Geometric optics for Rayleigh wavetrains in d-dimensional nonlinear elasticity

TL;DR

This paper proves that high-order approximate solutions for Rayleigh wavetrains in nonlinear elasticity closely match the exact solutions in any dimension, confirming the validity of the leading term approximation for small wavelengths.

Contribution

It establishes the closeness of approximate solutions to the exact solution for Rayleigh wavetrains in any dimension, extending previous results and providing rigorous error bounds.

Findings

High-order approximate solutions are close to exact solutions for small wavelengths.

The analysis applies to any space dimension $d \\geq 2$.

The method is adaptable to other wave types with high-order approximations.

Abstract

A Rayleigh wave is a type of surface wave that propagates in the boundary of an elastic solid with traction (or Neumann) boundary conditions. Since the 1980s much work has been done on the problem of constructing a leading term in an \emph{approximate} solution to the rather complicated second-order quasilinear hyperbolic boundary value problem with fully nonlinear Neumann boundary conditions that governs the propagation of Rayleigh waves. The question has remained open whether or not this leading term approximate solution is really close in a precise sense to the exact solution of the governing equations. We prove a positive answer to this question for the case of Rayleigh wavetrains in any space dimension $d\geq 2$. The case of Rayleigh pulses in dimension $d=2$ has already been treated by Coulombel and Williams. For highly oscillatory Rayleigh wavetrains we are able to construct high-order approximate solutions consisting of the leading term plus an arbitrary number of correctors. Using those high-order solutions we then perform an error analysis which shows (among other things) that for small wavelengths the leading term is close to the exact solution in $L^\infty$ on a fixed time interval independent of the wavelength. The error analysis is carried out in a somewhat general setting and is applicable to other types of waves for which high order approximate solutions can be constructed.