Reverberation between concentric spheres

Reverberation time T is widely used in architectural acoustics as a figure of merit for an auditorium. Only approximate formulas have been available for calculating T from auditorium dimensions. Recently H. Kuttruff and W. B. Joyce showed that T could be obtained exactly from a difficult integral equation. A solution is given here for a sphere containing a smaller concentric sphere. The radii and absorption coefficients of the two spheres can be varied independently to obtain a four-parameter family of examples with T known exactly. Concentric-sphere auditoria are used to test some well-known simple approximations to T and three new ones that have the advantage of being bounds on T.