Nonlinear interaction of waves in elastodynamics and an inverse problem

Abstract

We consider nonlinear elastic wave equations generalizing Gol'dberg's five constants model. We analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses. Using the boundary measurements of the nonlinear responses, we solve the inverse problem of determining elastic parameters from the displacement-to-traction map.

Determination of Space‐Time Structures from Gravitational Perturbations

“…In de Hoop-Uhlmann-Wang [11], the nonlinear responses of two scalar waves at an interface of different media was considered and the related inverse problem was addressed. Also, in de Hoop-Uhlmann-Wang [10], the problem for elastic systems is considered. In particular, the nonlinear interaction of two elastic waves is carefully analyzed and used to determine elastic parameters from boundary measurements.…”

Inverse problems for nonlinear hyperbolic equations

“…Nonlinear elastic wave equation. de Hoop, Wang and Uhlmann [7] considered the initial boundary value problem for a quasilinear elastic wave equation…”

“…The parameters λ and µ are called Lamé moduli and ρ is the density. This model is widely used and can be found in [21,8,7]. The inverse problem is to recover the parameters λ, µ, ρ, A , B, C from the displacement-totraction map…”

“…One can first recover the Dirchlet-to-Neumann map Λ lin for the linear elastic wave equation (11) by first order linearization of Λ (cf. [7])…”

“…The paper [37] considered a very general class of Einstein equations coupled with different fields as long as the microlocal linearization condition is satisfied. This condition is used to construct distorted plane waves for the system (7). The main result of [37] reads Theorem 4.1.…”