Energy decay estimates for an axially travelling string damped at one end

TL;DR

This paper analyzes the energy decay of a traveling string with damping at one end, providing explicit exponential decay estimates for the energy in a dynamic interval, extending existing results to moving boundary conditions.

Contribution

It introduces precise energy decay estimates for a traveling string with damping at one end, including explicit constants, in a setting with moving endpoints.

Findings

Derived explicit exponential decay bounds for energy

Established conserved functional for undamped case

Extended decay estimates to moving boundary scenarios

Abstract

We study the small vibrations of an axially travelling string with a dashpoint damping at one end. The string is modelled by a wave equation in a time-dependent interval with two endpoints moving at a constant speed $v$. For the undamped case, we obtain a conserved functional equivalent to the energy of the solution. We derive precise upper and lower estimates for the exponential decay of the energy with explicit constants. These estimates do not seem to be reported in the literature even for the non-travelling case $v=0$.