Low Frequency Asymptotic Analysis of a String with Rapidly Oscillating Density
Carlos Castro, Enrique Zuazua · SIAM Journal on Applied Mathematics · 2000
We consider the eigenvalue problem associated to the vibrations of a string with a rapidly oscillating bounded periodic density. It is well known that when the size of the microstructure $ \epsilon $ is small enough with respect to the wavelength of the eigenfunctions $ 1/\sqrt{\lambda^\epsilon},$ eigenvalues and eigenfunctions can be approximated by those of the limit system where the oscillating density is replaced by its average. 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 ($ \epsilon \sim 1/\sqrt{\lambda^\epsilon} $), singular phenomena may occur. In this paper we study the behavior of the eigenvalues and eigenfunctions when $ 1/\sqrt{\lambda^\epsilon} $ approaches the critical size $ \epsilon $. To do this we use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to $ \epsilon $. In particular, our analysis provides all order correction formulas for the limit eigenvalues and eigenfunctions below the critical size.