Geometric Perturbation Theory for a Class of Fiber Bundles

TL;DR

This paper develops a geometric perturbation theory for analyzing wave equations on time-varying domains, especially in acoustics, combining differential geometry, functional analysis, and physics to handle nonlinear, coupled PDEs with moving boundaries.

Contribution

It introduces a systematic perturbation framework using fiber bundles and semigroup techniques to analyze wave equations on dynamically changing domains, extending existing methods.

Findings

Stationary-domain approximation introduces only small errors.

Derived analytic simplifications via piston approximation.

Formalism applicable to scalar wave equations on time-varying domains.

Abstract

A systematic study of small, time-dependent, perturbations to geometric wave-equation domains is hardly existent. Acoustic enclosures are typical examples featuring locally reacting surfaces that respond to a pressure gradient or a pressure difference, alter the enclosure's volume and, hence, the boundary conditions, and do so locally through their vibrations. Accordingly, the Laplace-Beltrami operator in the acoustic wave equation lives in a temporally varying domain depending on the displacement of the locally reacting surface from equilibrium. The resulting partial differential equations feature nonlinearities and are coupled though the time-dependent boundary conditions. The solution to the afore-mentioned problem, as presented here, integrates techniques from differential geometry, functional analysis, and physics. The appropriate space is shown to be a (perturbation) fiber bundle. In the context of a systematic perturbation theory, the solution to the dynamical problem is obtained from a combination of semigroup techniques for operator evolution equations and metric perturbation theory as used in AdS/CFT. Duhamel's principle then yields a time-dependent perturbation theory, called geometric perturbation theory. It is analogous to, though different from, both Dirac's time-dependent perturbation theory and the Magnus expansion. Specifically, the formalism demonstrates that the stationary-domain approximation for vibrational acoustics only introduces a small error. Analytic simplifications methods are derived in the framework of the piston approximation. Globally reacting surfaces (so-called pistons) replace the formerly locally reacting surfaces and reduce the number of independent variables in the underlying partial differential equations. In this way, a straightforwardly applicable formalism is derived for scalar wave equations on time-varying domains.