Low and high frequency approximations to eigenvibrations in a string with double contrasts

We study eigenvibrations for inhomogeneous string consisting of two parts with strongly contrasting stiffness and mass density. In this work we treat a critical case for the high frequency approximations, namely the case when the order of mass density inhomogeneity is the same as the order of stiffness inhomogeneity, with heavier part being softer. The limit problem for high frequency approximations depends nonlinearly on the spectral parameter. The quantization of the spectral semiaxies is applied in order to get a close approximations of eigenvalues as well as eigenfunctions for the prime problem under perturbation.