Statistical wave field theory: Anisotropic wave fields under Robin's boundary condition

The statistical wave field theory mathematically establishes the statistical laws of the solutions to the wave equation in a bounded domain. It provides the closed-form expressions of the power distribution and the correlations of the wave field jointly over time, frequency, and space, which hold at high frequency and after many reflections, in terms of the geometry and the specific admittance of the boundary surface. This theory was originally developed in the particular case of mixing rooms, which are characterized by a diffuse wave field, based on the theory of dynamical billiards and on Weyl-like asymptotic laws. Then it was extended to the finite family of special polyhedra, where the wave field is anisotropic, based on a simpler geometric approach related to mathematical crystallography. In this paper, we develop a unified version of the theory dedicated to semi-mixing billiards. In the case of Robin's boundary condition, we show that such wave fields are characterized by a directional reverberation time that is independent of the receiver's position but depends on its orientation, and we provide its closed-form expression, which improves and generalizes Eyring's formula of the reverberation time in ergodic rooms.