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The assumption of diffuse reflection (Lambert's Law) leads to integral equations for the wall intensity in a reverberant sound field in the steady state and during decay. The latter equation, in the special case of a spherical enclosure with uniformly absorbent walls and uniform wall intensity, allows exponential decay with a decay time which agrees closely with the Norris-Eyring prediction. The sound-intensity and soundenergy density in the medium, during decay, are also calculated.

Continuing earlier work on this subject, a more rigorous discussion is given of the singular points of streamlines and the critical points of the stream function. The results for vortex and saddle points obtained earlier in piecemeal fashion and by way of examples are obtained systematically and by generally utilizing the applicable theory of differential equations and calculus. New results are also obtained. For example, a saddle point can occur when the phase of pressure and velocity differ by π/2, and in certain parts of the sound field, specifically inside a closed streamline, the number of vortex points and saddle points are related. Finally, the streamlines and singular point are considered for a discrete source: the line source.

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The effectiveness of spatial averaging in an ideal diffuse field is examined, with intended application to pure-tone reverberant sound power measurement. A theoretical measure of spatial averaging effectiveness is proposed, based on the variance of the spatial average, which is determined analytically from the shape and size of the intended path. Three cases, examined here in detail, are: continuous averaging on a line, a circle, and the surface of a disk. Results show that for greatest effectiveness, the path of averaging must extend over a region of space whose dimensions are large with respect to a wavelength. In particular, it is shown that averaging on a line segment of given length will be more effective if the line is straight rather than circular. Surprisingly, it is also shown that averaging on the surface of a disk will be less effective than averaging on its perimeter if the disk is small compared to a wavelength. This last result is employed to illustrate the dangers of oversampling (the use of highly correlated samples) which might otherwise go unrecognized.