High frequency asymptotics of global vibrations in a problem with concentrated mass

TL;DR

This paper analyzes the high-frequency behavior of global vibrations in an elastic system with a localized high-density region, deriving asymptotic expansions for eigenvibrations using WKB techniques.

Contribution

It provides the first complete asymptotic expansions for global eigenvibrations in a fourth-order differential operator with a concentrated mass.

Findings

Derived asymptotic formulas for global eigenvibrations

Identified two types of eigenvibrations: local and global

Applied WKB technique to obtain asymptotics

Abstract

We consider an elastic system containing a small region where the density is very much higher then elsewhere. Such system possesses two types of eigenvibrations, which are local and global vibrations. Complete asymptotic expansions of global eigenvibrations for ordinary differential operator of the fourth order are constructed using WKB--technique.