Nonexponential decay of sound in rooms

Abstract

(i) The above analysis of decay of sound in rectangular rooms under various boundary conditions shows that the overall decay is in general non?exponential because the normal modes corresponding to different angles of incidence on the material have different decay rates. Even the initial slope of a decay curve gives only the average value of decay rates of all the modes as given in Eq. (2.21) and not the value of absorption coefficient averaged over all angles of incidence. We shall illustrate this point as follows: Let a(?) be the absorption coefficient of the material for a wave incident upon it at an angle ?. Then by definition a(?) = 1 ? (Z/?c)cos?2??1(Z/?c)cos??+1\frac{(Z/?c)\cos^2 ? - 1}{(Z/?c)\cos ? + 1}(Z/?c)cos?+1(Z/?c)cos2??1? (2.38) Fig. 2?8 a(?) as a function of ? for various values of specific acoustic admittance ratio of the material. The statistical absorption coefficient a? is defined as a=2?0?/2a(?)sin??cos???d?a = 2 \int_{0}^{\pi/2} a(?)\sin ? \cos ? \, d?a=2?0?/2?a(?)sin?cos?d? (2.39) Whereas what is averaged in the initial slope is a?(?) as given in Eq. (2.19). The average value of a?(?), when all the modes are excited to the same amplitude, is given by a0=2?0?/2cos??ln?(Z/?c)cos??+1(Z/?c)cos???1sin???d?a? = 2 \int_{0}^{\pi/2} \cos ? \ln \frac{(Z/?c)\cos ? + 1}{(Z/?c)\cos ? - 1} \sin ? \, d?a0?=2?0?/2?cos?ln(Z/?c)cos??1(Z/?c)cos?+1?sin?d? (2.40) It is obvious that a? and a? are not in general equal. However, when a(?) ? 1, then ln?(Z/?c)cos??+1(Z/?c)cos???1=ln?(1+a(?))?a(?)\ln \frac{(Z/?c)\cos ? + 1}{(Z/?c)\cos ? - 1} = \ln (1 + a(?)) \approx a(?)ln(Z/?c)cos??1(Z/?c)cos?+1?=ln(1+a(?))?a(?) Hence Eqs. (2.39) and (2.40) give nearly the same result. This shows that the statistical absorption coefficient is obtained from the initial slope of the decay curve only if one wall is covered completely and the material has a small absorption. (ii) It is interesting to compare the dependence of the various coefficients plotted in Figs. (2.3), (2.5), (2.6) and (2.8) upon the specific acoustic susceptance of the material. The effect of susceptance is significant for large values of admittance in case of material covering a wall completely (see curves for ? = 0.8, ? = 0.1 and ? = 0.8, ? = 0.4 in Fig. 2.3). Also, the effect of increasing the susceptance in this case is to reduce the value of the coefficient. On the other hand, Figs. 2.5, 2.6 and 2.8 show that increasing the susceptance increases the value of the coefficient when the wall is partially covered. Moreover, in the latter case the effect of susceptance is significant for low values of admittance (see curves for ? = 0.1, ? = 0.4 and ? = 0.1, ? = 0.1 in Figs. 2.5, 2.6 and 2.8). This dependence of the coefficients (a?(?), a?(?) and a?(?)) on specific acoustic susceptance explains why normal coefficients are not suitable to describe the decay pattern in rooms even when the test material has high specific acoustic impedance. For example, in comparing the predicted and measured coefficients for four different materials, Morse, Bolt and Brown have shown (Fig. 2.9) that at low frequencies the measured absorption coefficients are far in excess of the normal coefficients for two of the materials, and they attribute this difference to the effect of sample size and mounting conditions. But it is obvious now that the discrepancy lies in the relative values of the resistive and reactive components of the specific acoustic impedance of the materials. (iii) Another point that emerges out of the analysis given in this chapter is the dependence of the coefficients upon the area of the material. The coefficient measured from the initial decay increases progressively with the area of the material. For materials having small value of conductance and large value of susceptance, however, the coefficient decreases slightly as the area of the material is increased (Fig. 2.7). (iv) It is also to be noted that the value of absorption coefficient a as measured in rooms without diffusers sometimes exceeds unity because the measured coefficient is not the Sabine absorption coefficient of the material.