Geometric optics for surface waves in nonlinear elasticity

TL;DR

This paper rigorously proves the existence of high-frequency Rayleigh surface waves in nonlinear elastic media and shows that approximate solutions derived from an amplitude equation closely match exact solutions over fixed time intervals.

Contribution

It extends the theory of the amplitude equation for surface waves and provides a rigorous 2D proof of the existence and approximation accuracy of Rayleigh waves in nonlinear elasticity.

Findings

Existence of exact Rayleigh wave solutions in 2D nonlinear elasticity.

Approximate solutions from the amplitude equation are close to exact solutions.

The method applies to both pulse and wavetrain surface waves.

Abstract

This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an \emph{approximate} Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which we refer to as "the amplitude equation", is an integrodifferential equation of nonlocal Burgers type. We begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^\eps$ to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength $\eps$, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to $u^\eps$ on a time interval independent of $\eps$. The paper focuses mainly on the case of Rayleigh waves that are \emph{pulses}, which have profiles with continuous Fourier spectrum, but our method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.