THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

252 11. Acoustics of Enclosed Spaces: Architectural Acousticsvalue of I ∞ = W/A. The ultimate values of the energy density and mean squareacoustic pressure are given byE ∞ = 4W Ac , p2 ∞ = 4W ρ 0cAA number of caveats pertain to the use of Equation (11.8). In order that the assumptionof an even distribution of acoustic energy be cogent, a sufficient time t musthave elapsed for the initial rays to undergo several reflections at the boundaries.This means approximately 1/20 of a second should have elapsed in a small chamber;and the time must approach nearly a full second for a large auditorium. Thefinal energy density, being independent of the size and shape of the room, shouldbe the same at all points of the room and dependent only upon the total absorptionA. But Equation (11.6) does not hold for spherical or curved rooms which canfocus sounds; neither is Equation (11.8) applicable to rooms having deep recessesnor to oddly shaped rooms or rooms coupled together by an opening, and nor torooms with some surfaces of extraordinarily high absorption coefficients α (thesecause localized lesser values of energy densities).11.7 Decay of SoundWe can now develop the differential equation describing the decay of uniformlydiffuse sound in a live room. The sound source is shut off at time t = 0, meaningW = 0 at that instant. E 0 denotes the uniformly distributed energy density at thatinstant. From Equation (11.4)AcEdt = dE (11.9)4Vand the solution to Equation (11.9) becomesE = E0 e −(Ac/4V )t (11.10)The intensity I at any time t after the cessation of the sound source is related tothe initial intensity I 0 byI= e −(Ac/4V ) (11.11)I 0Applying the operator 10 log to both sides of Equation (11.11) results inIL = 10 log e −(Ac/4V )t = 102.3 ln e−(Ac/4V )t =− 1.087Act (11.12)Vwhere IL denotes the intensity level change in decibels. The intensity level in alive room decreases with elapsed time at a constant decay rate D (in dB/s),D = 1.087Ac .V