19 th INTERNATIONAL CONGRESS ON ACOUSTICS

Sabine and Eyring equations are commonly used to estimate the reverberation time (RT) in concert halls. Derivations of the two equations are based on well-defined and different physical assumptions, and they are both slightly different from the realistic condition in actual halls. In many researches so far, comparatively simple room shapes have been utilized to study the question on which equation is preferable. In this study, the sound fields in rooms with more complex or wide-ranging shape are numerically analyzed by computer simulation techniques and CAD models for architecture. Focusing on the fundamental relation between the mean free path as a measure of the room shape and the RT, validity of the two equations in actual halls is discussed. Next, the influence of the room shape and/or the choice of the RT equation on the effective sound absorption is examined. INTRODUCTION Sabine and Eyring equations are commonly used to predict the reverberation time of concert halls. Both are based on simplified mathematical models so that the physical assumptions for the both equations deviate slightly from actual situation. The basic premises of Sabine and Eyring equations are summarized in Table I. Concerning to the sound particle’s behavior in a hall, the Sabine equation assumes each particle hits the wall at random timing, while the Eyring assumes each hits the wall simultaneously [1]. Regarding to the surface diffusivity, the Eyring requires all the surfaces are diffusive reflecting. On the other hand, for sphere, when 78% of the surfaces are specular reflecting, that is, 22% walls reflect the sound randomly, the Sabine coincides with the exact value [2]. When we focus on the free path length (FPL) of sound particles, the FPL should be deterministic value for the Eyring and all the free path length equal 4V/S, that is a value in the classical theory [3]. Here, V means the room volume, and S the total surface area. If FPL is treated as stochastic variable and it obeys the exponential distribution, the Sabine equation is obtained [4]. In this case, the average of the FPL equals 4V/S. Similarly, when we focus on the total reflection number N at time t, N is also deterministic value for Eyring situation. There is a famous relation by Schroeder [4] that when N obeys Poisson distribution the Sabine equation is derived. Generally, the absorption coefficient α = 1 means 100% absorption, which when used in the Eyring equation yields RT = 0. In the Sabine equation RT approaches zero only when this coefficient is very large. For the same reverberation time, the following equality is derived by equating the two equations, αsab = -2.3 log (1 – αey), and for each sub-surface Si in a room, the associated absorption coefficient is αeyi = (αey / αsab)*αsabi [5]. In actual concert halls the Sabine absorption coefficient αsab is less than 0.4 [6], which means that it can be used without experiencing large values, and we sometimes choose it here because the Sabine RT equation is simpler and in any particular case, the Eyring absorption coefficient can be obtained easily from the Sabine coefficient. The most important parameter of the statistical model of the reverberation process, probably, is either FPL or the total reflection number. Unfortunately, we can’t measure these quantities directly in actual sound field. Only the decay rate of sound pressure is measurable by direct method. But, by means of the computer simulation, the sound field in the “modeled” halls can be