Last time, we discussed how to derive equation (1) as the acoustic energy density of a room including up to n reflections, that is, the sum of the energy densities of the direct sound and the n reflected sounds, and how to derive the mean free path shown in equation (2).
α―: Average sound absorption coefficient of interior surface V: Room volume (m³)
p: Mean free path W: Output of sound source
From equation (1), we can find the steady-state energy density as E 0 = PW/cVa, which is equal to Sabine's theoretical formula E 0 = PW/cSa, and therefore the mean free path p is:
This is how it is calculated. To reiterate, this mean free path is the distance sound travels per reflection, and it is determined by the room volume and total surface area, regardless of the room shape.
This time, we will complete the remaining process and derive Eyring's reverberation formula.
First, we find the number of reflections at the interface in t seconds. In t seconds, the sound propagates a distance c t [m], and the mean free path p is the distance the sound travels per reflection. Dividing c t by the mean free path p gives c t / p, which can be expressed as c t / p = cS/4Vt from equation (2).
The energy density E after t seconds from when the sound source is stopped in a steady state, i.e., after n = cSt/4V reflections, is the value obtained by subtracting the previously calculated E n (total energy density of n reflected sounds + energy density of the direct sound) from the steady-state energy density E 0.
As shown in equation (1) above, E n = E 0 1-(1-α) n, so E after n reflections can be found using equation (3) below.
This equation can be described as the attenuation equation expressed in terms of the number of reflections n.
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Figure 1: Eyring's decay process
Figure 1 shows the damping equation (3).
Now, we replace the decay equation, which was expressed in terms of the number of reflections n, with an equation in terms of time t. Substituting n = cSt/4V into equation (3), we find t = T such that E/E 0 = 10- 6, and this T is the reverberation time.
This is Eyring's reverberation equation. The constant K is the same as in Sabine's case (0.161 at 20°C).
Up to this point, we have only considered sound absorption at interface surfaces as an element of sound absorption in a room. However, the energy of sound propagating through the air is absorbed by molecules in the air, especially water molecules (molecular absorption). When the volume of a room is large, such as in a hall or large conference room, the effect of this air absorption cannot be ignored. To take this into account, Knudsen applied a correction to Eyring's reverberation equation.
This is called the Knudsen-Eyring reverberation formula, and it is commonly used to calculate reverberation time in the design of halls and other acoustic spaces. m is the rate of decay due to air, which varies with frequency, temperature, and humidity, but in design, the following values are used, assuming standard conditions of 20°C and 60% humidity.
1000Hz：0.001、2000Hz：0.002、4000Hz：0.006
(Excerpt from the email newsletter issued on October 21, 2010)
