Last time, I introduced the origins of Sabine's equation not from a theoretical standpoint, but from the experimental approach of the time that formalized the concept of reverberation time. To reiterate, Sabine's equation is a simple formula that is proportional to the volume of the room and inversely proportional to the equivalent sound-absorbing area, as shown below.
.................................(1)
T: Reverberation time (seconds)
K = 0.161
V: Room volume (m 3)
A: Equivalent sound absorption area of the room (m²)
This formula is accurate when the sound decays exponentially in a diffuse sound field, and it agrees well with measured values in rooms with long reverberation times and good diffusion, such as reverberation chambers. However, in rooms with short reverberation times or poor diffusion (for example, rooms with low ceiling heights and a large ratio of width to depth, or rooms where sound-absorbing materials are unevenly distributed), the difference between the reverberation time calculated by formula (1) and the measured value tends to be large.
For example, if the average sound absorption coefficient α = 1.0, then the sound absorption coefficient of the interface that makes up the room is 1.0 (perfect sound absorption), so the reverberation time should be 0 seconds, but equation (1) still has a value.
The fact that such physical solutions do not always align in clear-cut cases is said to have greatly troubled Sabine in his later years.
Eyring solved this problem. Today, when considering the reverberation time of halls, cinemas, conference rooms, etc., it is common to use the Eyring-Knudsen equation, which adds a term for sound wave absorption (air absorption) to Eyring 's equation. In this and the next article, I would like to explain the derivation of the Eyring-Knudsen equation.
Eyring treated the indoor sound field as the sum of direct sound from the sound source and reflected sound from the interface. Furthermore, he interpreted the reflected sound as a wave radiated from the mirror image of the sound source with respect to the interface. Considering the changes over time, we consider the process in which sound is generated from a sound source in the room, the direct sound arrives first, and then the reflected sound arrives, and the energy grows step by step. This process corresponds to the growth process portion in Figure 1 of the 16th installment shown two installments ago.
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<16th Edition Figure 1 Time waveform of sound source and receiver in a room>
The distance a sound wave travels from a sound source to the next interface after being reflected once is not constant, but its average value is defined as the mean free path p. The time between reflections is p/c seconds (c: speed of sound), and the energy generated from the sound source with output W during that time is W・p/c .
Furthermore, if the average sound absorption coefficient of the wall is α-, then the output of the mirror image of the nth reflected sound is W ・ (1-α-) n, so an energy of W ・ (1-α-) n ・ p/c is generated between reflections.
Here, we consider the acoustic energy density of the room, including up to n reflections. This acoustic energy density is the sum of the energy densities of the direct sound and the n reflected sounds.
This can be expressed as equation (2) below.
.................................(2)
α―: Average sound absorption coefficient of interior surface V: Room volume (m³)
p:平均自由行路 W:音源の出力
Here, the energy density of the steady state E0 teeth, n → ∞ at that time (1 - α―) n → 0 Therefore E 0 = p W / c Vα―This is the case. 16 Circular type (5) The Sabine Similar to the theory E 0 = 4 W / c Sα―It must be equal to .
From this, the mean free path can be found by solving the equation p W / cVα = 4 W / c Sα, which gives p = 4 V / S.
That's all for today. Next time, we'll explain the process of deriving Eyring 's damping equation from this steady state.
(Excerpt from the email newsletter issued on September 22, 2010)
