High-frequency surface waves are considered in a linear anelastic medium of the Maxwell-Boltzmann-. Volterra type. The vertical dependences of the material ...
Journal of Mathematical Sciences, Vol. 96, No. 4. 1999
VISCOELASTIC WEAK LATERAL
LOVE WAVES IN A LAYERED INHOMOGENEITY
STRUCTURE
A . V. A r e f ' e v a n d A. P. K i s e l e v
WITH
UDC 517.95; 550.34; 534.2
High-frequency Love surface waves in a linear medium with Mazwell-Boltzmann-Volterra anelasticity are considered. Arbitrary vertical dependences of the material parameters are allowed. The weak lateral inhomogeneity and anelasticity of the medium, assumed small in the high-frequency range, are treated as perturbations. The leading term of the ray ezpansion, which corresponds to the balance of energy along real surface rays, is provide& The additional components, i.e., the Rayleigh-type components of the displacement, described by a higher-order correction, are discussed. Bibliography: 3 titles.
High-frequency surface waves are considered in a linear anelastic medium of the Maxwell-BoltzmannVolterra type. The vertical dependences of the material parameters are allowed to be arbitrary. The anelasticity of the medium, assumed small in the high-frequency range, and its weak lateral inhomogeneity. are both treated as perturbations. The leading term of the ray series, which obeys the equations o~energy balance along the surface rays corresponding to the case of perfect elasticity, is found. The additional components, i.e., the Rayleigh-type components of the displacement, described by the first-order term, are discussed. This note is a continuation of [1]. 1. The harmonic displacements U = U(r'), ~" = ( x l , x 2 , x 3 ) = ( x l , x 2 , z), of an isotropic viscoelastic half-space are described by the equations Ojamj(O)+pw2Um-----O,
z>O,
(1)
m=1,2,3,
where aj = ~'~, 0 p = p(r-) > O, ~rmj(O) -- M(r')(OmUj + OjUm) + 6mjA(r-') divU, and, for high frequencies, a s oJ ----+ OO,
A(r-3 = A ( ~ + - -- -=~ s- + . . . ,
M(r-) =#(r-') +
.
--Y.d
+ ....
(2)
Here, A, #, p, l, and m are real-valued functions independent of w; the parameters A, #, and p characterize the perfectly elastic reference medium, ), + 2# > # > 0, whereas l and m describe the effects of dissipation; m 0;
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L,D W + i AU c
e
iX O (cV) = O, DU + -i W = O, z = 0 ;
~ ao,
cJ On (cV)
[W]k---O, IUI + IWl--, o, z ~ , [U]k=0,
c
k
I~ DU + - W
= 0;
C
k
= 0;
(11)
k-'l,...,g;
Here, u = A :b 2#, and K = ~ ~ (~) is the curvature of the surface ray. This problem is identical to that for a perfectly elastic medium. 6. For arbitrary dependences of A, #, and p on z, we succeeded, as in [1], in finding simple expressions for U and W for two particular types of the dependence of A, #, and p on x 1 and x2. 6a. Consider the case of the laterally homogeneous reference medium where A,/z, and p are independent of xz,2, whereas l and m are allowed to depend on x l , x~., and z. The field of straight-line surface rays is arbitrary, but it is assumed that the observer is removed fzom the caustic. Then 1 0
V
u = 7 ~(c
),
w = 0.
(12)
In particular, for the central field of rays that correspondsto a point source placed at the origin of coordinates , a = arctan 1 ' J = r, and X l ---- 272 --" 0, we Can set r -- ~, ~ r = ~
w/dr
B= ~
~.
(13)
6b. Consider the case of "one-dimensionar' propagation, where, first, A,/z, and p are independent of x 2 and, second, the rays are straight lines parallel to the x axis. Here, l and m are allowed to depend on x a and x2. In this case, formula (12) remains valid, and we can set a = x2, J = 1, and
= f dxl ,
c'
s = w f dxl ~ &"
(14)
This work was supported in part by the Russian State Committee for Higher Education under grant 95-0-13.1--66. Translated by A. P. Kiselev.
REFERENCES 1. A. V: Aref'ev and A. P. Kiselev, variations," Zap. Nauchn. Semin. 2. V. M. Babi~ and V. S. Buldyrev, Russian], Naul~, Moscow (1972). 3. A. L. Levshin, T. B. Yanovskaya, Russian], Nauka, Moscow (1986).
"Viscoelastic Reyleigh waves in a layered structure with slow lateral POMI, 230, 7-13 (1995). Asymptotic Methods in Problems of Short-Wavelength Diffraction [in et al., Seismic Surface Waves in a Laterally Inhomogeneous Earth [in
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