Geometric optics for surface waves in nonlinear elasticity / Jean-François Coulombel, Mark Williams.

Format:

Book

Author/Creator:

Coulombel, Jean-François, author.

Williams, Mark (Professor of mathematics), author.

Series:

Memoirs of the American Mathematical Society ; no. 1271.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1271

Language:

English

Subjects (All):

Geometrical optics--Mathematics.

Geometrical optics.

Nonlinear difference equations.

Elasticity.

Elasticity--Mathematical models.

Physical Description:

v, 151 pages ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2020.

Summary:

This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. The authors 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 approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which is referred to as "the amplitude equation", is an integrodifferential equation of nonlocal Burgers type. The authors 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 superscript epsilon] to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength [epsilon], and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to [u superscript epsilon] on a time interval independent of [epsilon]. This paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but the authors' method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.

Contents:

Chapter 1. General introduction

Chapter 2. Derivation of the weakly nonlinear amplitude equation

Chapter 3. Existence of exact solutions

Chapter 4. Approximate solutions

Chapter 5. Error Analysis and proof of Theorem 3.8

Chapter 6. Some extensions

Appendix A. Singular pseudodifferential calculus for pulses.

Notes:

"January 2020; Volume 263; number 1271 (first of 7 numbers)."

Includes bibliographical references (pages 149-151).

ISBN:

9781470440374

1470440377

OCLC:

1132237716