Statistical wave field theory: Anisotropic wave fields under Neumann'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 introduce a unified version of the theory dedicated to a class of semi-mixing billiards. In the case of Neumann's boundary condition, we show that the wave field is stationary, but it is generally anisotropic. In particular, the correlation between two spatial positions at a given frequency is different from the well-known cardinal sine formula that characterizes diffuse acoustic fields.