Forward and inverse problems for a finite Krein-Stieltjes string. Approximation of constant density by point masses

TL;DR

This paper investigates inverse and forward problems for a Krein string system, deriving equations for density reconstruction and analyzing how point mass approximations affect wave propagation velocities.

Contribution

It introduces a Krein-type equation for restoring string density and studies the approximation of constant density using finite point masses.

Findings

Derived a Krein-type integral equation for density reconstruction

Analyzed the impact of point mass approximation on wave velocity

Provided methods for approximating constant density with finite point masses

Abstract

We consider a dynamic inverse problem for a dynamical system which describes the propagation of waves in a Krein string. The problem is reduced to an integral equation and an important special case is considered when the string density is determined by a finite number of point masses distributed over the interval. We derive an equation of Krein type, with the help of which the string density is restored. We also consider the approximation of constant density by point masses uniformly distributed over the interval and the effect of the appearance of a finite wave propagation velocity in the dynamical system.