Statistical wave field theory: Special polyhedra

The statistical wave field theory establishes mathematically the statistical laws of the solutions to the wave equation in a bounded volume. It provides the closed-form expression of the power distribution and the correlations of the wave field jointly over time, frequency, and space, in terms of the geometry and the specific admittance of the boundary surface. In a recent paper, we presented a mathematical approach to this theory based on the Sturm-Liouville theory and the theory of dynamical billiards. We focused on mixing billiards that generate an isotropic wave field, and we retrieved the well-known statistical properties of reverberation in room acoustics. In the present paper, we introduce a simpler geometric approach, dedicated to a particular class of non-ergodic billiards. Though limited to only a few polyhedra, this approach offers a precious insight into various aspects of the theory, including the first examples of anisotropic wave fields, whose statistical properties are related to mathematical crystallography. We also show that the formulas that we obtain in this anisotropic case are closely related to those of the mixing case, albeit based on a different mathematical approach.