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

11.8 Decay of Sound in Dead Rooms 255sound absorption coefficient of the room’s boundary materials. The growth ofacoustic energy at any point in the room results from the accumulation of successiveincrements from the sound source, from the first-order (single reflection) imageswith strengths W (1 – ᾱ), from the second-order (secondary reflection) imageswith strengths W (1 – ᾱ) 2 , and so on, until all the image sources of appreciablestrengths have rendered their contributions. When the true sound source is stopped,the decay of the sound occurs with all the image sources stopped simultaneouslyalong with the source. The energy decay in the room occurs from successive lossesof acoustic radiation from the source, then from the first-order images, the secondorderimages, and so on.Eyring derived the following equation for the growth in acoustic energy density:4W [E =−1 − ecS ln (1−α)t/4V ] (11.16)cS ln (1 − α)The above equation is very similar to Equation (11.5) excepting that the total roomabsorption is given byα =−S ln(1 − α) (11.17)Here S is the total area of the boundary surfaces of the room. In a like fashion theanalogy to Equation (11.6) for the decay of sound energy is given bycS ln(1−α)t/4VE = E0 eand the decay rate in dB/s is expressed as1.08cS ln(1 − α)D =−Vwith the reverberation time expressed byT = 0.161V−S ln(1 − α)For small values of absorption (α ≪ 1) the term ln(1 – α) may be replaced by α,the first term in an infinite series. This results in recovering the Sabine formulafor live rooms. It should also be noted that the coefficient 0.161 for the Sabineand the Eyring formulas, which is based on the speed of sound at 24 ◦ C, will varyaccording to air temperature. The coefficient becomes somewhat higher at lowerair temperatures and vice versa.Another formula for determining the reverberation time of a room lined withmaterials of widely ranging absorption coefficients was developed by Millingtonand Sette (Millington, 1932; Sette, 1993). The Millington–Sette theory indicatesthat the total room absorption is given byA = ∑ −S i ln(1 − ᾱ i )which yields the reverberation timeT =0.161V∑ − Si ln(1 − ᾱ i )