Frequency-Correlation Functions of Frequency Responses in Rooms

A mathematical study of the random interference of sound waves in large rooms requires statistical methods. “Statistical wave acoustics” is based on the random interference of many simultaneously excited normal modes of a room. In general, the random interference takes place for frequencies above 2000 (T60/V)12, where T60 is the reverberation time (in sec) and V is the volume (in m3) of the room. In the statistical theory, frequency responses between two points in a room are treated as random functions. The probability distributions, correlation functions, and “spectra” of these random functions are determined by physical parameters such as the distance between source and receiver, the volume and reverberation time of the room (or distribution of reverberation times), etc. In this paper, correlation functions of frequency responses are derived for rooms with uniform reverberation time, and negligible direct sound transmission between source and receiver. Analytic formulas for the following frequency correlation functions are found: the autocorrelation functions of the real and imaginary parts, the modulus and the squared modulus of the frequency response, and the cross correlation function between real and imaginary parts of the frequency response. The significance of these correlation functions in room acoustics is discussed. Measurement of the autocorrelation function of the real (or imaginary) part of the frequency response allows a precise determination of the distribution of reverberation times. The autocorrelation function of the modulus (or squared modulus) determines the required frequency shift in public address systems to improve their stability. For measurement of electroacoustic transducers in reverberation chambers, optimum bandwidths of noise or warble tones are obtained.