Low and high frequency approximations to eigenvibrations of string with double contrasts

TL;DR

This paper investigates eigenvibrations of an inhomogeneous string with contrasting stiffness and density, focusing on high frequency approximations in a critical case where inhomogeneity orders match, using spectral quantization for accurate eigenvalue and eigenfunction approximations.

Contribution

It introduces a novel analysis of high frequency eigenvibrations in a critical inhomogeneous string case with nonlinear spectral dependence, employing spectral quantization techniques.

Findings

Derived high frequency eigenvalue approximations for the critical inhomogeneity case.

Established nonlinear dependence of the limit problem on the spectral parameter.

Achieved accurate eigenfunction approximations using spectral quantization.

Abstract

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.