Theory of the Effect of Wall Irregularities on the Distribution of Sound in a Room

Abstract

The perturbation theory of Feshback and Clogston can be applied to determine the amount of wall irregularity required to produce adequate “randomness” in the sound waves in a room. When the second-order terms in the perturbation series are as large as the first-order terms, then each standing wave is a complete mixture of all the types of simple waves, and the decay rate approaches that predicted by the Sabine formula. From this criterion one can obtain an “index of randomness,” which is a measure of the ability of the room to “mix” sound of a given frequency adequately. If the index is much smaller than unity, the sound will not be well mixed, and the decay curve will be a curved line (Sabine formula will not hold). If the index is much larger than unity, the Sabine formula will hold. Approximate formulas for the index show that irregularies in wall shape are usually more effective (in mixing the sound) than irregular patches of absorbing material, that patches (or bumps) of the order of size of a half-wave-length are more effective than larger or smaller ones, and that rooms large compared to the wave-length are easier to “randomize” than smaller rooms. It is quite difficult to treat a room of dimensions smaller than ten wave-lengths so that the index of randomness comes out larger than unity.