Complex surface rays associated with inhomogeneous skimming and Rayleigh waves

Abstract

The Rayleigh wave, that propagates at the free surface of semi-infinite anisotropic medium, is composed of three inhomogeneous partial waves, each propagating along the surface with a different attenuation along the depth. Since this wave does not exhibit an attenuation on the surface, let us call it the homogeneous Rayleigh wave. The associated slowness corresponds to the real solution of the Rayleigh dispersion equation. Besides this classical solution, an infinite number of complex solutions of the Rayleigh dispersion equation exits. For such particular Rayleigh waves, the slowness vector, i.e. the identical component on the surface of the slowness of each partial waves, is taken to be complex. Thus, these Rayleigh waves are attenuated on the surface and as shown here, their attenuation is normal to the ray direction (or the energy velocity direction). Similarly to the infinite inhomogeneous plane waves which can be associated with complex rays, we call these waves, inhomogeneous Rayleigh waves. We use the inhomogeneous skimming waves, which are inhomogeneous plane waves, and the inhomogeneous Rayleigh waves to explain differently the usual diffraction phenomena on the free surface which cannot be explained by the real ray theory. For example, the arrival time of the wave packet observed beyond the cusp is in perfect accordance with the arrival time of some specific inhomogeneous Rayleigh waves. We show that these results are in agreement with the computation of the Green function. They apply to the theory of surface waves in linear elastodynamics with intrinsic anisotropy as well as to the theory of surface waves in linearized (incremental) elastodynamics with strain- A c c e p t e d m a n u s c r i p tPage 2 induced anisotropy (also known as small-amplitude waves superimposed on the large static homogeneous deformation of a nonlinear solid).

Ray splitting in the reflection and refraction of surface acoustic waves in anisotropic solids

“…One question already raised relates to the maximum number of outgoing SAW there can be for any given surface. From the degree of the secular polynomial function 47 it would appear that for a generally anisotropic solid there is an upper limit of 27, but this encompasses unphysical and other types of solutions as well, and so the true upper limit for the number of SAW must be considerably lower than this, and lower still where there is symmetry. Another question is as follows: When s x passes through a critical value where an incoming and outgoing solution on R merge, guided by the analogous situation for bulk waves, one might anticipate that beyond that point there should be a pair of complex conjugate solutions corresponding to Rayleigh waves which are exponentially growing and decreasing along the surface.…”

“…In this regard, Ref. 47, which includes a discussion of inhomogeneous Rayleigh waves, may be a useful starting point for dealing with this question. Another intriguing question revolves around the occurrence of configurations where, in conformity with Snell's law, an incident Rayleigh wave is reflected as a Rayleigh wave at a point where it degenerates with the bulk wave continuum.…”

“…The only published information we have been able to locate on polynomial equations for the angular dependent Rayleigh wave slowness s is from a recent paper by Deschamps and Huet. 47 The issue to us is rather unclear, but it appears that there are indeed polynomial expressions in the components s 1 and s 2 determining the Rayleigh wave angular dependent slowness, and these can be as high as 27 in the squared slowness in the general case. As with the case of the secular equation for a specific propagation direction, most of the solutions of this equation will pertain to Brewster reflection of bulk waves, lateral waves, leaky SAW, and nonphysical solutions which grow without bound into the depth of the medium.…”

Beam distortion detection and deflectometry measurements of gigahertz surface acoustic waves

“…5, the faster being the skimming longitudinal wave (marked L) and the slowest being the Rayleigh surface wave (marked R). 31 From the Rayleigh pulse half temporal period τ (see Fig. 5), the central frequency of the Rayleigh wave is calculated at f R = 1/2τ = 1.7 GHz.…”

Elastic surface waves in crystals – Part 2: Cross-check of two full-wave numerical modeling methods

“…Recently, Deschamps and his collaborators [49][50][51] showed that the cuspidal triangles of the qS wave extend beyond the edges or vertices of the cuspidal triangles, and that this phenomenon can be explained by inhomogeneous plane waves. In order to show this, we perform a simulation using a two-dimensional qP-qS modeling algorithm based on the staggered Fourier method to compute the spatial derivatives [52].…”