Sabine’s reverberation time and ergodic auditoriums

Abstract

It is shown in geometrical acoustics that ergodic specular enclosures do exist and that in such auditoriums, but not in general, 4V/S′ is the exact mean directed path length (V is volume and S′ is any part of surface area S). Sabine’s expression is then demonstrated to yield the exact reverberation time, provided the enclosure is mixing and provided the inhomogeneous anisotropic surface absorptivity is sufficiently weak. It is further proven that the functional form of Sabine’s expression cannot be modified so as to become correct for large absorption. In an attempt to reassign credit and to reconcile these results with influential findings to the contrary, a short historical account is added. Conditions imposed upon the surface reflectivity—whether the reflectivity be reversible (specular or other) or irreversible (statistical) —by the second law of thermodynamics and by the principle of detailed balance are evaluated. Extensions (e.g., mean length of curved paths in an ergodic auditorium with a thermal gradient) and other applications (electroluminescent-diode design) are noted. Subject Classification: 55.20, 55.30, 55.55.