Remark on an elastic plate interacting with a gas in a semi-infinite tube: periodic solutions

TL;DR

This paper studies an elastic plate interacting with a gas in a semi-infinite tube, revealing an infinite number of periodic solutions and challenging the expectation of wave energy decay in such systems.

Contribution

It demonstrates the existence of infinitely many periodic solutions in a coupled elastic-gas system, showing non-decay of wave energy contrary to typical tube domain behavior.

Findings

Infinite periodic solutions exist with frequencies tending to infinity.

Wave energy decay does not hold for this elastic-gas system.

The system's dynamics differ from classical tube wave energy decay expectations.

Abstract

We consider a conservative system consisting of an elastic plate interacting with a gas filling a semi-infinite tube. The plate is placed on the bottom of the tube. The dynamics of the gas velocity potential is governed by the linear wave equation. The plate displacement satisfies the linear Kirchhoff equation. We show that this system possesses an infinite number of periodic solutions with the frequencies tending to infinity. This means that the well-known property of decaying of local wave energy in tube domains does not hold for the system considered.