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.
Publisher
Cambridge University Press (CUP)
Cited by
19 articles.
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