Geometric Perturbation Theory and Acoustic Boundary Condition Dynamics

Abstract

Geometric perturbation theory is universal. A typical example is provided by the 3D wave equation, widely used in acoustics. We face vibrating eardrums as a binaural auditory input stemming from an external sound source. In the setup of internally coupled ears (ICE), which are present in more than half of the land-living vertebrates, the two tympana are coupled by an internal air-filled cavity, whose geometry determines the acoustic properties of the ICE system. The eardrums themselves are described by a 2-dimensional, damped, wave equation and are part of the spatial boundary conditions of the three-dimensional Laplacian belonging to the wave equation in the internal cavity that couples and internally drives the eardrums. In animals with ICE the resulting signal is the superposition of external sound arriving at both eardrums and the internal pressure coupling them. This is also the typical setup for geometric perturbation theory. In the context of ICE it boils down to acoustic boundary-condition dynamics (ABCD) for the coupled dynamical system of eardrums and internal cavity. In acoustics the deviations from equilibrium are extremely small (nm). Perturbation theory is therefore natural and shown to be appropriate. In doing so, we use a time-dependent perturbation theory à la Dirac in the context of Duhamel's principle. The relaxation dynamics of the tympanic-membrane system, which neuronal information processing stems from, is explicitly obtained in first order. Furthermore, both the initial and the quasi-stationary asymptotic state are derived and analyzed. Finally, we set the general stage for geometric perturbation theory where (d-1)-dimensional manifolds as subsets of the boundary of a d-dimensional domain are driven by their own dynamics with the domain pressure $p$ and an external source term as input, at the same time constituting time-dependent boundary conditions for $p$.