High frequency asymptotic analysis of a string with rapidly oscillating density

We consider the eigenvalue problem associated with the vibrations of a stri ng with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulae for the eigenvalues and eigenfunctions when the size o f the microstructure epsilon is shorter than the wavelength of the eigenfun ctions 1/root lambda (epsilon). On the other hand, it has been observed tha t when the size of the microstructure is of the order of the wavelength of the eigenfunctions (epsilon similar to 1/root lambda (epsilon)) singular ph enomena may occur. In this paper we study the behaviour of the eigenvalues and eigenfunctions when 1/root lambda (epsilon) is larger than the critical size epsilon. We use the WKB approximation which allows us to find an expl icit formula for eigenvalues and eigenfunctions with respect to epsilon. Ou r analysis provides all order correction formulae for the limit eigenvalues and eigenfunctions above the critical size. Each term of the asymptotic ex pansion requires one more derivative of the density. Thus, a full descripti on requires the density to be C-x smooth.