A power conservation approach to predict the spatial variation of the cross-sectionally averaged mean-square pressure in reverberant enclosures

Although it is commonly assumed that broadband mean-square pressure levels are spatially uniform in reverberant enclosures, there is a gradual spatial variation, especially if the room is long in one direction, and/or the acoustic absorption is not applied uniformly to the enclosure boundaries. An equation for predicting the average cross-sectional sound pressure levels in a lightly damped enclosure with absorption is derived based on conservation of acoustic power. The derivation involves a one-dimensional boundary value problem, the solution of which is an estimate of the average sound pressure level at cross-sections in the interior. In its simplicity, the resulting formula is reminiscent of the classical Sabine formulation; however, this prediction contains a spatially varying function that depends upon the distribution of absorption (side-wall versus end-wall). The formula is demonstrated on a model problem consisting of a rectangular acoustic enclosure with a source on one end-wall, absorption on the opposing end-wall, and a combination of hard and absorbing side-walls. Comparisons with an exact numerical simulation show that the prediction works well for a wide range of absorption levels and provides an improvement over classical diffuse field theory, where the levels are assumed to be spatially uniform. A formula for the volume-averaged, broadband, mean-square pressure (a modified Sabine formula) is also derived and shown to give excellent agreement with the numerical simulations.