| dc.description.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. | |
