Low Frequency Asymptotic Analysis of a String with Rapidly Oscillating Density

We consider the eigenvalue problem associated to the vibrations of a string with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulas for the eigenvalues and eigenfunctions when the size of the microstructure is shorter than the wavelength of the eigenfunctions 1/ √ λ . 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 ( ∼ 1/ √ λ ) singular phenomena may occur. In this paper we study the behavior of the eigenvalues and eigenfunctions when 1/ √ λ is larger than the critical size . We use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to . Our analysis provides all order correction formulas 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.