262 |
10 Room Acoustics and Acoustical Partitions |
Pd ¼ RMS pressure in a particular direction at a distance r from the source
P ¼ average RMS pressure over a sphere at a distance r from the source The directivity factor Qd for an isotropic acoustic source is:
Qd ¼ 1 [isotropic source]
In general, acoustic sources are not isotropic, and the direct energy density must be modified according to the directivity of the source.
Reverberant Energy Density
The energy density δ is related to the sound power w by an area S as:
δ ¼ cSw
The reverberant energy density is a summation of the reflected energy between two parallel walls as:
ð1 αÞ þ ð1 αÞ2 þ . . . þ ð1 αÞn ¼ ð1 αÞ ¼ S ; n ¼ 1
α R
Also, the reverberant energy density is a summation of the reflected energy. Therefore, the sound power w is increased by:
δr |
¼ cS ¼ cS ð |
α |
¼ cS |
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w Qrw |
1 αÞ |
4w |
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where S is the total area of the room and R is the room constant considering the reverberant effect and is related to the area S as shown in Sect. 10.2.2 as:
α
R ¼ 1 α S
And Qr is the reflectivity factor equals:
Qr ¼ 4
Total Energy Density
The total energy density δ is the sum of the direct and reverberant energy densities:
δ¼ δd þ δr
¼4wQπr2c þ 4cRw
10.3 Room Acoustics |
263 |
10.3.2 Sound Pressure Level due to an Acoustic Source
Equating the energy density expression above to the energy density from the sound
pressure δ ¼ P2RMS (see Sect. 12.1.3) yields a relationship between the RMS pressure
ρoc2
in a room and the acoustic source, as shown below:
δ¼ δd þ δr
!P2RMS ¼ wQ þ 4w ρoc2 4πr2c cR
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If we divide both sides of the square RMS pressure expression by the square of the international pressure reference Pr ¼ 20 10 6 [Pa], we obtain:
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P2 |
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wρ c |
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20 10 6 2 ¼ |
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ρoc |
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10 12 |
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4πr2 |
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Since ρoc 400: |
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20 10 6 2 |
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Take log base 10 of both sides of the above equation to arrive at:
10 log |
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P2 |
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10 6 2 |
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Because the 10 12 |
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in the equation above is the international sound power |
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reference, therefore, |
the equation above can be formulated in the form of levels as: |
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LP ¼ Lw þ 10 log 4πr2 |
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where LP is the sound pressure level at a distance r from the acoustic source considering both the direct and reverberant sound waves, Lw is the sound power of the acoustic source, Q is the directivity factor of the acoustic source and is equal to one for isotropic acoustic sources, and R is room constant that is related to the area and the absorption coefficient of the surface.