+CF,,; where

C-----((p")2);

F,,=fo"-2f;

p~-p/pc;

l=r

Bn=B,+iBi;

F~=fo~+f; 2f--f~-fo~-h"+fo";

B~=Bi;

s--s,~-so~-sV'+so"';

R== ( R p + i e i ~ ) f o " + (e4~+ iRi'Z)fo~+ 2(ea~+ iR4~)fIn+ (R4~+Rs(z)f~+ +Rs=(f~2-f~).; la~f,~-~hn+f~=; ~=--qn/q~; p = 0 , 1 , 2 .

Here, t h e c o e f f i c i e n t s B i n t r o d u c e d in ( 2 . 9 ) :

R~ and Bi d e t e r m i n e c e r t a i n

c o n v o l u t i o n s of t h e t e n s o r s A and ~

,

.

~n

,

~

ikpq

.

Rpqrs--R~(ZSpqOrs+ 2R2aJpqrs"FRsa(6pqn*rs+tt*pq 5r,s ) +4R4tZn*p)(r6s)(q-+-R5 n pqrs, Kpqrs~n' ijmnj~rs , n

"~

*

A

ihpq

Rpqrs+ iRpqr~---- rt kl,'~ars ,

(2.16)

Bi~m=B~6o6~z+ 2 B ~ l ~ ,

a

The i n t e g r a l s f p , s~, and hp a r e e x p r e s s e d t h r o u g h t h e c o n v o l u t i o n of t h e f u n c t i o n s G~c(k) in t h e k - s p a c e by t h e r e l a t i o n s =__!_l~ q2 f~a(x)-- 8n a ~(x-y)Q~(y)t=~dy; Q = ( y ) - _g2__ _ . q=2

T(k) and

'

s, ~ (x) = ~

j ~ ( x - y) Q: (y)xyt2,+ ' dy; y - - ak;

( 2. 1 7 )

1~ j'q~(x_y)t2Pdy; t = n - n , . h~(x) =-g~n Subsequent c a l c u l a t i o n

of t h e s c a t t e r i n g

coefficients

~a and t h e phase 5~ and t h e group ~

velocities entails the direct solution of dispersion relation (1.8), which by virtue of (2.13) and (2.14) reduces to the equation L~ = 0 or

(x = n, "~.

(2.18)

M= (x~) = M~, (x=).

(2.19)

7*~x~+ 2 S * ~ - p *~qe~ = O;

We find from this that

x~=qJ[l+M~(x=)];

Here, Ms(x) is the Fourier transform of the kernel of a certain operator playing the role of the mass operator in the theory of the scattering of electromagnetic waves [3, i0]. The roots x~ in Eq. (2.19), in the general case complex, will be written in the form xa-----x~(~)+fx~(~). 603

Equation (2.19) may have several roots in the upper half-plane for each =. Among these roots, only the root with the smallest imaginary part is important [3]. The Burr4 approximation used in (2.8) presumes the. presence of.a small parameter in the series expansion of the Neumann operator from (2.6) and sufficiently rapid convergence of this series. Without studying the conditions of its convergence in detail, we will use the fol~owing inequality as a sufficient condition

[M=(x=) I << 1'

(2.20)

which is similar to the corresponding inequalities in [3, 5, 8-I0]. The following approximation [i-6, 12] is often used in approximations which ignore spatial dispersion

M=(x~) =M~ (q~).

(2.21)

In accordance with (1.7) and (2.14), the real x~ I) and imaginary x~ 2) parts of the dimensionless wave number x=, being the solution of dispersion relation (2.18), determine the sought parameters of the averaged wave propagating in an inhomogeneous medium. Similarly to (2.12), we obtain the following for the dimensionless phase velocity v~

v*~ q ~ . 0"~----~ c= = x~O}'

am q~c= x~{ll . v*~ . . . v*~

(2.22)

The dimensionless group velocity is determined by the relations [2] e*=~ c= = [--d~--q=

; c*==

=c=

~

(2.23)

Taking into account the smallness of the quantity M s in accordance with (2.20), we find from (2.19) that

x~=qm ( 1+-~1M=- -~I M l) ' Inserting

(2.24)

M~---M~t,).+iM~t=,.

(2.24)

i n t o ( 2 . 2 2 ) and ( 2 . 2 3 ) , we o b t a i n I+-~-MJ I)

F i n a l l y , in a c c o r d a n c e w i t h ( 1 . 1 ) , mensionless scattering coefficient

g*==

( 1 . 7 ) , and ( 2 . 1 4 ) , ~a ~ aT~

a=21mx==ix='2)=q=MJ~)

+ -

(2~

we w i l l have t h e f o l l o w i n g f o r t h e d i 1

[1--~M="']

.

(2.26)

Equations (2.25) and (2.26) make it possible to use the Burr4 approximation to calculate the scattering coefficients for longitudinal and transverse waves, as well as their phase and group velocities. However, doing this requires information on the coordinate and tensor parts of all of the correlation functions from (2.9). Below, as the inhomogeneous medium, we examine a mechanical mixture of isotropic components. Such a medium can serve as a satisfactory model of a composite in which the reinforcing elements are either spherical on the average or are randomly oriented inclusions of arbitrary form. Both cases lead to the same results, since the approximation being examined, based on the description of an inhomogeneous medium by means of only the first two moments of the material characteristics, is insensitive to the form of randomly oriented inclusions. The correlation tensors A and B from (2.9) for a mechanical mixture of isotropic components have the form [7]

~jhz ~ , x ,,y "5. ,, Apqrs=Dx6ij6~t6pq6rs+ 2D~,~(bijbktlpqr~+ li.iht6pqfr~) +4D~lijhtlpqrs; D x , u--~ Bi~m= Dx$6i~6~t+ 2D ~,~lijht.

D~--~-Dxx; . ,

From h e r e , we o b t a i n t h e f o l l o w i n g in a c c o r d a n c e w i t h ( 2 . 1 5 ) and ( 2 . 1 6 ) f o r n o n t r i v i a l f i c i e n t s R~ and B~ Rl~=D~;

Ra~=2Dx'";

Rs~=8R4~=-2Rsx=4D~;

coef(2.27)

Now we proceed to take into account information on the coordinate relation q)(r) of correlation functions (2.9). It is evident from (2.18) that calculation of the characteristics of an

604

ultrasonic wave requires only the Fourier transform of the function ~(r). Below, we will examine a completely disordered, randomly nonuniform medium not possessing short- or longrange order. 3. We will examine an ideally disordered, randomly nonuniform medium possessing neither short- nor long-range order. The correlation function of such a medium will have the form [i]

~(r) =exp (--r/a). The F o u r i e r t r a n s f o r m

of t h i s

function

(3.1)

is e q u a l to [7] 8~

~(y)=

(1 +g2) 2

(3.2)

Insertion of (3.2) into (2.17) leads to the following results:

ho=l;

hp+~=(l+x-2)~p; p = O , l ; ~o= l - x - ~ a r c t g x ; g~= I/2(3h~-1); fo~ -

f:'=[@-t

q2 ; [l==fo~+ho--h~+f=; x~+ (1--iq=) "~ (l+x")(l+x~+3q~2) (ho-h~)-+ ( l + x ~ + q ~ ) " 2x~q~ 2x~q=~

(3.3)

2q= x=~==---1 +iqc~ + i --3"7 '

(3.4) ~+

1 +x~+q= 2 arctg x x 1-- iq-------~;

(

x) ;

So==q,=(i+q=)fo=+q= ~ 1 - x - ' a r c t g

l--iq~

(3.5)

s ~ = q a ( i +q=)f@+ q=2+ 3/4( l + x ~) (ho-h~) + 3/4( l + x2+q~2)~. The expressions obtained here, together with the corresponding formulas in Part 2, make it possible to calculate the quantities ?=, ~ , and fi~ for any values of q~ satisfying restriction (2.20). However, the transcendental character of Eq. (2.19) does not allow us to do this in general form. Thus, below we examine the asymptotes q~ ~ i and q~ ~ I. We will examine the case of long waves, when q~ ~ i. It follows from (2.24) that the inequality x~ I) ~ 1 is also satisfied and that, in solving Eq. (2.19), we can restrict ourselves to approximation (2.21). Then in accordance with (2.24) we obtain

x~q~

[ l+yMa(q~)

(3.6)

] .

For the exponential coordinate relation of correlation functions (3.1), the calculation leads to the following expressions f o r M = =M=(~ (3.6):

M~(') = t/3q~ 2 (2 + 132)D_~ _ 2/15q~ 2 (3 + 2132)D~, 7 + 1/105D~ [ 14 (3 + 2[32) + q2 (45 + 4132~+28134)] ; M~(m=q~a[2/3(2+13a)D~ +4/15(3+ 2135)D~ ]

(3.7)

for transverse waves and M~m=

I/3qn2 (1 + 213-2)D-; -2/15qn215D--o7 + 4 ( 1 - [32)D~.y ] +

+d~+q,:d2+8/lO5D~[7~2+q,~2(7+132)];

7=lilt;

~=~/~c;

(3.8)

I~2dI---D~+4/3D~,~+4/BD~; lO(d2-dl)~-8/15(D~,~+6/TD~); M~ (2) = 2/3qn3 ( 1+ 213-3) DF + 2qn3dl + 16/15q,~a~-lD-~ for longitudinal waves. Equations (3.7) and (3.8) can be simplified if we use the approximation of constancy of the Poisson's ratio 1 --2132

2(~,+ ~t)

--

2(1--132 )

= const.

(3.9)

Then we will have

6~=5(2+13a)D F + 2(3+213~)D~ ; 605

5n = 5 ( 1+ 2[~-3 ) D7 + (8[~-1 + ] 5

-

-

40[~2 + 32[~4).D-~.

In the special case when the Poisson's ratio is equal to 1/3, we obtain the parameter ~ = 1/2 from (3.9). This makes it possible to once more simplify the expressions for 6~ and 6n:

6~= 1/8(85D~ +49D-~) ; 6~ =85D-P +23D-. p~ Comparing the expressions obtained here, we find that with Dfi = 0 we have the relation 6 z = 6n/8. It should be noted that in the approximation of Poisson's ratio constancy, the parameters s introduced by Eqs. (2.27) and (3.8) will coincide at a = T, n. Inserting (3.7) into (3.8) and inserting (2.25) into (2.26), we obtain

(2/15)&zq~z4; a=z, n;

~==

5~=5(2+[33)D~ +2(3+2[~)D-~;

(3.10)

6n=5(l+2[~-~)D~-+ 15d]+8[~-lD-;

for the scattering coefficient and

~*:= l--a(z-b~q~;

a~= l/15(3+ 2(J~)D'g ;

==~,n;

2b~= (2+~)D 7 -2/5(3+ 2~2)D~;+ 1/3507(45+ 4~2+28~4); 2an=dl+8/15~2D-~;

2bn= (l+2p-2)D~ -2/515D~, T +4(1

_ ~2) D ~ , ~

(3.11)

]+3de+8/35(7+~2)D -

f o r t h e group v e l o c i t y , r e s p e c t i v e l y . E q u a t i o n s ( 3 . 1 0 ) and ( 3 . 1 1 ) c o i n c i d e w i t h t h e w e l l known a p p r o x i m a t i o n s f o r t h e longwave r e g i o n [1, 12, 13]. We w i l l now examine t h e case of s h o r t waves, when qa >> 1 and, by v i r t u e of ( 2 . 2 4 ) , x(z) >> 1. A f t e r p e r f o r m i n g some s i m p l e c a l c u l a t i o n s f o r t h e f u n c t i o n s g i v e n by Eqs. ( 3 . 3 ) ( 3 . 5 ) , we o b t a i n t h e a s y m p t o t i c e x p r e s s i o n s ~]=1---4x

2x'

~0 = 1

IL, l"q~,-';

hp+~=~v;

'

Ifp=-/o=l"'q'~-';

f~==f~

(3.12)

Is''=-s~

s#=so==q.=(i+q=)~o=+q=2; l,~r q, 2 ]+q~m=-2iq="

p=0,1;

1+~o=; p=O, 1, 2;

(3.13)

ms(x) =q=M~(x).

Changing over from M to m, we rewrite (2.21), (2.20), and (2.24) in the form

x2=q(q+m); Here and below, f o r s i m p l i c i t y

x=q+ym

I-TM

;

we omit t h e s u p e r s c r i p t

m~-mO)+imm);

ImI<
a = ~, n, w h i l e t h e p a r a m e t e r s of

the comparison medium are determined from the equalities <~'>=<7'~>=0. cumbersome calculations, we obtain the following expression for m

m=

q (D7[o+ OZ ~) - m[oD~F

(3.14)

; v~l--p.

After some simple but

(3.15)

1 - (l+fo)m T Inserting f0 from (3.13) into the above equation and ignoring D[ compared to unity, we arrive at the equation for m 1

m 2 - 2m ( i - ~--q + q D i ? ) -- [q237+ (1-2iq)DT.c] =0, X the solution of which yields 1

1

2

Here, we considered the inequalities

D72~DTDT<
606

(3.17)

]12<~; x----V; y----~" and t h e s m a l l n e s s of t h e f l u c t u a t i o n s . Using n o t a t i o n

(3.14),

f o r t h e sum o f t h e r o o t s we w r i t e ?++y_=2;

m+O~+m-~ll=2qD77--q-~.

(3.18)

S i n c e , as n o t e d a b o v e , t h e s o l u t i o n w i t h t h e l o w e s t ~ i s i m p o r t a n t [3] ( b e l o w , we w i l l examine only one solution, with the subscript "+" or "-" omitted), the first equation of (3.18) allows us to find the maximum value of the scattering coefficient for this solution. It is esily seen that the limiting value of 9 is equal to unity. The second equation of (3.18) turns out to be useful in examining the limiting value of the vawe number x(I) for longitudinal and transverse waves. We will study two asymptotes of solution (3.16): ~=--qD'~<<1 case, we find from (3.16) that

m=qDiT+lq 2 ( l + ~ -1q ) From h e r e , and

in accordance with (2.15),

(2.16),

andS>>1

.

In the first

q=--qDT~12.

;

and ( 3 . 1 4 ) ,

we o b t a i n t h e f o l l o w i n g f o r x/~}, 0 . G ,

x{O=q+l/imO)=q[l+l/8(DT+4Dzj)] ; ]m[<
O,=G=l_l/8(DT+4D),7); Equations 13].

1

(3.19)

~=_fqi;

(3.19) coincide with the well-known approximations

In the second case, by proceeding

for the shortwave region [i, 12,

in a similar manner we find from (3.16) that

Ill

m=i-- 2q--+qDz7+ which for x 0), v,, c , ,

q<
1>

and 9 gives

xO'=q (l+lDq'/2~'2/ '

? = ~ 5 * = g * = I - - 1 D K I/2'

(3.20)

q>>l, lml>>l. H e r e , in c a l c u l a t i n g x ( 1 ) , we c o n s i d e r e d i n e q u a l i t i e s (3.17). T h i s a l l o w e d us t o i g n o r e D[.7, compared t o D~. I t f o l l o w s from ( 3 . 2 0 ) t h a t a t q + ~ , ~ , = ~, = ~ = 1. 4. One of t h e p r o b l e m s t h a t a r i s e s when s o l v i n g p r o b l e m s of t h e t h e o r y o f wave p r o p a g a t i o n i n inhomogeneous media i s a l l o w i n g f o r s p a t i a l d i s p e r s i o n [3, 10]. In connection with this, it is often necessary to introduce limtations on different parameters that figure in the problem. As one such limitation we examine the inequality [i0]

N~__q2 I~dM

~= <<1,

(4.1)

the satisfaction of which allows us to ignore spatial dispersion in calculating the correlation contribution of M(x) in (2.19), assuming that M(x) = M(q). To find N in the shortwave approximation, when q >> I, we will use Eq. (3.15). In accordance with the latter, we have

Di-[o (x) + DT,I-

M(x) Inserting

(4.2)

into (4.1)

q2

9 f0(x)

I_DF.c[o(x ) '

and t a k i n g

(3.17)

=

( 4.2 )

l+xi_q,2_iiq

i n t o a c c o u n t , we o b t a i n

q~D~ N=

;

q>>l.

(4.3)

4 + qiD~-~ Inequality (4.1) takes the following form in the case of short waves

607

Ns = l/4qiD 7= 114#2<< I,.

(4.4)

where the subscript s d e n o t e s t h e r a n g e o f s h o r t w a v e s f o r w h i c h we c a n i g n o r e s p a t i a l d i s persion. A similar restriction, obtained in a different m a n n e r , was i n t r o d u c e d i n [3] i n the case of electromagnetic waves. Inequality ( 4 . 4 ) i s n e a r l y t h e same a s t h e c o n d i t i o n << 1 u s e d i n ( 3 . 1 9 ) and l e a d i n g t o t h e i n e q u a l i t y [ml << i . Thus, in accordance with the criterion (1.4), in the s-range (i<> i, inequality (4.1) is not satisfied and becomes the Opposite: N. =

q2

4 + q~D~ In the limit

q -* ~,

we f i n d

from (4.3)

>>1; 0>>1.

and ( 3 . 1 7 ) m~

N~--+ N = m - -

that v

>>1.

Thus, the problem of the propagation of waves in the u-range can be solved only with allowance for spatial dispersion, as was done in obtaining (3.20). The shortwave approximation (q ~ i) examined in [i, 12, 13] did not lead to results similar to (3.20) because of the use of the equality M(x) ~ M(q), which automatically narrows the interval of wave-number values in the u-range. The shortwave asymptote can be determined with the interval ql=~q=<<.q==, where the parameters q1~ and qa~ are determined by the conditions ?t(q,=)=?s(qm) and ~8(qia)=?u(qi=) Using Eqs. (3.10), (3.19), and (3.20), we obtain ql=2=-J~ - D 7 ~ - 1 ;

qi~2=DT-'(2-D~ln).

From here, we find a dimensionless parameter characterizing the width of the region of the asymptote ? ~ q=: ~ q2aJq]= = 2[D~ [6a/15 (2 - O~'12 ) ] ~12. It is evident that a decrease in the dispersion D 5 is accompanied by an increase in the width of the region in which the asymptote ? ~ q= is valid. Allowing for spatial dispersion makes it possible to obtain analytic expressions for ~, v,, and c, throughout the wave-number range. Here, the dependence of 7 on the wavelength can be represented in the form 7 ~ X-P, where 0~p~_4. This relation is substantiated by experimental results in [15], where it was found that ps = 4, Ps = 2, Pu = 0. It is also supported by the experimental results in [16, 17], where it was shown that the number p takes values of 4 and 0.5 in the case of scattering on microcracks in film polymers. Conclusions. i. The effective tensor operator of the wave equation of ideally disordered composites is an integrodifferential tensor operator containing terms with second derivatives with respect to the coordinates and time and mixed second-order derivatives. The kernels of the spectral operators on which the named differential operators act are components of tensors of the fourth, second, and third rank. Their dependence on the coordinates is determined by the specific form of the binary correlation functions of the material constants. 2. General formulas for the scattering coefficient and phase and group velocities of propagation of longitudinal and transverse waves can be obtained by using the Burr~ approximation, an exponential coordinate relation for the binary correlation functions, and the notion that the mean field in the composite, in the form of a monochromatic wave, is characterized by the effective wave vector. 3. Analytic expressions were obtained for one longwave and two shortwave asymptotes of the investigated parameters. It was shown that these asymptotes coincide with the literature data in the longwave case and are confirmed by experimental results in the shortwave case.

608

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2. 3. 4. 5. 6. 7.

8. 9. i0. ii. 12.

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A. A. Usov and T. D. Shermergor, "Dispersion of velocity and scattering of transverse ultrasonic waves in composite materials," Akust. Zh., 24, No. 2, 255-259 (1978). G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed., McGraw-Hill, New York (1967). L. G. Merkulov, "Study of the scattering of ultrasound in metals," Zh. Tekh. Fiz., 26, 64-75 (1956). V. M. Parfeev, I. V. Grushetskii, and V. P. Tamuzh, "Study of fatigue damages in a film polymer by the method of light scattering," Mekh. Kompozit. Mater., No. 5, 910-917

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609