density deviation and 6 Q ~ = 6C~t3~,~(r) is the elastic constant tensor deviation. We assume, o f course, that 6C~,t~ is invariant to the interchange o f a and /3, y and 6, and the interchange of a/3 and 76. It is to be noted that we have assumed that the property deviations are also i n d e p e n d e n t of frequency, implying that the inhomogeneity involves only ideal, lossless, elastic material. The unperturbed elastic constant tensor is of course given by C,~t~,s = A8,,~8~,~ +~(6,~6~,~ +6,~a6~) in a c c o r d a n c e with (2.3). It is useful to introduce the projection operators, P~ and P,, for longitudinal and transverse waves, respectively. We define the former by the relation P,~b = V 277 • ~,
where V -~ is an inverse o f the Laplacian V 2. It is desirable to choose a form of V 2 that is invariant to translation, i.e., one that c o m m u t e s with V. A satisfactory form is given by the familiar expression
V 2,t, = - ~
1
f -, d 3 r ' i r - r '] 'dp(r').
The commutivity of X7 with the above form of ~7 2 is discussed in A p p e n d i x A. It is clear that with this definition (2.5) can be rewritten in the alternative forms, Pt~h = ~TV 2V • ~b = ~TV • ~7 ~ .
e,,. 6 L ~ = 6poo2qb,~ +
3x~
r~C~t3~-qb~, ox~
(2.4)
(2.7)
It is easy to show that P~ has the projection o p e r a t o r property P~ = P~. The projection operator for transverse waves is defined by the relation P~ = I - P,
(2.8)
from which we readily deduce that Pt = Pt and P~P~= 0. To show h o w these operators work let us consider the special o p e r a n d ~b = a exp(ike, r),
3
(2.6)
(2.3)
where p is the density and where A and ~ are the Lain6 elastic constants characterizing the host medium. These quantities are o f course assumed i n d e p e n d e n t o f the position r. They are also a s s u m e d to be i n d e p e n d e n t o f the frequency ¢o as is appropriate for an ideal, lossless, elastic medium. In a m e d i u m containing an i n h o m o g e n e i t y the o p e r a t o r L is replaced by an operator L + 6L. Using indiciat notations on the right h a n d side, we can write O
(2.5)
(2.9)
where a, e, and k are constants and where e is a
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
unit vector. We obtain
ticular, we can rewrite (2.13) in the form
PI4' = ee. a exp(ike, r), Pt4" = ( l - ee) . a exp(ike, r),
(2.10)
LG = I,
G -
(2.11)
where I is the unit tensor and where ee is a dyadic. Green's operator G for the host m e d i u m is o f course defined by
1
A +2it
1
KjPt + - - KtPt, /z
(2.13)
where 1 f d3r,lr_r,l_ , K , 4" = - --4-~ D~ x e x p ( i k , Ir - r ' l ) 4 ' ( r ' ) ,
(2.14)
and where K, is defined by a similar expression with kt replacing k~ on the right h a n d side. In writing (2.14) we have assumed that the implicit time factor is e x p ( - i w t ) . The quantities k~ and k, are the wave numbers associated with longitudinal and transverse waves, respectively, both with frequency to. These are given by O)
kt = --,
l
a +2#
l PjK, + - - P t K t , /~
(2.19)
which is useful in certain contexts. In deriving the consequences o f inversion symmetry in Section 5, it is o f course necessary to introduce the inversion operator R defined by
(2.12)
with the usual radiation conditions at infinite distance. In A p p e n d i x B it is shown that G =
327
R4"(r) = 4 ' ( - r ) .
(2.20)
I't is perhaps reasonable to consider an alternative definition in which the right h a n d side is - 4 ' ( - r ) rather than 4 ' ( - r ) . We will find it expedient not to consider such an alternative, in which case R can be applied also to scalar and even tensor functions of r on a simple uniform basis. If a function 4' is invariant to R then it is said to have even parity and if R causes it to change sign it is said to have odd parity. It is useful to introduce the even and odd parity projection operators P+ and P_ by the definitions P+=½(I +R),
(2.21)
P_ = ½(I - R).
(2.22)
It is easy to deduce that P+ + P_ = I, P+2 = P+, p 2 = P_ and P+P_ = 0. It is obvious that if 4' has even parity then , 0 4 ' = 0 and P+4' = 4'. C o r r e s p o n d ingly, if 4' has odd parity then P+4' = 0 and P_4' =
(2.15)
4,.
(2.16)
3. F o r m u l a t i o n
(2.17)
The small amplitude motion of a perfect iossless elastic m e d i u m is given by the partial differe~itial equation (p.d.e.)
¢1 it)
kt = - - , Ct
where
c,=(A+2~]
i/2
\
p
/
,
(L+SL)u
(2.18) are the longitudinal and transverse propagation velocities. Because the operators K~, Kt, Pt and Pt are all translationally invariant they are mutually c o m m u t i n g as discussed in A p p e n d i x A. In par-
=0,
(3.1)
where u = u(r, to) is the displacement field at the frequency ¢o and where the operators L and 8 L are defined by (2.3) and (2.4), respectively. The relation (3.1) is assumed to be valid everywhere in the infinite d o m a i n Do~. We assume that the m e d i u m contains an inhomogeneity o f finite
J.M. Richardson / Scattering from symmetric inhomogeneities
328
extension in all directions. In other words, there exists a scatterer domain Ds of finite extension, outside of which (i.e., r ~ D~) the deviations of material properties vanish (i.e., 6 L = 0). In the complementary domain D~, defined by D~ = D ~ - D~, the above p.d.e, reduces to L u = O,
r e D~.
(3.2)
In formulating the boundary conditions, it is desirable to divide the displacement field into incident and scattered parts, namely u = u i + u ~,
(3.3)
where the incident field satisfies the unperturbed p.d.e, everywhere, i.e.,
Another important feature of the integral equation (3.8) or (3.9) is that, if desired, r can be confined to the scatterer domain D~ and thus it can be solved without explicit reference to the displacement field outside. In the far field (3.8) can be written in the form u ~
I[e~e~ exp(ikjr)
= -G~Lu~r
+ ( I - e~e s) exp(iktr)] • H6Lu,
(3.10)
where the operator H is defined by
H~-
1
f ] d~r'[ct 2eSe" e x p ( - i k l e ~- r')
4~rp J
+ c t 2 ( l - e~e ~) e x p ( - i k t e ~. r')]. qS(r'),
(3.4)
(3.11)
The boundary conditions now consist of the usual radiation conditions applied to u t When applied explicitly to the longitudinal and transverse parts, these take the form
which obviously depends on r only through the unit vector e". In (3.11) the arbitrary function 4~(r) can, if desired, be required to vanish outside of Ds since in (3.10) 6L does. Let us now assume that the incident field u ~ is composed of longitudinal a n d / o r transverse plane waves propagating in the incident direction given by the unit vector e ~. We can then write
L u i = O,
r ~ D,~.
(V - ik~e~)PlU~-~ O,
(3.5)
(V - i kteS) Pt u" ---,O,
(3.6)
a s r = l r ] ~ o o . In the above expression e s = r 1 r i s the unit vector in the r-direction, i.e., the scattered or observer direction, and the quantities k~ and kt are the wave numbers for longitudinal and transverse polarizations, respectively, defined by (2.15), (2.16), (2.17) and (2.18). We assume that the origin of the coordinate system in which r is defined is placed somewhere within the scatterer domain Ds. Substitution of (3.3) into (3.1) yields Lu ~ = - ~ L u ,
(3.7)
and application of Green's operator (2.13) gives the integral equation [1], [2], u s = - G6Lu,
(3.8)
u = u i - G6Lu.
(3.9)
or
Since G satisfies the radiation conditions (3.5) and (3.6), it follows that u s given by (3.8) will also
U i = [ e i e i exp(ik, e i
+(l-eie
• r)
i) exp(ikt ei. r)]. a,
(3.12)
where a is a constant vector (possibly complex). It is obvious that e ~ . a is the amplitude of an incident longitudinal wave and ( I - e~e~) • a is the vector amplitude of an incident transverse wave. It is conventional to define the scattering amplitude tensor A ( e ~, e~; w) by the relation u~,,rg~r l[e~e~ exp(iklr)
+ ( I - e~e ~) exp(iktr)] • A ( e ~, ei; to). a. (3.13) By comparison with (3.10) we deduce that when r is very large, the scattering amplitude sensor is given by A ( e s, ei; to)= - H 6 L u ,
(3.14)
329
J.M• Richardson / Scatteringfrom symmetric inhomogeneities
where it is understood that u is a linear function of a and, in general, a nonlinear function of e ~. It is of interest to define the phase shift y by the expression
y = y(e s, b; e i, a ; to) = Arg b. A(e S, el; to) • a, (3.15) with the phase convention 0 ~< y(e s, b; d, a; 0) < 2~r,
(3.16)
i.e., the phase shift for any scattering process at zero frequency must be non-negative and less than 2-rr. The unit vector b defines the measured polarization of the scattered wave. For other values to the phase shift ambiguities associated with unknown multiples of 2~r can be resolved by the requirement of continuity except perhaps in singular cases in which b. A . a vanishes at a non-zero real value of w. It is easy to show that the solution of the integral equation (3.9) must satisfy the reality condition, i.e.,
u(r, to)* = u(r, -to).
(3.17)
We note the operators G and 6L satisfy similar relations
G(to)* = G ( - t o ) ,
(3.18)
6L(to )* = 6L(-to ).
(3.19)
ui(r, w) = ~ u,(r)to i , n
(4.1)
u(r, w ) = 2 u,(r)w",
(4.2)
rt=0
rt--O
where r c Ds. From its definition (2.4) it is clear that 6L contains only zeroth and second degree terms, i.e.,
6L = ~Lo + 6Lzw 2.
(4.3)
On the other hand the expansions of G and H contain all powers, namely
G = Y. G, to",
(4.4)
n--0
and ct3
The first of the above relations follows from (2.13) in which it is to be reemphasized that k~ and k t are proportional to to not Ito[. The second of the above relations follows trivially from the fact that ~L is real and an even function of to. Finally, noting that u i satisfies the reality condtion, we readily see that (3.9) implies (3.7). It then follows that the scattering amplitude A satisfies the reality condition, i.e.,
A(e S, ei; to)*= A(e ~, ei; --to).
in a convergent power series in the temporal frequency to with a nonvanishing circle of convergence. A similar expansion procedure has been considered by.Dassios [4] in the acoustical (scalar wave) case. In the infinite domain D o the assumption of convergence is obviously untrue since there we can always make r much larger than the wavelengths of longitudinal and transverse waves regardless of how large these are chosen to be. We assume here that the same statements are valid for the scattered displacement field u s and hence for the total field u. We accordingly write
H = Y~ H , to".
The coefficients u, are, of course, yet to be determined. The other coefficients are already known and are given by the expressions
i
1
i
. i r,
.o
1 [eie i
(3.20)
The more difficult questions of unitarity and reciprocity will not be discussed here.
(4.5)
n=O
ie" n ] . )
"
1--eie i]
- n !L c~ + - ~ J 1
G, - - - P ~ K , , y+2/x
' a(iei " r)", 1
+--PtKt,,
(4.6) (4.7)
where 4. E x p a n s i o n in powers o f frequency l
We note that in the scatterer domain Ds the incident displacement field u i can be expanded [3]
f
d3r'lr-rl" l ~ ( r ' ) , (4.8)
J.M. Richardson / Scattering from symmetric inhomogeneities
330
and Kt~ is given by an analogous expression with c, replacing q. Correspondingly, from (3.11) we infer that
1 [e"e ~ l-e~e ~] / ~ + ~ /
/-/.4,
4wpnILq
f
"
c,
"
J
d3r'(-ie ~. r')'~b(r').
(4.9)
D~
Finally, from (2.4) we obtain
involving the combination G~ 8Lo must also vanish. This result can be easily proved by noting that KH and K,, reduce, aside from constant factors, to the operation of integration on r over the domain D~. This integration applied to 8L0~b gives zero because of the fact that this quantity is a divergence. With these simplifications (4.12-1), (4.12-2), etc., reduce to ul = uit
e,~. 8Lod# =
(4.10)
8Co43.y~7--qS~, (gx t3
axe,
-
(4.14-1)
Go6Loul,
u, = u i - GoSLou2-
(4.14-2)
Go~L2uo,
u3 = u ~ - G o S L o u 3 - G o S L 2 u l - G i 8 L 2 u o
and
6L~4, = at,,/,.
Expanding the integral equation (3.9) in powers of w and equating coefficients of equal powers we obtain the set of equations i
Uo = Uo - G o S L o u o ,
(4.12-0)
ul = uil - G o 8 L o u l - G i S L o u o ,
(4.12-1)
i
u2 = u2 - G o S L o u 2 - G o 8 L 2 u o -
GI ~ L o u l
(4.12-2)
- G26Louo, U3 = U ~ - G o 8 L o u 3 - Gl8L2uo-
GoSL2ul - Gi 6Lou2 G28Loul - G38Louo,
(4.12-3) u4 = u i 4 - G o S L o u 4 -
GoSL2u~_- Gi 8Lou3
- G i 8L2Ul - G 2 8 L o u 2 - G 2 8 L e u o -
G38LoUl - G48Louo,
(4.12-4)
etc. It is understood in all the equations of this section that r e D~. There are several simplifications that can be made on the above results. First of all, it is easily seen that the solution of the 0th order integral equation is simply
(4.14-3)
- G28Lout,
(4.1 I)
u4 = u i 4 - G o S L o u 4 -
GofL2u2-
Gi 8Lzul
- G28Lou~ - G28L2uo - G38Loul,
(4.14-4) etc. A few remarks are in order concerning the meaning of the first two members of the above sequence of equations• In (4.14-1) the first-order incident displacement field u~ represents a uniform applied strain and u~ represents the total (i.e., u~ + u~) displacement field arising from the quasi-static response to this applied strain. In (4.14-2), u2 would represent a similar quasi-static response to an applied strain field with a uniform gradient if the term G o b L z u o were absent. This last term gives the inertial effect of a uniform displacement field with a periodic time dependence. We turn now to a discussion of the scattering amplitude. This tensor function can also be expanded in a power series in oJ as follows oc
a ( e ' , e ~• w)
Z a.(e,
=
n
"
~
e ) w .
(4.15)
"
0
From (3.14) we obtain the following expressions for the coefficients
(4.13)
-Ao
• a = Ho6Louo,
(4.16-0)
since according to (4.10) 8Lo applied to a constant operand gives a vanishing result. Therefore all terms involving the combination 8 L o u o must vanish. Furthermore, we can show that all terms
-Ai
• a = HoSLoul + HtSLouo,
(4.16-1)
-A2
" a = Ho8Lou2 + HoSL2uo + Hi8Loul
Uo= u~ = const.
+ H28Louo,
(4.16-2)
J.M. Richardson / Scattering from symmetric inhomogeneities - A 3 • a = H o 6 L o u 3 + HoBL2ul + H i 6Lou2 + H i 6L2uo + H2~Loul + Ha6Louo,
(4.16-3) - A 4 " a = Hot~Lou4 + not~L2u2 + H I t~tou 3 + Hi ~L2ul + H28Lou2 + Hz6L2uo + H36Loul + H46Louo,
(4.16-4)
- A s • a = Ho6Lou5 + H o S L 2 n 3 + HiSLon4 + H I (~L2u 2 + H2~Lou 3 + H 2 ~ L 2 u I
331
corresponds to the low frequency scattering theory of Gubernatis et al. [5]. It follows generally from (3.20) and specifically from the above results that A, is real when n is even and imaginary when n is odd. Let us consider the behavior of the phase shift y as a power series in the frequency. According to the definition (3.15) y = A r g b. A. a-= I m l o g b. A . a .
(4.18)
Letting
+ H 3 ~ L o u 2 + H3($L2uo + HM$Lou I
b.A,.a=B,, + H56Louo,
etc. Again, there are several simplifications that can be introduced into the above results. As before, all terms involving the combination 6Louo must vanish. Since the operator Ho, aside from a constant tensor factor, is an integration on r over the domain D~, it follows that all terms involving the combination Ho6Lo must vanish because of the divergence nature of 6Lo. With these simplifications (4.16-0), (4.16-1) etc. reduce to -Ao" a=0,
(4.17-0)
-AI" a=0,
(4.17-1)
- A 2 • a = HoSL2uo + H j S L o u l ,
(4.17-2)
- A 3 • a = HotSL2u j + Hi ~SLou2 + HI tSL2uo + H26Loul,
=iB,,
n odd,
(4.19)
where the B, = B , ( e S, b; e ~, a) are real, we obtain y = Im log(B2w 2 +iB3to 3 +
B4to 4 • • .)
= Arg B2 + I m log(1 +iB21B3to + BftB4to 2 +...)
= Arg B2 + B21B3to + ( B 2 1 B5 - B22B3B4 - ~Bf3B3)w 3 + O ( w 5).
(4.20)
Since B2 is real, the quantity Arg B2 is equal to 0 or ~r depending on whether B2 is positive or negative.
5. Some consequences of inversion symmetry
(4.17- 3)
- A 4 • a = HoSL2u2 + HI tSLou3+ Hi 6L2u~ + H28Lou2 + H2~SL2uo + H3t~LoUl ,
(4.17-4) - A 5 • a = Ho6L2u3 + Hi tSLou4 + H1 t~L2n2 + H26Lou3 + H z S L 2 u l + H3t~Lon2 + H38L2uo + H4BLoul.
neven;
(4.16-5)
(4.17-5)
It is to be noted that A , - a depends at most upon u,_l and lower orders. Since the vector a is arbitrary it follows from (4.17-0) and (4.17-1) that Ao and At, must vanish. We note that (4.17-2)
The results derived in previous sections are valid for a general localized inhomogeneity regardless of the presence or absence of symmetry properties. In the present section we consider some of the consequences of inversion symmetry. In particular, we consider only the consequences relative to the possible vanishing of some of the coefficients A,. A more comprehensive analysis of the consequences of symmetry will be given in a later communication. The inversion symmetry of the scatterer is represented by the requirement that the material property deviations 6p and 6 C ~ be invariant to
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
332
inversion, i.e.,
Application of P+ to (4.14-3) yields
R6p(r) = 8p(-r) = 6p(r)
(5.1)
R6C~t3r~(r ) = 6 C , ~ ( -
(5.2)
P+u3 = - G o 6 L o P + u 3 -
(5.7-3a)
Gi6L2uo,
and r) = 6 C , , ~ ( r ) .
These relations imply that R commutes with 6L, that is
which implies that in general P+u3 ¢ O. Furthermore, with P replacing P+ we obtain P_u3 = u~ - Go6LoP_u3 - Go6L2ul
-
G26Lout,
(5.3)
(5.7-3b)
and hence R commutes with 6Lo and 6L2. It is easy to show from (2.13) that G and R commute, i.e.,
which, again, implies that P u 3 ~ 0. Thus, in general, u3 has parts of both odd and even parities. It can be shown that the same statement is true of all higher order un's. To summarize, if the scatterer has inversion symmetry, Uo and u2 have even parity, u~ has odd parity, but u~, u4. . . . in general have parts with both even and odd parities. We now must examine quantities of the form H m 6 L , up where, of course, n = 0 or 2. From (4.9) we see that H,~4~ is proportional to the integral
Rt~L = 6 L R ,
(5.4)
RG = GR.
This follows from the fact that R commutes with P~ and Pt and that in the definitions of K~ and Kt (e.g., (2.14)) r and r' enter only through the quantity J r - r ' I. It then follows trivially that R commutes with the G,, the coefficients in the frequency expansion of G. Thus, we can say that operators t~Lo, 6L2 and G,, n = 0, 1 , . . . , preserve parity. The coefficients u~, in the frequency expansion of the incident field have even or odd parity depending on whether n is even or odd, i.e.,
Rui. = (-1)"ui.
(5.5)
or, equivalently, P+ui, = 0 ,
n odd,
P_u~, = 0 ,
n even.
(5.6)
We turn now to the consideration of the consequences of inversion symmetry in the sequence of integral equations (4.14-1), (4.14-2), etc. Before doing this we note that, since Uo= u~, it is obvious that Uo must have even parity. Application of P+ to (4.14-1) gives P+ul = -- G o f L o P + u l ,
(5.7-1 )
from which we infer that P+u~ = 0, i.e., u~ has odd parity. Application of P to (4.14-2) yields P - u2 = - G o 6 L o P _ u2,
(5.7-2)
which implies that P - u 2 = 0, i.e., u2 has even parity.
J,,--- f
d 3 r ' ( - i e ~- r')md~(r').
(5.8)
It is obvious that J,, will vanish if the entire integrand has odd parity because the contribution at r' will cancel the contribution at - r ' . Thus, since the factor ( - i e s. r) m has even parity if m is even and odd parity if m is odd, it follows that J,, and hence /-/,nO will vanish if m is even and ~ has odd parity and vice versa. The above results imply that in the sequence (4.17-0), (4.17-1), etc. the r.h. side of the third member, i.e., (4.17-3), vanishes because all of the terms involve integrands of odd parity and thus we obtain A3 = 0. The r.h. sides of the even numbers are of course nonvanishing because none of the terms involve integrands of odd parity. The r.h. sides of the higher order odd numbers, i.e., fifth and higher, are in general nonvanishing because each of the quantities u3, us, etc. contains parts of both even and odd parity. Thus, to summarize, Ao and At vanish because of general considerations, A3 vanishes because of inversion symmetry, but the higher order An do not vanish in general.
J.M. Richardson / Scatteringfrom symmetricinhomogeneities The phase shift 3,, previously given by (4.20), now takes the form
operators defined by
x,(r) = D,~b(r) = f
y = Im log(B2to 2 + Bato 4 +iBsto 5 +- • .)
n = 1. . . . , N,
+ i B f l B s t o 3 • . .) (5.9)
We obtain the following frequency derivative o f the phase shift (5.10)
which is proportional to the distance between the effective scattering center and the geometrical center. It is clear that this distance approaches 0 quadratically as to-* 0. In other words, for a scatterer with inversion s y m m e t r y the effective scattering center is the geometrical center in the limit o f zero frequency. The results holds independently o f the p r o p a g a t i o n directions and polarizations o f incident and scattered waves. It is clear that the low-frequency analysis o f the phase shifts obtained from three i n d e p e n d e n t scattering measurements can yield a unique determination of the position o f the geometrical center o f the scatterer with inversion symmetry [6].
(A1)
where ~b(r) is an arbitrary scalar operand. In Sections 2 and 3 we used a vector o p e r a n d but here a scalar one suffices. That the operators D , are translation invariant is easily d e m o n s t r a t e d by making the substitutions
x.(r) ~ x ' ( r ) 0_fly= 3B2~ Bsto 2 + O(to4), ato
dar ' D , ( r - r')~b(r'),
J D~
-- Arg B 2 + I m log(l +Bf~B4to 2
-- Arg B2 + B21Bsto 3 +O(tos).
333
=
x . ( r - a),
~b(r) ~ 0'(r) = 0 ( r - a)
(A2)
in (A1) with the result that the primed functions X" and 0 ' are related in the same way as the u m p r i m e d functions X, and 0. We turn now to the consideration o f the commuo tivity o f two translation-invariant operators, say D~ and D2. We obtain
(D~ D2 - D2Di)~b = f
J D~
dar ' A12(r -- r')~b(r'), (A3)
where Al2(r-- r") = fo~ d3 r'[Dj(r- r')D2(r'- r")
- D2(r - r')Dl(r'- r")]. Acknowledgement The a u t h o r is indebted to R.K. Elsley for previously advancing the hypothesis that the results o f this p a p e r are valid for a spherical void and to E.R. C o h e n for helpful comments.
Appendix A. Commutation of translation invariant operators
A sufficient condition for the existence o f Ai2 is that the integrals ID
d 3 r , Din(r- r t)Dn(r t - r")
(A5)
exist for all r and r" in Doo and for the combinations m = 1, n = 2 and m = 2, n = 1. In turn, a sufficient condition for the existence o f the integrals immediately above is that
f Here we confront the problem o f proving the commutivity of translation-invariant operators in the mathematical context o f scattering theory. To be more explicit let us consider examples o f such
(A4)
d3rD2,(r)
m = 1,2,
(A6)
DR
i.e., D~(r)c L2(Doo), n = 1, 2, i.e., the functions D,(r) are quadratically integrable in the infinite d o m a i n Do. We wish to emphasize that it is also
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
334
assumed that t)(r)c L2(D~,) which in combination with (A6) provides a sufficient condition for the existence of D~ D2t) and D2D~ t). We now show that under the above assumptions we obtain the relation At2(r)= 0, r c D~. Consider the integral
l12(r-- r") = f d3r ' D l ( r - r')D2(r'- r").
(A7)
Letting r' = r + r" - r'" we obtain r - r' = r"' - r" and r ' - r " = r - r'". With r'" replacing r' as the variable of integration we obtain Ii2(r - r") = f d3r ''' Dffr'"- r")D2(r- r"')
= f d3r ''' D2(r- r'")Dl(r'"- r")
= 121(r -- r"),
(A8)
from which it follows that Ai2(r-- r") = lt2(r-- r") -- I,_l(r-- r") = 0.
(A9)
Thus, if (but not only if) D 6 r ) , D2(r) and t)(r) are quadratically integrable, then
Dt D2tb - D2Dt t) = 0
(Al0)
DID2
(A1 I)
and -
D2DI
=
0.
However, as we will see, the quadratic integrability condition is not satisfied by any of the translation-invariant operators of interest to us here and thus more subtle arguments must be invoked to prove commutivity. Examples of such operators are V, V 2, PI, Pt, Ki and Kt. A conceptually simple approach is to substitute each of these operators by modified operators that are quadratically integrable in the sense that the corresponding D(r) functions are quadratically integrable. It then requires that the modified operators approach the original ones as some small-valued parameter e tends to zero. In terms of the general purpose scalar
operand t)(r), one could make the substitutions; Vt)~fdr'V~exp( t,~Tre)-/-
\
]r--r'].~ 4e / t ) ( r ' ) ' (AI2)
V 20~-1
f dr'eXp(l-re_lr-r'l)~(r),
(AI3)
1 [ e x p ( ( i k , - e ) l r - r'l) Kit) -, -4-~ a dr ]r - r'l t)(r'), (AI4) with a similar expression for K,. The modified forms of the operators P~ and P, are to be constructed by appropriate combinations of the modified forms of V and V -2 given by (AI2) and (A13). The reader can easily verify that the D(r) functions corresponding to the r.h. sides of (A 12), (AI 3), and (AI4) are quadratically integrable for e > 0. Furthermore, the reader can also readily verify that the D(r) functions corresponding to the original operators on the l.h. sides of (AI2), (AI3), and (A14) are regained in the limit e ~ 0 . Clearly, the same results apply to P~, P,, and K~. Thus the manipulations in Sections 2 and 3, involving the assumed commutivity of translation invariant operators, can be justified on a rigorous basis by substituting them by modified operators with the quadratic integrability property and then taking the limit e --, 0.
Appendix B. Derivation of the elastodynamic green function Here we present a detailed derivation of Green's operator G given by (2.13) in the text. Furthermore, we present various alternative forms of G in the usual explicit representation. We start by noting that the operator L defined by (2.3) can be rewritten in the form L = (A +2/~)L~PI +/xLtPt,
(B1)
where P~ and Pt are respectively the longitudinal and transverse projection operators defined by (2.5)-(2.8) and where the operators L~ and L, are
J.M. Richardson / Scatteringfrom symmetric inhomogeneities defined by
where
L~tb = (V 2 + k~)4~
(a2)
Lt~
(B3)
and =
(~72 -]-kt2)~
The longitudinal and transverse wave numbers k~ and k, are defined by (2.15)-(2.18). The correctness of (B1) can be verified by direct substitution. It is to be noted that all of the operators on the right hand side of (BI) are mutually commuting and hence the products can be rewritten in the reverse order if desired. The verification of the validity of the form of Green's operator given by (2.13) or (2.19), namely
G =
335
1
1
KIP I + - - K t P t y+2# ~x 1
=y+2~
1
PtKI +-- PtKt, /x
(B4)
now becomes an almost trivial exercise. We obtain the desired result
1 R l exp(ikiR), Kj( R ) = 4---~
(B9)
in which
R=lr-r'l.
(BIO)
The definition for K t is the same as K~ except that t replaces 1 everywhere. In terms of the more explicit notation G can be expressed as the following form of Green's function.
G ( r - r') -
1 -
-
y +2~t
VVV-2KI(R)
1
2
+ - - ( I - VVV- )K~(R).
(Bll)
It is sometimes useful to rewrite the above expression in a form from which the operator V 2 is eliminated. With a trivial rearrangement we obtain
G(r - r') =
1
pro
2[k~VVV-2KI(R)
+ k2t(l - VVV-2)Kt(R)]
L G = [(Y + 2/tz)L,P, +/.tLtPt] -
1
pw
2[Ik~K,(R)
+VVV 2(k2K~(R)- k2tKt(R))].
= L1PIKI + LtPtKt
(Bl2) From (B6) we obtain the obvious result
= PILIKI + P, LtKt
LIK~- LtKt = 0, = P, + Pt = I,
(BI 3)
(B5) which corresponds, in more explicit notation, to
using the fact that K~ and K~ defined by (2.14) etc. are Green's operators for L~ and Lt, respectively, i.e.,
LiKi = LtKt = L
(B6)
We now proceed to rewrite Green's operator in terms of more explicit notation. We first consider the definitions
G#p(r) = I
d3r ' G(r - r') . dp(r')
(B7)
D~
and
(V2 + k ~ ) K , ( R ) - ( V 2 +k~)Kt(R)=O. From which we deduce the result
k~ K,( R) - k 2tKt( R) = - V 2 ( K , ( R ) - Kt( R)).
(BI5) Substitution of (Bl5) into (Bl2) yields the more familiar form [7]:
G ( r - r') =
1
pro -
K, rb(r) = f
d3 r ' Dx
K,(Ir- r'l)q,(r'),
(B8)
(B14)
2[Ik2,K,(R)
V V ( K , ( R ) - K,(R))],
(B16)
from which the inverse Laplacian operator V -2 is absent.
J.M. Richardson/Scattering from symmetric inhomogeneities
336
References [1] J.E. Gubernatis, E. Domany, J.A. Krumhansl, and M. Huberman, "The fundamental theory of elastic waves scattering by defects in elastic materials--Integral equation methods for application to ultrasonic flaw detection",
Materials Science Center, Cornell University, Tech. Rpt., 2654 (1975), Also ERDA Tech. Rpt. C00-3161-42 (1975). [2] J.E. Gubernatis, E. Domany, and J.A. Krumhansl, "Formal aspects of the theory of the scattering of ultrasound by flaws", J. AppL Phys. 48, 2804 (1977). [3] J.E. Gubernatis, "Long wave approximations for the scattering of elastic waves from flaws with applications to ellipsoidal voids and inclusions", J. Appl. Phys. 50, 4046 (1979).
[4] G. Dassios, "'Convergent low frequency expansions for penetrable scatterers", J. Math. Phys. 18 126-137 (1977). [5] J.E. Gubernatis, J.A. Krumhansl, R.M. Thomson, "Interpretation of elastic wave scattering theory for analysis and design of flaw characterization experiments: I. Long wavelength limit", Los Alamos Scientific Laboratory Report, LA-UR-76-2546 (1976). [6] J.M. Richardson, "'Low frequency behavior of amplitude and phase shift in elastic wave scattering", paper presented at the NBS Symposium on Materials Characterization, Gaithersburg, Maryland, June 7-9, 1978. [7] P.M. Morse, in Handbook of Physics, edited by E.V. Condon and H. Odishaw (McGraw-Hill, New York, 1967).
SCA'I'TERING OF ELASTIC AT LOW FREQUENCIES
325
WAVES
FROM
SYMMETRIC
INHOMOGENEITIES
J.M. R I C H A R D S O N Rockwell International Science Center, Thousand Oaks, California 91360, USA
Received 25 March, 1983, Revised 6 January 1984
The problem of the scattering of elastic waves is treated in the low-frequency regime by a systematic expansion in powers of the frequency in the spatial domain of the scatterer. The zeroth and first degree scattering amplitudes vanish if the scatterer is localized in all directions. The second degree scattering amplitude corresponds to the so called Rayleigh regime in which the quasi-static result of Gubernatis et al. is valid. In general, the third and higher degree scattering amplitudes are nonvanishing. However, in the case where the scatterer has inversion symmetry about the origin, it is shown that the third degree scattering amplitude vanishes identicallly for all incident and scattered directions and for all polarizations. This result implies that the frequency derivative of the phase shift approaches zero at least quadratically as the frequency goes to zero. In other words, at sufficiently low frequencies the effective scattering center of a scatterer with inversion symmetry is its geometrical center. The use of this result in the processing of experimental scattering data is discussed.
I. Introduction The m a i n p u r p o s e o f this p a p e r is the d e r i v a t i o n o f certain c o n s e q u e n c e s o f inversion s y m m e t r y o f the scatterer as m a n i f e s t e d in the low f r e q u e n c y b e h a v i o r o f the p h a s e shift o f the scattered wave. In m o r e explicit terms it is p r o v e d that if all o f the p r o p e r t y d e v i a t i o n s (i.e., d e v i a t i o n s o f d e n s i t y a n d the elastic c o n s t a n t m a t r i x from their values in the host m a t e r i a l ) are i n v a r i a n t to the o p e r a t i o n o f inversion t h r o u g h the origin o f the c o o r d i n a t e system, then in the f r e q u e n c y e x p a n s i o n o f the scattering a m p l i t u d e the t h i r d degree term vanishes i d e n t i c a l l y (i.e., i n d e p e n d e n t l y o f the directions a n d p o l a r i z a t i o n s o f the i n c i d e n t a n d scattered waves). It is also p r o v e d that the h i g h e r o r d e r terms o f o d d d e g r e e d o not vanish i d e n t i c a l l y ; at least this is p r o v e d e x p l i c i t l y for the fifth d e g r e e term. In this p a p e r we do n o t a t t e m p t to derive all o f the c o n s e q u e n c e s o f i n v e r s i o n s y m m e t r y ; these will be d i s c u s s e d in a l a t e r p a p e r . The d e r i v a t i o n o f the a b o v e results requires the setting u p o f a relatively c o m p l e x f o r m a l i s m for c a r r y i n g o u t the f r e q u e n c y e x p a n s i o n o f the scat-
tering a m p l i t u d e . The s e c o n d degree term (the lowest o r d e r n o n v a n i s h i n g term in the case o f scatterers that are s p a t i a l l y l o c a l i z e d in all directions) r e p r e s e n t s the scattering process in the Rayleigh regime (low f r e q u e n c y or long wavelength), a case that has b e e n i n v e s t i g a t e d in slightly less generality b y G u b e r n a t i s et al. [1]. In Section 2, we i n t r o d u c e essential definitions a n d c o n s t r u c t a c o m p a c t n o t a t i o n that will significantly facilitate the d e r i v a t i o n o f results in subsequent sections. In Section 3, we set u p the f o r m a l ism d e s c r i b i n g the scattering o f elastic waves from a g e n e r a l lossless i n h o m o g e n e i t y a n d i n t r o d u c e an a p p r o p r i a t e definition o f phase-shift. In Section 4, we a d d r e s s the p r o b l e m o f e x p a n d i n g the scattering a m p l i t u d e in a p o w e r series in the t e m p o r a l f r e q u e n c y to a n d e l i m i n a t e terms that vanish for a general l o c a l i z e d scatterer. F i n a l l y , in S e c t i o n 5 we derive the c o n s e q u e n c e s o f inversion s y m m e t r y - n a m e l y , that the c u b i c term in the f r e q u e n c y e x p a n s i o n vanishes i d e n t i c a l l y if the origin o f the c o o r d i n a t e system is p l a c e d at the center o f symm e t r y - a n d we discuss its i m p l i c a t i o n s for signal processing.
0165-2125/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
326 2. N o t a t i o n
and definitions
The present section is devoted to miscellaneous preliminary matters. Because of the complexity of the later analysis, it is necessary to introduce a c o m p a c t notation in order to simplify the manipulations. This notation does not represent a higher level o f abstraction, only a shorthand version of the conventional notation e m p l o y e d in elastodynamic problems. To accomplish this, we first introduce a general purpose o p e r a n d , i.e., an arbitrary complex vector function o f position q~(r). The position vector r is defined by
r = elxl +e2x 2 + e3x3,
(2.1)
where x~, x2 and x3 are the usual Cartesian coordinates and where e,, e2 and ¢3 are the unit vectors directed along the coordinate axes. In this section we will take the d o m a i n of r to be all o f physical space, i.e., D ~ ; later, we will introduce various subdomains. We first introduce the identity operator I defined by l~b = ~.
(2.2)
We next introduce the o p e r a t o r L involved in the description of the p r o p a g a t i o n o f waves in a h o m o g e n e o u s , isotropic host m e d i u m at the frequency (temporal) w, namely
L ~ = pw2d~ +(A +p~)V~7 • q~ +/~V2~,
where here each of the Greek indices takes the values 1, 2, 3 and repeated Greek indices imply summation. In the above expression 6p = 6p(r) is the density deviation and 6 Q ~ = 6C~t3~,~(r) is the elastic constant tensor deviation. We assume, o f course, that 6C~,t~ is invariant to the interchange o f a and /3, y and 6, and the interchange of a/3 and 76. It is to be noted that we have assumed that the property deviations are also i n d e p e n d e n t of frequency, implying that the inhomogeneity involves only ideal, lossless, elastic material. The unperturbed elastic constant tensor is of course given by C,~t~,s = A8,,~8~,~ +~(6,~6~,~ +6,~a6~) in a c c o r d a n c e with (2.3). It is useful to introduce the projection operators, P~ and P,, for longitudinal and transverse waves, respectively. We define the former by the relation P,~b = V 277 • ~,
where V -~ is an inverse o f the Laplacian V 2. It is desirable to choose a form of V 2 that is invariant to translation, i.e., one that c o m m u t e s with V. A satisfactory form is given by the familiar expression
V 2,t, = - ~
1
f -, d 3 r ' i r - r '] 'dp(r').
The commutivity of X7 with the above form of ~7 2 is discussed in A p p e n d i x A. It is clear that with this definition (2.5) can be rewritten in the alternative forms, Pt~h = ~TV 2V • ~b = ~TV • ~7 ~ .
e,,. 6 L ~ = 6poo2qb,~ +
3x~
r~C~t3~-qb~, ox~
(2.4)
(2.7)
It is easy to show that P~ has the projection o p e r a t o r property P~ = P~. The projection operator for transverse waves is defined by the relation P~ = I - P,
(2.8)
from which we readily deduce that Pt = Pt and P~P~= 0. To show h o w these operators work let us consider the special o p e r a n d ~b = a exp(ike, r),
3
(2.6)
(2.3)
where p is the density and where A and ~ are the Lain6 elastic constants characterizing the host medium. These quantities are o f course assumed i n d e p e n d e n t o f the position r. They are also a s s u m e d to be i n d e p e n d e n t o f the frequency ¢o as is appropriate for an ideal, lossless, elastic medium. In a m e d i u m containing an i n h o m o g e n e i t y the o p e r a t o r L is replaced by an operator L + 6L. Using indiciat notations on the right h a n d side, we can write O
(2.5)
(2.9)
where a, e, and k are constants and where e is a
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
unit vector. We obtain
ticular, we can rewrite (2.13) in the form
PI4' = ee. a exp(ike, r), Pt4" = ( l - ee) . a exp(ike, r),
(2.10)
LG = I,
G -
(2.11)
where I is the unit tensor and where ee is a dyadic. Green's operator G for the host m e d i u m is o f course defined by
1
A +2it
1
KjPt + - - KtPt, /z
(2.13)
where 1 f d3r,lr_r,l_ , K , 4" = - --4-~ D~ x e x p ( i k , Ir - r ' l ) 4 ' ( r ' ) ,
(2.14)
and where K, is defined by a similar expression with kt replacing k~ on the right h a n d side. In writing (2.14) we have assumed that the implicit time factor is e x p ( - i w t ) . The quantities k~ and k, are the wave numbers associated with longitudinal and transverse waves, respectively, both with frequency to. These are given by O)
kt = --,
l
a +2#
l PjK, + - - P t K t , /~
(2.19)
which is useful in certain contexts. In deriving the consequences o f inversion symmetry in Section 5, it is o f course necessary to introduce the inversion operator R defined by
(2.12)
with the usual radiation conditions at infinite distance. In A p p e n d i x B it is shown that G =
327
R4"(r) = 4 ' ( - r ) .
(2.20)
I't is perhaps reasonable to consider an alternative definition in which the right h a n d side is - 4 ' ( - r ) rather than 4 ' ( - r ) . We will find it expedient not to consider such an alternative, in which case R can be applied also to scalar and even tensor functions of r on a simple uniform basis. If a function 4' is invariant to R then it is said to have even parity and if R causes it to change sign it is said to have odd parity. It is useful to introduce the even and odd parity projection operators P+ and P_ by the definitions P+=½(I +R),
(2.21)
P_ = ½(I - R).
(2.22)
It is easy to deduce that P+ + P_ = I, P+2 = P+, p 2 = P_ and P+P_ = 0. It is obvious that if 4' has even parity then , 0 4 ' = 0 and P+4' = 4'. C o r r e s p o n d ingly, if 4' has odd parity then P+4' = 0 and P_4' =
(2.15)
4,.
(2.16)
3. F o r m u l a t i o n
(2.17)
The small amplitude motion of a perfect iossless elastic m e d i u m is given by the partial differe~itial equation (p.d.e.)
¢1 it)
kt = - - , Ct
where
c,=(A+2~]
i/2
\
p
/
,
(L+SL)u
(2.18) are the longitudinal and transverse propagation velocities. Because the operators K~, Kt, Pt and Pt are all translationally invariant they are mutually c o m m u t i n g as discussed in A p p e n d i x A. In par-
=0,
(3.1)
where u = u(r, to) is the displacement field at the frequency ¢o and where the operators L and 8 L are defined by (2.3) and (2.4), respectively. The relation (3.1) is assumed to be valid everywhere in the infinite d o m a i n Do~. We assume that the m e d i u m contains an inhomogeneity o f finite
J.M. Richardson / Scattering from symmetric inhomogeneities
328
extension in all directions. In other words, there exists a scatterer domain Ds of finite extension, outside of which (i.e., r ~ D~) the deviations of material properties vanish (i.e., 6 L = 0). In the complementary domain D~, defined by D~ = D ~ - D~, the above p.d.e, reduces to L u = O,
r e D~.
(3.2)
In formulating the boundary conditions, it is desirable to divide the displacement field into incident and scattered parts, namely u = u i + u ~,
(3.3)
where the incident field satisfies the unperturbed p.d.e, everywhere, i.e.,
Another important feature of the integral equation (3.8) or (3.9) is that, if desired, r can be confined to the scatterer domain D~ and thus it can be solved without explicit reference to the displacement field outside. In the far field (3.8) can be written in the form u ~
I[e~e~ exp(ikjr)
= -G~Lu~r
+ ( I - e~e s) exp(iktr)] • H6Lu,
(3.10)
where the operator H is defined by
H~-
1
f ] d~r'[ct 2eSe" e x p ( - i k l e ~- r')
4~rp J
+ c t 2 ( l - e~e ~) e x p ( - i k t e ~. r')]. qS(r'),
(3.4)
(3.11)
The boundary conditions now consist of the usual radiation conditions applied to u t When applied explicitly to the longitudinal and transverse parts, these take the form
which obviously depends on r only through the unit vector e". In (3.11) the arbitrary function 4~(r) can, if desired, be required to vanish outside of Ds since in (3.10) 6L does. Let us now assume that the incident field u ~ is composed of longitudinal a n d / o r transverse plane waves propagating in the incident direction given by the unit vector e ~. We can then write
L u i = O,
r ~ D,~.
(V - ik~e~)PlU~-~ O,
(3.5)
(V - i kteS) Pt u" ---,O,
(3.6)
a s r = l r ] ~ o o . In the above expression e s = r 1 r i s the unit vector in the r-direction, i.e., the scattered or observer direction, and the quantities k~ and kt are the wave numbers for longitudinal and transverse polarizations, respectively, defined by (2.15), (2.16), (2.17) and (2.18). We assume that the origin of the coordinate system in which r is defined is placed somewhere within the scatterer domain Ds. Substitution of (3.3) into (3.1) yields Lu ~ = - ~ L u ,
(3.7)
and application of Green's operator (2.13) gives the integral equation [1], [2], u s = - G6Lu,
(3.8)
u = u i - G6Lu.
(3.9)
or
Since G satisfies the radiation conditions (3.5) and (3.6), it follows that u s given by (3.8) will also
U i = [ e i e i exp(ik, e i
+(l-eie
• r)
i) exp(ikt ei. r)]. a,
(3.12)
where a is a constant vector (possibly complex). It is obvious that e ~ . a is the amplitude of an incident longitudinal wave and ( I - e~e~) • a is the vector amplitude of an incident transverse wave. It is conventional to define the scattering amplitude tensor A ( e ~, e~; w) by the relation u~,,rg~r l[e~e~ exp(iklr)
+ ( I - e~e ~) exp(iktr)] • A ( e ~, ei; to). a. (3.13) By comparison with (3.10) we deduce that when r is very large, the scattering amplitude sensor is given by A ( e s, ei; to)= - H 6 L u ,
(3.14)
329
J.M• Richardson / Scatteringfrom symmetric inhomogeneities
where it is understood that u is a linear function of a and, in general, a nonlinear function of e ~. It is of interest to define the phase shift y by the expression
y = y(e s, b; e i, a ; to) = Arg b. A(e S, el; to) • a, (3.15) with the phase convention 0 ~< y(e s, b; d, a; 0) < 2~r,
(3.16)
i.e., the phase shift for any scattering process at zero frequency must be non-negative and less than 2-rr. The unit vector b defines the measured polarization of the scattered wave. For other values to the phase shift ambiguities associated with unknown multiples of 2~r can be resolved by the requirement of continuity except perhaps in singular cases in which b. A . a vanishes at a non-zero real value of w. It is easy to show that the solution of the integral equation (3.9) must satisfy the reality condition, i.e.,
u(r, to)* = u(r, -to).
(3.17)
We note the operators G and 6L satisfy similar relations
G(to)* = G ( - t o ) ,
(3.18)
6L(to )* = 6L(-to ).
(3.19)
ui(r, w) = ~ u,(r)to i , n
(4.1)
u(r, w ) = 2 u,(r)w",
(4.2)
rt=0
rt--O
where r c Ds. From its definition (2.4) it is clear that 6L contains only zeroth and second degree terms, i.e.,
6L = ~Lo + 6Lzw 2.
(4.3)
On the other hand the expansions of G and H contain all powers, namely
G = Y. G, to",
(4.4)
n--0
and ct3
The first of the above relations follows from (2.13) in which it is to be reemphasized that k~ and k t are proportional to to not Ito[. The second of the above relations follows trivially from the fact that ~L is real and an even function of to. Finally, noting that u i satisfies the reality condtion, we readily see that (3.9) implies (3.7). It then follows that the scattering amplitude A satisfies the reality condition, i.e.,
A(e S, ei; to)*= A(e ~, ei; --to).
in a convergent power series in the temporal frequency to with a nonvanishing circle of convergence. A similar expansion procedure has been considered by.Dassios [4] in the acoustical (scalar wave) case. In the infinite domain D o the assumption of convergence is obviously untrue since there we can always make r much larger than the wavelengths of longitudinal and transverse waves regardless of how large these are chosen to be. We assume here that the same statements are valid for the scattered displacement field u s and hence for the total field u. We accordingly write
H = Y~ H , to".
The coefficients u, are, of course, yet to be determined. The other coefficients are already known and are given by the expressions
i
1
i
. i r,
.o
1 [eie i
(3.20)
The more difficult questions of unitarity and reciprocity will not be discussed here.
(4.5)
n=O
ie" n ] . )
"
1--eie i]
- n !L c~ + - ~ J 1
G, - - - P ~ K , , y+2/x
' a(iei " r)", 1
+--PtKt,,
(4.6) (4.7)
where 4. E x p a n s i o n in powers o f frequency l
We note that in the scatterer domain Ds the incident displacement field u i can be expanded [3]
f
d3r'lr-rl" l ~ ( r ' ) , (4.8)
J.M. Richardson / Scattering from symmetric inhomogeneities
330
and Kt~ is given by an analogous expression with c, replacing q. Correspondingly, from (3.11) we infer that
1 [e"e ~ l-e~e ~] / ~ + ~ /
/-/.4,
4wpnILq
f
"
c,
"
J
d3r'(-ie ~. r')'~b(r').
(4.9)
D~
Finally, from (2.4) we obtain
involving the combination G~ 8Lo must also vanish. This result can be easily proved by noting that KH and K,, reduce, aside from constant factors, to the operation of integration on r over the domain D~. This integration applied to 8L0~b gives zero because of the fact that this quantity is a divergence. With these simplifications (4.12-1), (4.12-2), etc., reduce to ul = uit
e,~. 8Lod# =
(4.10)
8Co43.y~7--qS~, (gx t3
axe,
-
(4.14-1)
Go6Loul,
u, = u i - GoSLou2-
(4.14-2)
Go~L2uo,
u3 = u ~ - G o S L o u 3 - G o S L 2 u l - G i 8 L 2 u o
and
6L~4, = at,,/,.
Expanding the integral equation (3.9) in powers of w and equating coefficients of equal powers we obtain the set of equations i
Uo = Uo - G o S L o u o ,
(4.12-0)
ul = uil - G o 8 L o u l - G i S L o u o ,
(4.12-1)
i
u2 = u2 - G o S L o u 2 - G o 8 L 2 u o -
GI ~ L o u l
(4.12-2)
- G26Louo, U3 = U ~ - G o 8 L o u 3 - Gl8L2uo-
GoSL2ul - Gi 6Lou2 G28Loul - G38Louo,
(4.12-3) u4 = u i 4 - G o S L o u 4 -
GoSL2u~_- Gi 8Lou3
- G i 8L2Ul - G 2 8 L o u 2 - G 2 8 L e u o -
G38LoUl - G48Louo,
(4.12-4)
etc. It is understood in all the equations of this section that r e D~. There are several simplifications that can be made on the above results. First of all, it is easily seen that the solution of the 0th order integral equation is simply
(4.14-3)
- G28Lout,
(4.1 I)
u4 = u i 4 - G o S L o u 4 -
GofL2u2-
Gi 8Lzul
- G28Lou~ - G28L2uo - G38Loul,
(4.14-4) etc. A few remarks are in order concerning the meaning of the first two members of the above sequence of equations• In (4.14-1) the first-order incident displacement field u~ represents a uniform applied strain and u~ represents the total (i.e., u~ + u~) displacement field arising from the quasi-static response to this applied strain. In (4.14-2), u2 would represent a similar quasi-static response to an applied strain field with a uniform gradient if the term G o b L z u o were absent. This last term gives the inertial effect of a uniform displacement field with a periodic time dependence. We turn now to a discussion of the scattering amplitude. This tensor function can also be expanded in a power series in oJ as follows oc
a ( e ' , e ~• w)
Z a.(e,
=
n
"
~
e ) w .
(4.15)
"
0
From (3.14) we obtain the following expressions for the coefficients
(4.13)
-Ao
• a = Ho6Louo,
(4.16-0)
since according to (4.10) 8Lo applied to a constant operand gives a vanishing result. Therefore all terms involving the combination 8 L o u o must vanish. Furthermore, we can show that all terms
-Ai
• a = HoSLoul + HtSLouo,
(4.16-1)
-A2
" a = Ho8Lou2 + HoSL2uo + Hi8Loul
Uo= u~ = const.
+ H28Louo,
(4.16-2)
J.M. Richardson / Scattering from symmetric inhomogeneities - A 3 • a = H o 6 L o u 3 + HoBL2ul + H i 6Lou2 + H i 6L2uo + H2~Loul + Ha6Louo,
(4.16-3) - A 4 " a = Hot~Lou4 + not~L2u2 + H I t~tou 3 + Hi ~L2ul + H28Lou2 + Hz6L2uo + H36Loul + H46Louo,
(4.16-4)
- A s • a = Ho6Lou5 + H o S L 2 n 3 + HiSLon4 + H I (~L2u 2 + H2~Lou 3 + H 2 ~ L 2 u I
331
corresponds to the low frequency scattering theory of Gubernatis et al. [5]. It follows generally from (3.20) and specifically from the above results that A, is real when n is even and imaginary when n is odd. Let us consider the behavior of the phase shift y as a power series in the frequency. According to the definition (3.15) y = A r g b. A. a-= I m l o g b. A . a .
(4.18)
Letting
+ H 3 ~ L o u 2 + H3($L2uo + HM$Lou I
b.A,.a=B,, + H56Louo,
etc. Again, there are several simplifications that can be introduced into the above results. As before, all terms involving the combination 6Louo must vanish. Since the operator Ho, aside from a constant tensor factor, is an integration on r over the domain D~, it follows that all terms involving the combination Ho6Lo must vanish because of the divergence nature of 6Lo. With these simplifications (4.16-0), (4.16-1) etc. reduce to -Ao" a=0,
(4.17-0)
-AI" a=0,
(4.17-1)
- A 2 • a = HoSL2uo + H j S L o u l ,
(4.17-2)
- A 3 • a = HotSL2u j + Hi ~SLou2 + HI tSL2uo + H26Loul,
=iB,,
n odd,
(4.19)
where the B, = B , ( e S, b; e ~, a) are real, we obtain y = Im log(B2w 2 +iB3to 3 +
B4to 4 • • .)
= Arg B2 + I m log(1 +iB21B3to + BftB4to 2 +...)
= Arg B2 + B21B3to + ( B 2 1 B5 - B22B3B4 - ~Bf3B3)w 3 + O ( w 5).
(4.20)
Since B2 is real, the quantity Arg B2 is equal to 0 or ~r depending on whether B2 is positive or negative.
5. Some consequences of inversion symmetry
(4.17- 3)
- A 4 • a = HoSL2u2 + HI tSLou3+ Hi 6L2u~ + H28Lou2 + H2~SL2uo + H3t~LoUl ,
(4.17-4) - A 5 • a = Ho6L2u3 + Hi tSLou4 + H1 t~L2n2 + H26Lou3 + H z S L 2 u l + H3t~Lon2 + H38L2uo + H4BLoul.
neven;
(4.16-5)
(4.17-5)
It is to be noted that A , - a depends at most upon u,_l and lower orders. Since the vector a is arbitrary it follows from (4.17-0) and (4.17-1) that Ao and At, must vanish. We note that (4.17-2)
The results derived in previous sections are valid for a general localized inhomogeneity regardless of the presence or absence of symmetry properties. In the present section we consider some of the consequences of inversion symmetry. In particular, we consider only the consequences relative to the possible vanishing of some of the coefficients A,. A more comprehensive analysis of the consequences of symmetry will be given in a later communication. The inversion symmetry of the scatterer is represented by the requirement that the material property deviations 6p and 6 C ~ be invariant to
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
332
inversion, i.e.,
Application of P+ to (4.14-3) yields
R6p(r) = 8p(-r) = 6p(r)
(5.1)
R6C~t3r~(r ) = 6 C , ~ ( -
(5.2)
P+u3 = - G o 6 L o P + u 3 -
(5.7-3a)
Gi6L2uo,
and r) = 6 C , , ~ ( r ) .
These relations imply that R commutes with 6L, that is
which implies that in general P+u3 ¢ O. Furthermore, with P replacing P+ we obtain P_u3 = u~ - Go6LoP_u3 - Go6L2ul
-
G26Lout,
(5.3)
(5.7-3b)
and hence R commutes with 6Lo and 6L2. It is easy to show from (2.13) that G and R commute, i.e.,
which, again, implies that P u 3 ~ 0. Thus, in general, u3 has parts of both odd and even parities. It can be shown that the same statement is true of all higher order un's. To summarize, if the scatterer has inversion symmetry, Uo and u2 have even parity, u~ has odd parity, but u~, u4. . . . in general have parts with both even and odd parities. We now must examine quantities of the form H m 6 L , up where, of course, n = 0 or 2. From (4.9) we see that H,~4~ is proportional to the integral
Rt~L = 6 L R ,
(5.4)
RG = GR.
This follows from the fact that R commutes with P~ and Pt and that in the definitions of K~ and Kt (e.g., (2.14)) r and r' enter only through the quantity J r - r ' I. It then follows trivially that R commutes with the G,, the coefficients in the frequency expansion of G. Thus, we can say that operators t~Lo, 6L2 and G,, n = 0, 1 , . . . , preserve parity. The coefficients u~, in the frequency expansion of the incident field have even or odd parity depending on whether n is even or odd, i.e.,
Rui. = (-1)"ui.
(5.5)
or, equivalently, P+ui, = 0 ,
n odd,
P_u~, = 0 ,
n even.
(5.6)
We turn now to the consideration of the consequences of inversion symmetry in the sequence of integral equations (4.14-1), (4.14-2), etc. Before doing this we note that, since Uo= u~, it is obvious that Uo must have even parity. Application of P+ to (4.14-1) gives P+ul = -- G o f L o P + u l ,
(5.7-1 )
from which we infer that P+u~ = 0, i.e., u~ has odd parity. Application of P to (4.14-2) yields P - u2 = - G o 6 L o P _ u2,
(5.7-2)
which implies that P - u 2 = 0, i.e., u2 has even parity.
J,,--- f
d 3 r ' ( - i e ~- r')md~(r').
(5.8)
It is obvious that J,, will vanish if the entire integrand has odd parity because the contribution at r' will cancel the contribution at - r ' . Thus, since the factor ( - i e s. r) m has even parity if m is even and odd parity if m is odd, it follows that J,, and hence /-/,nO will vanish if m is even and ~ has odd parity and vice versa. The above results imply that in the sequence (4.17-0), (4.17-1), etc. the r.h. side of the third member, i.e., (4.17-3), vanishes because all of the terms involve integrands of odd parity and thus we obtain A3 = 0. The r.h. sides of the even numbers are of course nonvanishing because none of the terms involve integrands of odd parity. The r.h. sides of the higher order odd numbers, i.e., fifth and higher, are in general nonvanishing because each of the quantities u3, us, etc. contains parts of both even and odd parity. Thus, to summarize, Ao and At vanish because of general considerations, A3 vanishes because of inversion symmetry, but the higher order An do not vanish in general.
J.M. Richardson / Scatteringfrom symmetricinhomogeneities The phase shift 3,, previously given by (4.20), now takes the form
operators defined by
x,(r) = D,~b(r) = f
y = Im log(B2to 2 + Bato 4 +iBsto 5 +- • .)
n = 1. . . . , N,
+ i B f l B s t o 3 • . .) (5.9)
We obtain the following frequency derivative o f the phase shift (5.10)
which is proportional to the distance between the effective scattering center and the geometrical center. It is clear that this distance approaches 0 quadratically as to-* 0. In other words, for a scatterer with inversion s y m m e t r y the effective scattering center is the geometrical center in the limit o f zero frequency. The results holds independently o f the p r o p a g a t i o n directions and polarizations o f incident and scattered waves. It is clear that the low-frequency analysis o f the phase shifts obtained from three i n d e p e n d e n t scattering measurements can yield a unique determination of the position o f the geometrical center o f the scatterer with inversion symmetry [6].
(A1)
where ~b(r) is an arbitrary scalar operand. In Sections 2 and 3 we used a vector o p e r a n d but here a scalar one suffices. That the operators D , are translation invariant is easily d e m o n s t r a t e d by making the substitutions
x.(r) ~ x ' ( r ) 0_fly= 3B2~ Bsto 2 + O(to4), ato
dar ' D , ( r - r')~b(r'),
J D~
-- Arg B 2 + I m log(l +Bf~B4to 2
-- Arg B2 + B21Bsto 3 +O(tos).
333
=
x . ( r - a),
~b(r) ~ 0'(r) = 0 ( r - a)
(A2)
in (A1) with the result that the primed functions X" and 0 ' are related in the same way as the u m p r i m e d functions X, and 0. We turn now to the consideration o f the commuo tivity o f two translation-invariant operators, say D~ and D2. We obtain
(D~ D2 - D2Di)~b = f
J D~
dar ' A12(r -- r')~b(r'), (A3)
where Al2(r-- r") = fo~ d3 r'[Dj(r- r')D2(r'- r")
- D2(r - r')Dl(r'- r")]. Acknowledgement The a u t h o r is indebted to R.K. Elsley for previously advancing the hypothesis that the results o f this p a p e r are valid for a spherical void and to E.R. C o h e n for helpful comments.
Appendix A. Commutation of translation invariant operators
A sufficient condition for the existence o f Ai2 is that the integrals ID
d 3 r , Din(r- r t)Dn(r t - r")
(A5)
exist for all r and r" in Doo and for the combinations m = 1, n = 2 and m = 2, n = 1. In turn, a sufficient condition for the existence o f the integrals immediately above is that
f Here we confront the problem o f proving the commutivity of translation-invariant operators in the mathematical context o f scattering theory. To be more explicit let us consider examples o f such
(A4)
d3rD2,(r)
m = 1,2,
(A6)
DR
i.e., D~(r)c L2(Doo), n = 1, 2, i.e., the functions D,(r) are quadratically integrable in the infinite d o m a i n Do. We wish to emphasize that it is also
J.M. Richardson / Scatteringfrom symmetric inhomogeneities
334
assumed that t)(r)c L2(D~,) which in combination with (A6) provides a sufficient condition for the existence of D~ D2t) and D2D~ t). We now show that under the above assumptions we obtain the relation At2(r)= 0, r c D~. Consider the integral
l12(r-- r") = f d3r ' D l ( r - r')D2(r'- r").
(A7)
Letting r' = r + r" - r'" we obtain r - r' = r"' - r" and r ' - r " = r - r'". With r'" replacing r' as the variable of integration we obtain Ii2(r - r") = f d3r ''' Dffr'"- r")D2(r- r"')
= f d3r ''' D2(r- r'")Dl(r'"- r")
= 121(r -- r"),
(A8)
from which it follows that Ai2(r-- r") = lt2(r-- r") -- I,_l(r-- r") = 0.
(A9)
Thus, if (but not only if) D 6 r ) , D2(r) and t)(r) are quadratically integrable, then
Dt D2tb - D2Dt t) = 0
(Al0)
DID2
(A1 I)
and -
D2DI
=
0.
However, as we will see, the quadratic integrability condition is not satisfied by any of the translation-invariant operators of interest to us here and thus more subtle arguments must be invoked to prove commutivity. Examples of such operators are V, V 2, PI, Pt, Ki and Kt. A conceptually simple approach is to substitute each of these operators by modified operators that are quadratically integrable in the sense that the corresponding D(r) functions are quadratically integrable. It then requires that the modified operators approach the original ones as some small-valued parameter e tends to zero. In terms of the general purpose scalar
operand t)(r), one could make the substitutions; Vt)~fdr'V~exp( t,~Tre)-/-
\
]r--r'].~ 4e / t ) ( r ' ) ' (AI2)
V 20~-1
f dr'eXp(l-re_lr-r'l)~(r),
(AI3)
1 [ e x p ( ( i k , - e ) l r - r'l) Kit) -, -4-~ a dr ]r - r'l t)(r'), (AI4) with a similar expression for K,. The modified forms of the operators P~ and P, are to be constructed by appropriate combinations of the modified forms of V and V -2 given by (AI2) and (A13). The reader can easily verify that the D(r) functions corresponding to the r.h. sides of (A 12), (AI 3), and (AI4) are quadratically integrable for e > 0. Furthermore, the reader can also readily verify that the D(r) functions corresponding to the original operators on the l.h. sides of (AI2), (AI3), and (A14) are regained in the limit e ~ 0 . Clearly, the same results apply to P~, P,, and K~. Thus the manipulations in Sections 2 and 3, involving the assumed commutivity of translation invariant operators, can be justified on a rigorous basis by substituting them by modified operators with the quadratic integrability property and then taking the limit e --, 0.
Appendix B. Derivation of the elastodynamic green function Here we present a detailed derivation of Green's operator G given by (2.13) in the text. Furthermore, we present various alternative forms of G in the usual explicit representation. We start by noting that the operator L defined by (2.3) can be rewritten in the form L = (A +2/~)L~PI +/xLtPt,
(B1)
where P~ and Pt are respectively the longitudinal and transverse projection operators defined by (2.5)-(2.8) and where the operators L~ and L, are
J.M. Richardson / Scatteringfrom symmetric inhomogeneities defined by
where
L~tb = (V 2 + k~)4~
(a2)
Lt~
(B3)
and =
(~72 -]-kt2)~
The longitudinal and transverse wave numbers k~ and k, are defined by (2.15)-(2.18). The correctness of (B1) can be verified by direct substitution. It is to be noted that all of the operators on the right hand side of (BI) are mutually commuting and hence the products can be rewritten in the reverse order if desired. The verification of the validity of the form of Green's operator given by (2.13) or (2.19), namely
G =
335
1
1
KIP I + - - K t P t y+2# ~x 1
=y+2~
1
PtKI +-- PtKt, /x
(B4)
now becomes an almost trivial exercise. We obtain the desired result
1 R l exp(ikiR), Kj( R ) = 4---~
(B9)
in which
R=lr-r'l.
(BIO)
The definition for K t is the same as K~ except that t replaces 1 everywhere. In terms of the more explicit notation G can be expressed as the following form of Green's function.
G ( r - r') -
1 -
-
y +2~t
VVV-2KI(R)
1
2
+ - - ( I - VVV- )K~(R).
(Bll)
It is sometimes useful to rewrite the above expression in a form from which the operator V 2 is eliminated. With a trivial rearrangement we obtain
G(r - r') =
1
pro
2[k~VVV-2KI(R)
+ k2t(l - VVV-2)Kt(R)]
L G = [(Y + 2/tz)L,P, +/.tLtPt] -
1
pw
2[Ik~K,(R)
+VVV 2(k2K~(R)- k2tKt(R))].
= L1PIKI + LtPtKt
(Bl2) From (B6) we obtain the obvious result
= PILIKI + P, LtKt
LIK~- LtKt = 0, = P, + Pt = I,
(BI 3)
(B5) which corresponds, in more explicit notation, to
using the fact that K~ and K~ defined by (2.14) etc. are Green's operators for L~ and Lt, respectively, i.e.,
LiKi = LtKt = L
(B6)
We now proceed to rewrite Green's operator in terms of more explicit notation. We first consider the definitions
G#p(r) = I
d3r ' G(r - r') . dp(r')
(B7)
D~
and
(V2 + k ~ ) K , ( R ) - ( V 2 +k~)Kt(R)=O. From which we deduce the result
k~ K,( R) - k 2tKt( R) = - V 2 ( K , ( R ) - Kt( R)).
(BI5) Substitution of (Bl5) into (Bl2) yields the more familiar form [7]:
G ( r - r') =
1
pro -
K, rb(r) = f
d3 r ' Dx
K,(Ir- r'l)q,(r'),
(B8)
(B14)
2[Ik2,K,(R)
V V ( K , ( R ) - K,(R))],
(B16)
from which the inverse Laplacian operator V -2 is absent.
J.M. Richardson/Scattering from symmetric inhomogeneities
336
References [1] J.E. Gubernatis, E. Domany, J.A. Krumhansl, and M. Huberman, "The fundamental theory of elastic waves scattering by defects in elastic materials--Integral equation methods for application to ultrasonic flaw detection",
Materials Science Center, Cornell University, Tech. Rpt., 2654 (1975), Also ERDA Tech. Rpt. C00-3161-42 (1975). [2] J.E. Gubernatis, E. Domany, and J.A. Krumhansl, "Formal aspects of the theory of the scattering of ultrasound by flaws", J. AppL Phys. 48, 2804 (1977). [3] J.E. Gubernatis, "Long wave approximations for the scattering of elastic waves from flaws with applications to ellipsoidal voids and inclusions", J. Appl. Phys. 50, 4046 (1979).
[4] G. Dassios, "'Convergent low frequency expansions for penetrable scatterers", J. Math. Phys. 18 126-137 (1977). [5] J.E. Gubernatis, J.A. Krumhansl, R.M. Thomson, "Interpretation of elastic wave scattering theory for analysis and design of flaw characterization experiments: I. Long wavelength limit", Los Alamos Scientific Laboratory Report, LA-UR-76-2546 (1976). [6] J.M. Richardson, "'Low frequency behavior of amplitude and phase shift in elastic wave scattering", paper presented at the NBS Symposium on Materials Characterization, Gaithersburg, Maryland, June 7-9, 1978. [7] P.M. Morse, in Handbook of Physics, edited by E.V. Condon and H. Odishaw (McGraw-Hill, New York, 1967).