Acoustic waves interacting with non--locally reacting surfaces in a Lagrangian framework

TL;DR

This paper investigates the well-posedness, relationships, and stability of acoustic wave models interacting with non--locally reacting surfaces within a Lagrangian framework, connecting various formulations and proving asymptotic stability under damping.

Contribution

It introduces a unified analysis of multiple acoustic wave models in a Lagrangian setting, establishing their well-posedness, interrelations, and stability properties.

Findings

Proved well-posedness of the models.

Established relations between Lagrangian and Eulerian formulations.

Demonstrated asymptotic stability with linear damping.

Abstract

The paper deals with a family of evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. They are all derived in a Lagrangian framework. We study well-posedness of these problems, their mutual relations, and their relations with other evolution problems modeling the same physical phenomena. They are those introduced in an Eulerian framework and those which deal with the (standard in Theoretical Acoustics) velocity potential. The latter reduce to the well--known wave equation with acoustic boundary conditions. Finally, we prove that all problems are asymptotically stable provided the system is linearly damped.