Computed Frequency and Angular Distribution of the Normal Modes of Vibration in Rectangular Rooms

This paper describes the mathematical model used for computing both the frequency and angular distribution of the normal modes in rectangular rooms. The criteria adopted were computed for each half‐octave band over the first 4 octaves of normalized frequency for rooms with dimension ratios ranging from 313:1 to 1:1. Considerable variation in the frequency‐spacing criterion exists not only for changes in room dimensions but also from one half‐octave band to the next. No clearly defined optimum room dimension, as predicted by Bolt, emerges from this study. The angular‐distribution index is more regularly behaved with rather definite stratification apparent as a function of the room height/length ratio, when the height direction is taken as the angular reference. When both the frequency and angular criteria are combined, only a few small regions of dimension ratios appear to be good. From these regions, p = 0.69, q = 0.43; p = 0.83, q = 0.65; p = 0.82, q = 0.72, together with p = 1/21/3, q = 1/41/3, appear to be among the best. For rooms having satisfactory mode distribution, an approximate formula has been developed for determining the lowest midband frequency for which a room may be used for measurements of continuous spectrum sounds. The formula turns out to be a constant divided by the cube root of the volume, where the constant is a function of the measuring bandwidth and the number of normal modes required therein. For 20, 12, and 9 modes in 1‐, 12‐, or 13‐oct bands, the constants are 1150, 1280, and 1355 cps, respectively.