エルゴード仮定に基づくSabine理論の解釈

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Sabine's theory stands on the following two assuptions : 1. Uniform, diffuse distribution of sound energy throughout a room at any instant, 2. equal probability of propagation of sound in all directions. Obviously, the Sabine's theory does not follow the wave theory of normal modes. Therefore, although the results obtained from the Sabine's theory are very useful and practical, the theoretical background seems to be very vague. Morse and Bolt assumed that the sound field developed by Sabine is based on the assumption of the ergodicity. It seems however that the correspondence of the ergodicity with the two assumptions has not been explained fully. On the other hand, the problem of room acoustics was studied statistically by many investigators, e. g. 3), 4), 5), based on the assumption of the randomness of the sound field. But, it seems also that the relation between the ergodicity and the Sabine's theory is not yet explained fully. Therefore, we discuss the relation between the Sabine's theory and the ergodicity in the stedy state. If the non-deterministic sound field is made up in a room by use of the band noise, we can regard the sound energy density at every point in the room as non-deterministic. The time average of the energy density is shown by equation (1). Introducing a constant quantity shown by equation (2) and the ergodicity, the time average equals the ensemble average given by equation (3). It shows that the assumption 1 by Sabine, that is, ''uniform, diffuse distribution of sound energy throughout a room at any instant", is satisfied by the ergodicity. But, the assumption 2 is not always satisfied by the ergodicity. For, the asumption 2 depends on the ensemble average only, as shown by equations (5, 6, 7, 8). Therefore, we can find that the Sabine's theory in the steady states stands on two assumptions, that is, the ergodicity and equal probability of propagation of sound in all directions essentially. Both assumptions are independent, so the assessment of the diffusibility must be performed taking the both assumptions into account.
社団法人日本音響学会の論文
1977-07-01

エルゴード仮定に基づくSabine理論の解釈

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