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

11.6 Growth of Sound with Absorbent Effects 251that converts into heat through friction. Ordinarily the values of α should fallbetween zero for a perfect reflector and unity for a perfect absorber. Measurementsof α>1.0 have been reported, owing possibly to diffraction at low frequenciesand other testing condition irregularities.Let α 1 , α 2 , α 3 , ...α i denote the absorption coefficient of different materials ofcorresponding areas S 1 , S 2 , S 3 ,....S i forming the interior boundary planes (viz.the walls, ceiling and floor) of the room as well as any other absorbing surfaces(e.g. furniture, draperies, people, etc.). The average absorption coefficient α for anenclosure is defined byα = α 1 S1 + α 2 S2 + α 3 S3 +···+α i Si= A (11.3)S1 + S2 + S3 +···+Si Swhere A represents the total absorptive area ∑ α i S i , and S the total spatial area.11.6 Growth of Sound with Absorbent EffectsThe rate W of sound energy being produced equals the rate of sound energy absorptionat the boundary surfaces of the room plus the rate at which the energyincreases in the medium throughout the room. This may be expressed as a differentialequation governing the growth of acoustic energy in a live room:V dEdt + AcE ≡ W (11.4)4The solution for E in Equation (11.4) isE = 4W Ac(1 − e−(Ac/4V )t ) (11.5)with the initial condition that the sound source begins operating at t = 0. From therelationship of Equation (11.2) the intensity becomesI = W A(1 − e−(Ac/4V )t ) (11.6)and from Equation (3.58) the energy density isE =(11.7)2 ρ 0 c 2The mean square acoustic pressure becomesp 2 = 4W ρ 0 c (1 − e−(Ac/4V )t ) (11.8)AEquation (11.8) is analogous to the one describing the growth of direct current inan electric circuit containing an inductance and a resistance. The time constant ofthe acoustic process is 4V/Ac. If the total absorption is small and the time constantis large, a longer time will be necessary for the intensity to approach its ultimatep2