A Mixed Problem for the Wave Equation in a Curvilinear Half-Strip with Discontinuous Initial Data

TL;DR

This paper solves a wave equation problem with discontinuous initial data in a curvilinear half-strip, modeling impact on an elastic rod, by explicitly constructing solutions and analyzing their uniqueness and existence conditions.

Contribution

It introduces an explicit analytical solution method for a wave equation with discontinuous initial data in a curvilinear domain, including uniqueness and existence proofs.

Findings

Explicit solution constructed using method of characteristics

Proved uniqueness of the solution

Established conditions for classical solution existence

Abstract

For a one-dimensional wave equation, we consider a mixed problem in a curvilinear half-strip. The initial conditions have a first-kind discontinuity at one point. The mixed problem models the problem of a longitudinal impact on a finite elastic rod with a movable boundary. We construct the solution using the method of characteristics in an explicit analytical form. For the problem in question, we prove the uniqueness of the solution and establish the conditions under which its classical solution exists.