Exact effect of surface roughness on the reverberation time of a uniformly absorbing spherical enclosure

We assess the validity of the Sabine, Eyring, and Kuttruff reverberation‐time expressions, and of their underlying mean‐path‐length formula 〈l〉=4V/S, by determining the exact reverberation time and asymptotic sound distribution for a uniformly absorbing spherical enclosure with any amount of air absorptivity and with a surface that can be continuously varied from nonabsorbing to completely absorbing and from specular to randomly reflecting. In disagreement with an extensive literature that presents the Eyring formula as a correction to that of Sabine, we find, on the one hand, that Sabine’s formula and the applicability of 4V/S are vindicated in this application under Sabine’s stated conditions of weak absorptivity and (any nonzero amount of) irregular reflection, while, on the other hand, we find that Eyring’s and Kuttruff’s expressions are less accurate than Sabine’s unless the roughness and absorptivity of the surface exceed certain levels which are evaluated. Related concepts are discussed in some detail. Geometrical acoustics is assumed throughout, and its limitations are not considered here. The analysis is equally applicable to light in LEDs (light‐emitting diodes) and to mechanical particles which stick or experience elastic reflection at the surface of their container.