# The transmission problem in linear isotropic elasticity

## TL;DR

This paper analyzes the behavior of elastic waves in a bounded domain with interfaces, revealing how wave transmission, reflection, and mode conversion depend on boundary conditions and geometric properties, and establishing unique determination of wave speeds.

## Contribution

It provides a detailed microlocal analysis of elastic wave transmission across interfaces and derives conditions for unique wave speed recovery from boundary measurements.

## Key findings

- Wave reflection, transmission, and mode conversion are characterized at interfaces.
- Knott's equations are recovered in the context of elastic wave transmission.
- Unique determination of P and S wave speeds from boundary data under convexity conditions.

## Abstract

We study the isotropic elastic wave equation in a bounded domain with boundary with coefficients having jumps at a nested set of interfaces satisfying the natural transmission conditions there. We analyze in detail the microlocal behavior of such solution like reflection, transmission and mode conversion of S and P waves, evanescent modes, Rayleigh and Stoneley waves. In particular, we recover Knott's equations in this setting. We show that knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the P and the S waves if there is a strictly convex foliation with respect to them, under an additional condition of lack of full internal reflection of some of the waves.