  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.
