Softcover ISBN:  9781470440374 
Product Code:  MEMO/263/1271 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
eBook ISBN:  9781470456504 
Product Code:  MEMO/263/1271.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
Softcover ISBN:  9781470440374 
eBook: ISBN:  9781470456504 
Product Code:  MEMO/263/1271.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $102.00 $76.50 
Softcover ISBN:  9781470440374 
Product Code:  MEMO/263/1271 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
eBook ISBN:  9781470456504 
Product Code:  MEMO/263/1271.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
Softcover ISBN:  9781470440374 
eBook ISBN:  9781470456504 
Product Code:  MEMO/263/1271.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $102.00 $76.50 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 263; 2020; 143 ppMSC: Primary 35; 74; 78
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a halfspace. 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 VenantKirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and wellposedness 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 \(u^{\varepsilon} \) to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength \(\varepsilon \), and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to \(u^{\varepsilon}\) on a time interval independent of \(\varepsilon \).
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.

Table of Contents

Chapters

1. General introduction

2. Derivation of the weakly nonlinear amplitude equation

3. Existence of exact solutions

4. Approximate solutions

5. Error Analysis and proof of Theorem 3.8

6. Some extensions

A. Singular pseudodifferential calculus for pulses


Additional Material

RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a halfspace. 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 VenantKirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and wellposedness 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 \(u^{\varepsilon} \) to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength \(\varepsilon \), and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to \(u^{\varepsilon}\) on a time interval independent of \(\varepsilon \).
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.

Chapters

1. General introduction

2. Derivation of the weakly nonlinear amplitude equation

3. Existence of exact solutions

4. Approximate solutions

5. Error Analysis and proof of Theorem 3.8

6. Some extensions

A. Singular pseudodifferential calculus for pulses