Hideo Soga

Asymptotic solutions of the elastic wave equation and reflected waves near boundaries

In the first half of this paper, we construct asymptotic solutions of linear anisotropic elastic equations. In the latter half, we investigate waves reflected by boundaries for plane incident waves in terms of these solutions. Especially, it is examined whether or not the mode-conversion occurs near points where the incident waves hit the boundaries perpendicularly.

Relation Between Scattering Theories of the Wilcox and Lax–Phillips Types and a Concrete Construction of the Translation Representation

Abstract In this article we extract characteristic points from scattering theories of the Lax–Phillips type and the Wilcox one, and in view of those points we show that the both settings can be translated into each other. Furthermore, we construct concrete translation representations of the Lax–Phillips type for the elastic equation in the half-space, in order to show the good selection from admitted constructions in the abstract arguments.

ASYMPTOTIC SOLUTIONS OF THE ELASTIC EQUATION IN THE CASE OF TOTAL REFLECTION

In this paper we construct outgoing asymptotic solutions of the anisotropic elastic wave equation coinciding with given data on the boundary when total reflection happens. In this case the solutions consist of the usual hyperbolic parts and the elliptic parts exponentially decreasing in the normal direction to the boundary. The main assertion is that sum of those parts can cover any boundary data because of the elasticity. The proof is reduced to the complex analysis of the symbol of the elastic operator. By means of the asymptotic solutions we can make parametrices and study propagation of the singularities in the case of total reflection. *Partly supported by Grant-in-Aid for Sci. Research (c) 10640151, the Ministry of Educ., Japan.In this paper we construct outgoing asymptotic solutions of the anisotropic elastic wave equation coinciding with given data on the boundary when total reflection happens. In this case the solutions consist of the usual hyperbolic parts and the elliptic parts exponentially decreasing in the normal direction to the boundary. The main assertion is that sum of those parts can cover any boundary data because of the elasticity. The proof is reduced to the complex analysis of the symbol of the elastic operator. By means of the asymptotic solutions we can make parametrices and study propagation of the singularities in the case of total reflection. *Partly supported by Grant-in-Aid for Sci. Research (c) 10640151, the Ministry of Educ., Japan.

Scattering theory for the elastic wave equation in perturbed half-spaces

In this paper we consider the linear elastic wave equation with the free boundary condition (the Neumann condition), and formulate a scattering theory of the Lax and Phillips type and a representation of the scattering kernel. We are interested in surface waves (the Rayleigh wave, etc.) connected closely with situations of boundaries, and make the formulations intending to extract this connection. The half-space is selected as the free space, and making dents on the boundary is considered as a perturbation from the flat one. Since the lacuna property for the solutions in the outgoing and incoming spaces does not hold because of the existence of the surface waves, instead of it, certain decay estimates for the free space solutions and a weak version of the Morawetz arguments are used to formulate the scattering theory. We construct the representation of the scattering kernel with outgoing scattered plane waves. In this step, again because of the existence of the surface waves, we need to introduce new outgoing and incoming conditions for the time dependent solutions to ensure uniqueness of the solutions. This introduction is essential to show the representation by reasoning similar to the case of the reduced wave equation.