Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T07:31:48.977Z Has data issue: false hasContentIssue false

High frequency asymptotic analysis of a string with rapidly oscillating density

Published online by Cambridge University Press:  05 March 2001

C. CASTRO
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
E. ZUAZUA
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

Abstract

We consider the eigenvalue problem associated with the vibrations of a string with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulae for the eigenvalues and eigenfunctions when the size of the microstructure ε is shorter than the wavelength of the eigenfunctions 1/√λε. On the other hand, it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions (ε ∼ 1/√λε) singular phenomena may occur. In this paper we study the behaviour of the eigenvalues and eigenfunctions when 1/√λε is larger than the critical size ε. We use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to ε. Our analysis provides all order correction formulae for the limit eigenvalues and eigenfunctions above the critical size. Each term of the asymptotic expansion requires one more derivative of the density. Thus, a full description requires the density to be C smooth.

Type
Research Article
Copyright
2000 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)