Journal of Mathematical Sciences. Vol. 91. No. 2. 1998

P R O P A G A T I O N OF R A Y L E I G H W A V E S OF S V T Y P E I N TRANSVERSELY ISOTROPIC ELASTIC MEDIA Z. A. Y a n s o n

UDC 550.344.55

The asymptotics of high-frequency surface waves in elastic media is studied for a special case of anisotropy, namely, for transversely isotropic media (where the parameters of elasticity are invariant with respect to rotations about one of the coordinate azes). In the zeroth asymptotic approzimation, the slow Rayleigh waves (of S V type) under study are polarized in the plane of the normal section of the surface. The principal term of the asymptotics (which has the form of a space-time {caustic) expansion) is found, and calculations related to the necessity of introducing two additional faster waves with complez eikonals are carried out. The conditions on the elasticity parameters of the medium that insure the origination of the surface waves in question are obtained. Due to the specific structure of the elasticity tensor under consideration, the boundary of the medium is necessarily plane. For appropriate values of elastic parameters, the resulting formulas coincide with the corresponding expressions in the isotropic case. Bibliography: 8 titles.

This p a p e r considers asymptotic expansions of high-frequency surface Love waves in an anisotropic elastic medium, which were previously studied in the case of isotropic media in [1-4]. T h e constructions are carried out for the case of a transversely isotropic medium (also called the hexagonal system), which is a special type of s y m m e t r y of elastic media. In posing and solving the boundary-value problem for Love waves, we refer directly to the results obtained in [5-6], by using the technique of space-time (ST) ray expansions, which are commonly applied in constructing .asymptotics of nonstationary wave fields. However, when seeking the coefficients of the ray expansions in question, we use the results in [1], due to which the transition from an anisotropic (in our case, a transversely isotropic) medium to an isotropic one, as the elasticity p a r a m e t e r s specifying anisotropy assume appropriate values, becomes transparent. Physically, we aim at constructing asymptotics of surface waves near a surface E in the case where, in contrast to the case of SH Love waves, the principal term if0 of the asymptotics (which is a transverse SV wave on the surface) has a nonzero component z~~ on E, where g is the normal to the surface. We will refer to such waves as Rayleigh waves of SV type. In this paper, we construct the principal term of the asymptotics of Rayleigh waves (sought as an ST ray series involving the Airy functions) and carry out relevant computations, caused by the necessity of introducing two additional faster waves with complex eikonals. These computations d e m o n s t r a t e that the higher terms of the asymptotics of surface waves can be constructed recursively while successively determining the coefficients of ST ray expansions for the two faster waves, which a t t e n u a t e with depth. In addition, we also obtain relations for the entries of the m a t r i x of elasticity that are necessary for the surface waves under s t u d y to propagate in a transversely isotropic medium. 0. T h e desired solution ff is the displacement vector that satisfies the equations of m o t i o n for an elastic medium, which, in tensor notation, are of the form

(iff)j =

O2uj =

-p--~

0,

i,j,a=l,2,3,

(I)

where p(ql, q2, q3) is the density of the medium; t is time; ql, q2,q3 is a curvilinear coordinate system1; G i~ is the metric tensor; Vi is the symbol of the covariant derivative. The stress tensor aij and the strain tensor crs are interrelated as follows: aij = ao~,#GarG#~r

~r, = ~ l ( v ~ u ~ + V , u ~ ) ,

r,s,a,/3=l,2,3.

(ii)

i In what follows, we attach the coordinate system to the surface E of an elastic body, assuming that q3 = n and that r~ is the inward normal to E. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 278-293. Original article submitted November 20, 1995. 1072-3374/98/9102-2883520.00 9 1998 Plenum Publishing Corporation

2883

In (II), aij~ is the elasticity tensor of a medium with known symmetry properties, due to which the tensor can be represented as a 6 x 6 symmetric matrix with entries c~s (see [7-8]). We write the no-stress boundary conditions on the surface P. in the form

~rsjl ~ = aaj~,~a~'aVhU~ls = 0,

(III)

j = 1,2,3.

The wave field ff of a surface wave is sought in the form of an asymptotic expansion

~(r t) =

~(-p'/'m) +

ip-~"

k=0

~

(-p, ~)~e, I,

(IV)

k=0

where v(r) and v'(r) are the Airy function and its derivative, respectively, which decrease exponentially as r ~ +oo; p is a large parameter. By analogy with [1], [5], and [6], we require that rain = 7 ,

7=const>0-

(V)

Substituting (IV) into boundary conditions (III) and assuming (see [51) that

v(-p21S7)=O,

7=7,=q,,/p213,

v = l,2,... ,

(where -q~ are the roots of the Airy function v(r)), we see that the expressions 0.(,~) 3j

n=O:

i_-l/3fr~ P

'[[~3j~

ic,/~kXZ_a ~ Orn(Ak3~ l ~,~ 1 , _ 2/3 7 , ) e x p [ipl(ql,q2,t)] ID ) "1- 3 j . ~ p ~ ' ~ n t" / l +aaj(B - )},~=0 v ( p

j = 1,2,3;

k = 0, 1 , . . . ;

A - " = B - " - 0, r = 1,2,

(IIIa)

must vanish on E. In (Ilia), o3j _(k) are the coefficients of the expansion of a 3 / l , = 0 in powers of 1/ip; i ~ = Gaj ~-q~. ol The parameter 7~ determines the distance of the caustic of the ray field of the Rayleigh mode with number v from the surface 32,. In the sequel, we will omit the subscript v a n d regard 7 as the small parameter of the problem. The substitution of expansion (IV) into the equations of motion (I) yields a recurrent system for the coefficients ~k a n d / ~ k of the form

(iV, + ~ m ) d

~ + ~ 2 d , m g ~ = - [A?,X ~-' + ~ m g ~-1 + . ~ ? m g ~-' + s

.~2~7,md~ + (-~, + m ~ . . ) g ~ = _ [ ~ m X k-1 + Fr,g ~-~ + Lgk-~], k=0,1,2,...; where we denote 2

-

A-'=/~,'=0,

(VI)

r=1,2,

o. Cj, 2 ( G . d ) s --- ~,s~ (~'.~ + ~ ~ 1 6 2 - 2p o~ ~r (-Gd)j = a , . ~ X ~ ~ - p(~)~ cj,

-

(via) O~

(M~C)j - G c,i (V,~(aij~[ ~))C ~ + [aij,~iGh6VhC ~ + aij6~'GaiVoC ~] - 2p-~ {i

= G,,~ O~

Oq,~'

r

= G~. 077

OCj Ot

02~ p - ~ Cj, (VIb)

C" = G ' ~ G , .

Oq~,'

1. In the case of a transversely isotropic medium, the elasticity tensor aijc~fl (whose components are, in general, functions of the coordinates) reduces to a symmetric matrix (c,s), r, s = 1 , 2 , . . . ,6, which is specified by 5 independent elasticity parameters, whereas in the isotropic case the corresponding matrix is 2In (I)-(VI), (Vla), (Vlb), etc., summation over all repeating indices is implied. 2884

specified by just two independent parameters. Recall that the zero entries of the latter matrices coincide. These elasticity (material) parameters are invariant with respect to rotations about the axis of symmetry L, the existence of which is the characteristic property of transversely isotropic media. In the Cartesian coordinate system x~, x2, xa, where 0za is the axis of symmetry L, the nonzero entries of the matrix (c~,) are as follows: ell

=

C22 ,

C33 ,

C12 =

C21 ,

C13 =

C23 :

C31 =

C32,

C44 - - C 5 5 ,

C66 =

(Oil

--

C12)/2.

(VII)

The positive definiteness of the quadratic form (which represents the density of the potential energy of strain) 1

W = -aijaflcijcotfl > 0 2

implies, in particular, that c11>0, c~s>0, c44>0, c33>0, c11>c12, (c12+c66)c33>c~a.

(VIIa)

Note that the correspondence between the pairs of indices (i j) and (aft) of the components aij,~# and the indices r, s of the entries c~ is given by the following diagram [7]:

(11),

, 1, (22),

,2, (33),

,3, (23)=(32),

,4, (13)=(31),

,5, (12)=(21),

In order to compare the formulas derived below with those in the isotropic case, along with also use the notation c l l = c22 = A + 2#, c33 = A + 2p - ~, C12 =

C21 -~" A ,

C13 ~--- C23 =

A --~

C44 =

C66 - -

CSS =

,6. (VlIb)

(vii),

we will

~2 - - m ,

c l l - c12 2 - #'

(VIIc)

exploited in [7]. In concluding this section, we present the expression for the operator/Y~ from (Via) in terms of (VIIc), which is of importance for the sequel:

~7,d = (~ + , ) ( d v r 1 6 2 -

{

+

+ (,(vr

- ;r

+ ((# - ~ ) ( v r

~-

pr

(viii)

7)cov, +

+

~(p - -

+

where C = C~ + G f f , (C~, if) = 0, V{ = V3~ + ~3n, {a = ~ , Vs~ is a vector with components ~oql and Oq2, the coordinate system ql, q2, q3 = n in (VIII) is Cartesian, which implies that the surface E is a plane. w

A CONDITION OF THE SOLVABILITY OF THE HOMOGENEOUS SYSTEM FOR THE COEFFICIENTS d ~ AND ~o. THE EIKONAL EQUATION

1.1. Set k = 0 in (VI). This yields a homogeneous system for the coefficients .~0 and/~0. From the first equation of the system derived, taking into account (VI), we obtain that on the surface E,

(#t

=

o ----0.

( I.I)

For the second equation of this system to be solvable, it is necessary that

(2#,rod ~ d ~ ~=o = 0 7=0

(1.2a)

or, by virtue of (V) (0q l~ _ aq 2~ on E), that

(aaj~,~( d~

A~

= O.

(1.2) 2885

Relation (1.2) is a necessary condition for the origination of Love waves near the surface of an elastic body in the general anisotropic case. In the case of the transversely isotropic medium we are interested in, in view of (VII) and the relations a3j,cd#'

= a33,3a'

~

a',fl',j'

0,

= 1,2,

(1.3a)

on E Eq. (1.2) takes the form

A= j = - A ~ 1 7 6

5[r176

(1.3)

where 13 = ~o,' n ' Xo = Xo + A O . f f , ( d 0 , r T ) = 0 . Taking into account that in an isotropic medium from (1.1) and (V) it follows that 13 = 0 and ( A ~ = 0 on E, and that in our special case of anisotropy alI directions on the surface E are equivalent (because, without loss of generality, the plane (ql, q2) may be chosen orthogonal to the axis .L if the coordinates qJ are Cartesian) as in the isotropic case, we assume that (see [1])

A~

7s O,

(,4~

= 0,

7 = 0.

(1.4a)

Then, since c44 > 0 and c33 > 0 (see (VIIa)), from (1.3) it follows that

Ol

-13lz= ~-nlz;=O, 7=0.

(1.4)

The coefficient X ~ on surface E (for 7 = 0) is an eigenvector of problem (1.1), where A = pl 2 are eigenvalues of the matrix operator (aij,~#l~K'); in our case, the operator N~ is of the form (VIII) with ~ = l and d = ~0. By projecting (1.1) onto ~ and using (1.4a), (1.4), and (VIII), we obtain t h a t on E (3' = 0), C44(Vs/) 2 - -

pl 2 =

0

or

(#

--

m)(V,/) 2

-- pl 2 =

O.

(1.5)

Equation (1.5) is precisely the space-time eikonal equation (with eikonal lI,~=0,3"=0) for the ray field on E. As will be clear from subsequent calculations (see the Appendix), for the excitation of Rayleigh waves in the case considered ( A ~,~[,,=0 :~ 0), it is necessary that r~ > _ 0, i.e., c44 _< c66 in (1..5). Introducing three orthogonal vectors g, Vsl, and ( = that (~0, ()~=0 = 0 if ~ # O.

fix Vsl on E and projecting (1.1) onto (, we see

3,=0

As was indicated in [1, 6], equality (1.2) is directly related to the group velocity ~" of the Love waves, the components of which can be expressed in the form

gJ = GJla'"~#(rn~/4Ur)(rn~/4U#)r~

j

=

1,2,3,

(1.6)

p(-T),m'/2101 and ~r m-l'/4A0 :::t:rnll4B~ are the eikonals and amplitudes of waves approaching where r = 1 :F 2_,~312 3"'" the surface E (with minus sign) and reflected from it (with plus sign), respectively. On E we have m = 3', and, for j = 3 and 7 = 0, the numerator of the right-hand side of (1.6) coincides with the left-hand side of equality (1.2), which implies that the normal component of ] vanishes on E. In our case, the tangent components of ~"on E are =

gj,

#-~ -

Ol oqJ'

,,=0'

j'=

1,2.

(1.6a)

,=0

Taking into account the original formulation of the problem, we preserve superscripts and subscripts in several formulas (e.g., in (1.6)). 2886

1.2. Now we turn to the determination of the components of the vector/~0. From boundary conditions (III) and (IIIa), using (VII) and (1.3a), we derive that, for k = 0 and j = 3, [c'3(/~~

+

c33rn3A~

= 0,

Om m 3 - On'

(1.7)

whereas, for j = 1,2, oOI ~ o c44Bn0--~-+c44 3Ay]n=0 = 0 ,

j,

= 1,2.

(1.8)

By projecting relations (1.8) onto ( = ~ x Vfl, we obtain that (~0, ~'),,=0 = 0 as above, and thus on E ~=0

(~ = o)

(l.9a)

A~~ = A~ Now from (1.8) it is obvious that Bn0 n=0 = 0. '7----0

Consider the second vector equation of system (VI) for k = 0 and find its projections onto ( a n d Vfl. By using (VII) and (1.3a) as above, for 7 = 0 in the case r~ ~ 0 we obtain (/~0, (),~=0 = 0, and the projection of the above equation onto Vfl is

[(cxx - c44)(/~~ Thus, for the components (/3~

+ (c,3 + c44)rn3A ~ n=O = O. 7=8

i ==00 and Anln=0, 0 "t----0

(1.9)

the recurrent system (VI) and boundary conditions

(III) yield the homogeneous system (1.7), (1.9) with the determinant

D-=(Cll-C44)c33-Cls(C13+c44).

(1.10)

For an isotropic medium, D = 2#(A +/~) 2> 0. In our case, if Cll k c33 (and c44 < c66), one can prove that D > 0, provided that 0_
(if l" > 0 (l" < 0), then c13 < c12 (Cl3 2> c12)), where p" and • are defined by relations (Vile). Under assumptions (l.10a), for 7 = 0 system (1.7), (I.9) has only the zero solution, i.e., A~ -- 0, which contradicts the initialformulation of the problem. A remedy to this situation (see [I, 5]) is to supplement the field of the surface wave with exponentially attenuating body waves (in general, there are two of them) with complex eikonals. w

WAVES WITH COMPLEX EIKONALS. T H E TRANSPORT EQUATION FOR fl~0

2.1. To the field ff (IV) of a surface wave we add the summands oo d k l , 2

~1,2 =

aeipOt'2 E

(2.1)

(ip)k "

k=0

On the eikonals

01,2 of

these waves on E we impose the following conditions:

o~,2l~

= If=,

(2.2)

001 2 Im--~ ~ > O.

The substitution of (2.1) into Eq. (I) yields the recurrent relations . +. ~.o ~ , r fo, .~d~2

+ iOF~ 2 = o, d x,2 -~ = d -x,2 ' ~-0,

k=O, 1,2,.... 2887

The coefficients ffk1,2 are sought in the form rffl,1,2 = ffl),,2X',2 k o q'- ~ k1,2'

ffoo 1,2 = O,

k = 0 , 1 , "'" ,

(2.3)

and for k = 0 we obtain d~

= r176

pl~,,~l 2 -- 1,

(2.3a)

where C~, ~0 r = 1,2, are eigenvectors of the matrix A = (aijc, BOrO i r ), and X,- = P( O~)t2 are its eigenvatues. The polarization vectors ;q,2 are solutions of the linear system (2%, ;,,)j =

-

= O,

(2.4a)

d = 1, 2, 3,

and are uniquely determined by (2.2) and the normalization condition [t.~',,2 12 = 1. The functions ~5~ and Ck,,2 from (2.3) and (2.3a) are to be determined. ',2xk For the coefficients /ka3j ) of the expansion of the tensor craj(ffl,2) in powers of 1/ip on Y] we obtain the expressions

,. k

^f

00,.

k ~

-~-~

(~ra/)"=~ = a l a 3 J a ~ - - q g ( C r ) P + t r 3 j ( C r

}

) ~=oe i#(''''''O,

r=1,2,

k=O,1,...,

(2.4)

which must be added to expressions (IIIa). Let ffl from (2.1) be analogous to the longitudinal wave, so that (~,,Vfl),,=0 ~ 0 and ~ # 0, and let, in accordance with (2.2), I m - ~ 1~-0 > 0 (see the Appendix). We add (2.4) to expressions (IIIa) and equate the sum obtained to zero, setting k = 0 and 7 = 0. Taking into account the specific structure of the tensor aij,~ (namely, relations (VII) and (1.3a)), we see that on E

C13(gOvsl) q- ~ 0[C13(Xl 1

V s l) q- C33( 0 l ) n ( x l ) n ]

}

~-

--(C33T'I~3nOn)n=O'

(2.5a)

n=O

{ oO, ,o, ~Oqj----v+~~

o},~=o=0,

0--~+(0,),,(x,)j,] +msAj,

j':

1,2,

(2.5b)

where (0,), = "~n; K'l = (Kq)s + (x,),~ ~, (;'1), is a vector with components ( x , ) j , , j ' = 1,2; ~0 = SO0,

= -ipl/a/v'(-p2/aT). We regard (2.5a) and (1.9) as a linear inhomogeneous system for (/?~ a n d (~ ~ the righthand sides of which are proportional to A~ if 7 = 0. The determinant of this system is of the form b = -(c,1 -

;=o.

+

(2.6)

Since, for n = 0 and 7 = 0, we have the relation (see (2) in the Appendix) C33(Ol)rt(Xl)n

: --(C,3 -1- C44)(~1Vsl),

(2.6) takes the form 5 = c44(Cl, - c-)(~,Vfl)l,=0,7=0-

(2.6a)

Taking into account that (~iVsl) ~ O, c44 > 0, and (Cll - c44) = A + # + r~ > 0 (because r~ > 0, see (6) in the Appendix), we conclude that D ~ 0, which insures the solvability of the original problem. Projecting (2.5b) onto Vsl and using the equality (A~ = 0 (see (1.4a)), we express B~ 7-- 0

in terms of A ~ 7=0

(K'l, 0,,=0 = 0 (see the Appendix), we conclude that ( d ~ 0 , = o = 0, as above, i.e., (1.9a) remains valid. 3,=0

2888

---

Then, projecting (2.5b) onto ( = ff x V~l and taking into account the equality ~=0

However, in higher approximations in 7 (k = 0), it is possible to satisfy the boundary conditions (i.e., the no-stress condition on E) only by adding the wave if2 from (2.1), which has not yet been accounted for, to the field ff of the Rayleigh wave. To this end, expressions ([IIa) and (2.4) must be differentiated with respect to 7. In our case, the wave if2 must be of order 0(3'), i.e., ~~ = 0, whereas the polarization vector -~2 for 7 = 0 and n = 0 has the only nonzero component E'2[~- # 0 (see the Appendix). Set k = 0 and add the term G[-~~ a33~ ~--~(x2)~],,=0 to the left-hand side of Eq. (2.5a) and the terms ^a {(I.~0~5~ [c,3' ( x 2 ) , ~ _ i , +qc", OI (O2),(~2)j,]

}

,

j, = 1,2,

(2.6b)

n=0

to the left-hand sides of Eqs. (2.5b), respectively. Then equate the coefficients at @ in the relations obtained to zero. By differentiating Eqs. (VI) for k = 0 with respect to 3', we arrive at an inhomogeneous linear 0~t ~ t system for the coefficients (OoB-~~ Vfl)~--0) and --o~ I_-_0'~-~with the same determinant 5

(see (2.6), (2.6a)).

0, OA~ " . then the right-hand sides of this system are now proportional to A ~ and "O"f By projecting the relations deduced in this way (i.e., the boundary conditions) onto V~l and (, respec-

If n = 0 and 7

tively, we obtain a second pair of inhomogeneous linear equations for the coefficients ~

In=0 and ~ .r= 0

0"f

In=0

13,=0

because, for n = 0 and 3' = 0, the projections fOA~ ,-~-~, V ~t~ , and ( ~ 0- , ~) are found from system (VI). Thus, the original problem proves to be solvable, and the construction of the coefficients of the expansions of/~0 and @~ in powers of 7 ~ on N. can be carried out recursively, provided that A ~ and O'(A~) 03,~ , r : 1 , 2 , . . . ,r0, can be determined on ~ for any r = r0. 2.2. In order to find A 0n ]~, we need (see [1, 5]) to know the eikonal equation for the ray field on the surface ~. Such an equation is already deduced and is precisely Eq. (1.5) for the eikonal l(q ~, q2, t). In addition, we need to find the coefficients of the expansions of the functions l and m from (IV) in integer powers nit k near the surface IE. We present the results of computing the following most important coefficients: mlo -----m3

[.~=o = ~Oml.~--g, =

101 :

0'[

8--47n=o, and 12o . .21 0.021 , , 2.

Differentiating the first equation of system (IV) with respect to n and 3' and projecting it onto ~0 (i.e., onto rT), we derive M ( T n l 0 ) 3 = -- ~ '~33/~''~

/~=0'

(--~00)t

yml~

(2.7)

Mo

where G2 = ( o In ~,-,ao)n=0 (Re is the effective radius of curvature), ass = "css/p, c's3 = c33 100 = tin=~

(c,3+c44) 2 r _r ,

integration is performed along the ST ray, and ds = dr. As is easily seen, in (2.7) ~ss > (v-No "q

A+~ .... if 0 < :~< #, l ' > O, and ~ > O, where q = x+.+,a"

In order to compute the coefficient t2o, we project the first and second equations of system (VI) onto ]~0 and ~0, respectively. On subtraction of the projections obtained, we have

(2d, mdO, do) _

go) = o.

(2.8)

The differentiation of (2.8) with respect to n yields dml0 2t20

33

-

-,10

ml0

P(-t')--d-i- + iAOt2 IdoF [3-s176176162 i' = 1,2; a, fl = 1,2,3,

+ (- 1)(#'m g~ g~ (2.Sa)

/ i ~d~./_~_~ at is the derivative along the ST ray. In a transversely where a3a - (a3~30(A~176176176 ~ iX012 -r=0 and ~-.tj isotropic medium, a'aa = c33. Relation (2.8a), expressed in invariant form, which is independent of the 2889

type of anisotropy, is of importance in finding A ~ as will be seen below. Obviously, the computation of 120 reduces to the computation of the sum in brackets occurring in (2.8a). Its first summand involves the projection (~176 , Vfl) (for n = 0 and 7 = 0), which is determined from the first equation of system (VI) (by differentiating the projection of this equation onto V,l with respect to n). The second summand of the above sum is transformed with the help of (1.9) to the form c13 § C44)2 0 0 : m o[ ~ J 7---0 -r=0 k Cll - c44 .

.

.

(2.9a)

.

As a result for the coefficient 12o we obtain the expression

-1 2120

-

-

drnlo

7nloa33

(--lo0)t dt

'

(2.95)

where ~33 = ~'a3/P, 100 = ll.~-0, and ~'33 is defined in (2.7). In an isotropic medium, we also have F33 =/~. Consider the first equation of system (VI) for k = 1. On the surface E (for 7 = 0), the solvability condition for the vector equation obtained for ,~i is as follows:

(a?,d ~ X~

+ (#rag ~ d~

= o.

(2.9)

In view of (VIb), we obtain

{

O (a3j6#,(A~176

V~,(G~'i'ai,j~#(A~176

6) +

,§176

~ § N(-

o

= -t- ( N m B ~

}

(2.10)

p1,lA~ =) .=o = 0. 7----0

Introducing .4 2 = IX~ and using relation (2.8a) and expressions (1.6), (1.6a) for the group velocity f, we ml0 write (2.10) in the form { - ( / o o ) t [ d i v ~ ( ~ . 4 2 ) + ~ - . 4 2] + A 2 E 1 .

o=0,

(2.11)

) 3,----0

where div,(~A 2) = Vi,(gi'A2), i ' = 1, 2; l

tt

(2.12)

= if-~-lmlolac44(c,l L c44)

(---~V~l-~'~)2

;

D is of the form (1.10), and ~'33 is the same as in (2.7). In deriving relation (2.12), we have used (2.9a) and the equality (Nm/3 ~176 = (ca3A~176 and then, in accordance with boundary conditions (2.5a) ~'=0

~:0

and (2.5b), we have expressed B ~ in terms of A ~ Equation (2.11) is the transport equation for the amplitude d ~ of the surface wave on E. Integrating this equation along the ST ray, we derive the following modified formula of the ray method: M a~ n=O = x O ( o ~ I , ~ 2 ) ( ] y / I l O t ~ I / 2

~=o where

X~

c~2) is

e- - i ~ ( I m E ) d s

,

(2.13)

\ pZX ] . = o

v(t'q~'q2) v(s,c~,c~2) is the Jacobian of the transformation of the coordinates ql,q2,t into the ray coordinates s, al,a2; the integral in the phase of the exponent is taken along the ST ray with ds = dt.

2890

constant along the ST ray; &

-

We use expression (2.7) for m~o and also relations (2) and (3) (see the Appendix) to represent the integrand in (2.13) in the form

(-too)t

i

2(-/00)--------~I m S =

D2

nel(0,).l

-

(2.14)

where (-/00)t > 0, R~ > 0, c i , - c 4 4 > 0, D is of the form(1.10), (01),~ = - ~ is found for n = 0 and 7 = 0 from (3) (see the Appendix), and I(0,).1 = [Im(0x),l r 0 (in the case Im(0~)n = 0, the problem of constructing the asymptotic formulas has no solution). Using relation (2.14) and setting i ' = ff~ = ~ = 0 (see (VIIc)), it is easy to ascertain that (2.13) coincides with the expression for A ~ ,=0 in an isotropic medium (see [1]). -,t=0

In conclusion, we note that by differentiating the first equation of system (VI) for k = 1 with respect to 7, one can determine oa~ ov I.~-00 from the condition of solvability of the equation obtained. Thus, the boundary conditions are satisfied up to @ (see 2.1), and the process of expanding the coefficients ~ 0 , / ~ 0 , and ~01,2 in powers 7", s = 1, 2, "" " , proves to be recursive.

APPENDIX We describe a procedure for constructing (in any approximation in 3') both the eikonals 01,2 of expansions (2.1) and the polarization vectors ffl,2 of the eigenvectors (2.3a) on E. In accordance with (2.2), (2.4a), on E we have =

O,

(I)

O0 = K,~ + x,~K, V0 = V , l + 0,,7, 0, = ~nn"

(la)

In (1), ~ and 0 stand for ff~ and (o~), ,- = 1 , 2 . We project (1) onto if, make use of (VIII), and take into account (la) and the eikonal equation (1.5). Assuming t h a t 0,~ r 0, as a result we obtain (2)

First we consider the wave fix from (2.1) under the assumption that ( X l ) n r 0 (consequently, we also have ( ~ l V f l ) r 0). Now we project (1) onto (V,/) and take into account (2). In this way, for n = 0 and 7 = 0, we derive

(0~f.-

P~ (V~l) 2,

(3)

C33C44

- (c,,

(c,3 + c4,)

If P~ > 0, then il(01),~[ determines the attenuation of the wave if1. In the case ~ > 0, the inequality P~ > 0 can be written in the form (3a) (c,2+c66)c33 > (c,3 +c66) ~. As is easily seen, if ct3 > 0, c66 > 0, and (3a) holds true, then inequalities (VIIa), which insure the positivity of the potential energy W, 3 are satisfied. The inequality P~ > 0 is obviously valid if c11 > c33 a n d c l 2 > c13 > 0, provided that 0_< ~_< /~, 0 < ~ < A, andrTz > 0. Furthermore, we observe that the positivity of P~ implies D > 0, where D is of the form (1.10). 3 T h e i n e q u a l i t y Pn2 > 0 c a n also be o b t a i n e d as a c o n s e q u e n c e of t h e inequality I,V > 0 by u s i n g a specific e x p r e s s i o n for W valid for t h e t y p e of a n i s o t r o p y considered (see [7]). A l o n g with c o n d i t i o n s ( V l l a ) , we also o b t a i n t h e i n e q u a l i t y e12e33 > c23 nt- 2c12c66 ( p r o v i d e d t h a t cle > 0 a n d c33 > 2c66), w h e n c e it follows t h a t P~ > 0.

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The projection of (1) onto ~'= r7 x .Vd yields (4) Equality (4) implies two possible cases. (1) Letting 0 = 01 in (4), we find that ((;71)~, (') = 0, which allows us to regard t~1 a s a n analog of the longitudinal wave. (2) Let ;7 = ;72 and assume that (x2, ~)~=00 # 0. Then, for n = 0 and 3' = 0, f r o m (4) it follows t h a t r~

(02) 2 = - - - - ( V s / )

2

9

(5)

C44

The equation obtained determines (02),~ = i ( ~ ) 1 / ~ [V,ll, provided titat (6)

= C66 - - C44 > 0.

Inequality (6) is the condition that insures the origination of Rayleigh surface waves. The case where ff~ = 0 (c44 = c66) is not excluded from our considerations and is, in a sense, similar to the isotropic case [1] because it does not require the addition of the attenuating "transverse" wave if2. Setting 0 = 02 and x = x2, projecting (1) onto V,/, and using (5), we derive

{(C12

-{-

C66)( ~f2 V sl) -~- (C13 -{'-C44 )( 02 )n( a'C2)n } n=O -~" O.

(7)

7=0

This equality and relation (2) (where it is assumed that 0 = 02 and ~" = ;72) form a linear homogeneous system for (K'2V,I) and (02),(x2),, with the determinant

5 ~--. (C12"~LC66)C33-- (C13"~C44)2; in addition, if 0 < i~ _< #, "/> 0, and ~ > 0, t h e n / 9 > 0 (see (3a)). In this case, for n = 0 and 7 = 0, we have (;72V,/) = 0 and (x2),~ = 0. Therefore, we can regard ff2 as an analog of the transverse SH wave. Thus, under the normalization conditions pl;Trl 2 = 1, r = 1 , 2 , the polarization vectors (;71,2),,= o axe -1=0

completely determined. Higher approximations in 7 for 01,2 and ;71,2 on E can be obtained by successively differentiating (1) with 0-c o-c ' s = 1 , 2 , . . . , can be found recursively respect to 7- As a result (for n = 0 and 7 = 0) 0'(0t.2)_____z, and 0",~t.2 from the already determined values of (01,2).17=o= and (;71,2)n=0-3,=0 The author wishes to thank V. M. Babich for helpful discussions and valuable comments. This work was supported by the Russian Foundation for Basic Research (Grant No. 93-011-16148). Translated by Z. A. Yanson. REFERENCES 1. Z. A. Yanson, "Nonstationary waves of Rayleigh type near the surface of an inhomogeneous elastic body," Zap. Nauchn. Semin. LOMI, 156, 168-183 (1986). 2. I. V. Mukhina and L. A. Molotkov, "On the propagation of Rayleigh waves in an elastic half-space," hr. Akad. Nauk SSSR, Fiz. Zemli, No. 4, 3-8 (1967). 3. N. Yd. Kirpichnikova, "On the propagation of surface waves concentrated near rays in an inhomogeneous elastic body," Trudy Math. Inst. Akad. Nauk SSSR, C X V , 114-130 (1971). 4. P. V. Krauklis and N. V. Tsepelev, "On the construction of the high-frequency asymptotics of a wave concentrated near the boundary of an elastic body of arbitrary shape," Zap. Nauchn. Semin. LOMI, 34, 72-92 (1973). 2892

5. V. D. Azhotkin and V. M. Babich, "On the propagation of Love waves along the surface of an anisotropic body of arbitrary shape," Zap. Nauchn. Semin. LOMI, 165, 9-14 (1987). 6. Z. A. Yanson, "On nonstationary Love waves near the surface of an anisotropic elastic body," Zap. Nauchn. Semin. LOMI, 203, 166-172 (1992). 7. G. I. Petrashen, Wave Propagation in Anisotropic Elastic Media [in Russian], Leningrad (1980). 8. J. N. Sneddon and D. S. Berry, The Classical Theory of Elasticity, Springer-Verlag, Berlin-G6ttingenHeidelberg (1958).

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