On the Propagation of Love Waves along the Surface of an Inhomogeneous Elastic Body of Arbitrary Shape

The present paper continues the work carried out by the author for the case of an isotropic elastic medium [1,2], and also for a special case of an anisotropic medium, namely, for a transversally isotropic one [3]. The study of surface waves in anisotropic elastic media is a topical problem because of its practical applications to seismology and seismic prospecting.In this note, to construct asymptotic expansions for nonstationary whispering gallery waves we use the same approach as in [1][2][3][4]. It consists of making use of space-time (ST) ray expansions in the presence of a simple caustic. Conditions are deduced which stipulate the existence of surface waves similar to Love waves, so that the normal component of the principal asymptotic term is zero on the surface S. A class of anisotropic media where this situation holds with surface modes resembling SH Love waves is specified. The components of the elasticity tensor are assumed to be smooth coordinate functions. The formulas are derived in a curvilinear coordinate system, and they are also of methodical interest as compared to those for the isotropic case.1. In an anisotropic elastic medium where a curvilinear coordinate system ql, q2, q3 is introduced, the vector of displacements ff satisfies the system of equationswhere p is the density of the medium; t is time; the relationship between the stress tensor (~rij) and the strain tensor (e~s) is given by the expressions (Hooke's law)where (aije.~) is the elasticity tensor with known symmetry properties [6]; ~7i is a covariant derivative symbol; (G 'J) (i,j = 1, 2, 3) is a metric tensor. The matrix (G ij) is inverse to (G~j). Summation is implied over all repeating indices. The coordinate system associated with the surface S is as follows: ql,q2 are curvilinear coordinates on S, qZ = n is the distance between a point M and the surface S; fi is an inwardly directed normal. We impose the boundary conditions which express the absence of stress on S, i.e., 0"3j [ n=OWe look for the wave field ff in the form ff (ql,q2,q3,t)

Section: "3j [ N=omentioning

confidence: 99%

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confidence: 99%

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In this note, to construct asymptotic expansions for nonstationary whispering gallery waves we use the same approach as in [1][2][3][4].

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confidence: 92%

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confidence: 99%

See 2 more Smart Citations

On nonstationary love waves near the surface of an anisotropic elastic body

Yanson¹

1996

“…For many years the ray method has remained a classical method for finding the principal term of an expansion of the solution in acoustic, electromagnetic, elastic, and other media. However, an evaluation of even the principal term of the ray series for nonhomogeneous media requires cumbersome analytic calculations (see, e.g., [7,8]). In calculating the subsequent terms of the ray asymptotic expansion in a nonhomogeneous three-dimensional medium, these difficulties increase many times.…”

mentioning

confidence: 99%

On calculating the second term of the series of the ray method for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media

Kirpichnikova¹

1997

Self Cite

This paper continues the series of papers [1][2][3][4][5][6] devoted to the development of algorithms for computing the second and subsequent terms of the ray series for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media. For many years the ray method has remained a classical method for finding the principal term of an expansion of the solution in acoustic, electromagnetic, elastic, and other media. However, an evaluation of even the principal term of the ray series for nonhomogeneous media requires cumbersome analytic calculations (see, e.g., [7,8]). In calculating the subsequent terms of the ray asymptotic expansion in a nonhomogeneous three-dimensional medium, these difficulties increase many times. Thus, we arrive at the problem of finding algorithms such that their comparison could provide an opportunity for simplifying the solution process in problems related to wave propagation in different media. In this paper, for determining the second term of the ray expansion of the displacement vector in a three-dimensional nonhomogeneons elastic medium, we suggest an algorithm similar to that of [1,2]. Simultaneously, we compare the methods of [1,2] and [3,4]. In the latter papers, the need for finding the second term of the ray series for the solutions of problems on wave propagation in different media was thoroughly justified.Consider a nonhomogeneous isotropic elastic medium with the Lam~ parameters A(~ ), #(Z ) and density p(~, ). The functions A,/~, and p are assumed to be smooth enough as functions of the point ~. The ray expansion of the vector of displacements U is considered in the formwhere w is the frequency, v is the eikonal, t is time, and s = (x, y, z). The principal (first) term of the ray series (0.1) corresponds to k = 0, whereas the correction (second) term, which is the subject of our considerations, corresponds to k = 1. We analyze the case of longitudinal vibrations, for which the recurrent relations of the ray method are known (see [9,10]). These relations were used in [3,4] as a starting point for determining the second (k = 1) term of the ray expansion (0.1) of the displacement vector U. For the convenience of comparing the algorithm for finding tl presented in this paper with that in [3,4], we recall these relations: t0 = ~0Vr, q~0 = ~b0(V1,72 9 (0.2)Here, VT" is the gradient of the eikonal r; 71,7~ are the coordinates specifying the ray; ~bo is a constant along the ray that describes the initial data for t0 and is dependent upon the type of source in question; a = (()~ + 2#)/p) 1/2 is the velocity of longitudinal waves; J is the geometrical spreading of the ray tube equal to I J= 77(7.71,72 "

Behavior of surface waves at transition through a junction line on the boundary of an elastic homogenous isotropic body

Kirpichnikova¹,

Philippov²

1998

Self Cite

Consider a convex homogeneous isotropic elastic body ~ C R~_ = {(z, y, z)lz >_ 0} bounded by a stress-free surface Of}. The surface 0f~ consists of a smooth cylindrical surface 0f/-(with z _> 0, x < 0) with generator parallel to the y-axis and of the half-plane 0~ + (z = 0, x >_ 0). The junction line 10 = {(x, y, z)lx = z = 0} coincides with the y-axis. The plane tangent to the surface Oft is continuous on the junction line, but the curvature of the normal section of the surface 012 has a discontinuity of the first kind at each point of the y-axis (see Fig. 1). The value of the discontinuity depends on the y-coordinate.Let L be a ray lying in a plane coplanar to the x, z plane and belonging to the surface 012. By definition, rays are extremals of the Fermat functional f ds T' where ds is an element of the arc length of the curve along which the integration is carried out, and b is the velocity of propagation of the transversal elastic waves in ft.In this paper, we consider high-frequency asymptotic expansions of the solutions of the equations of elasticity theory concentrated in the vicinity of the boundary Oft-. These solutions determine waves which propagate along the ray L with a velocity close to b and overrun on the line I0 of junction of the surfaces Of/-and 0f~ +.