9.2 Sound Pressure Levels |
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235 |
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! ½dB& |
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Lp ¼ 10 log 10 |
PRMS |
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¼ 20 log 10 |
PRMS |
¼ 20 log 10 |
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p2 |
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Pr |
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Pr |
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10 6 |
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9.2.5Sound Pressure Level Calculated in Frequency Domain
In the previous section, sound pressure level (SPL) was defined and calculated in the time domain. However, there is a limitation to calculating SPL in the time domain: weighted SPL cannot be calculated in the time domain because the weighting is frequency-dependent. For this reason, SPL is commonly formulated and calculated in the frequency domain:
TIME DOMAIN |
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FREQUENCY DOMAIN |
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FFT |
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Pressure in Time Domain |
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Pressure in Frequency Domain |
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+ ∑ |
cos |
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Unweighted RMS Pressure |
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Unweighted RMS Pressure |
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Sound Pressure Level (SPL)
Parseval’s Theorem Relates RMS Pressure
= 10 log
in Time Domain and Frequency Domain
Parseval’s theorem is used to formulate the RMS pressure in the frequency domain. Parseval’s theorem is shown and proved as follows:
Parseval’s Theorem
1 p2RMS ¼ P2o þ 12 X P2k
k¼1
Proof of Parseval’s Theorem
Based on the Fourier transform, a pressure function in time domain p(t) can be formulated in the frequency domain with the frequency contents Pk as:
X1
pðtÞ ¼ Po þ k¼1Pk cos ð2π f kt þ θkÞ
The RMS pressure of this function can be formulated with Parseval’s theorem as:
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9 Sound Pressure Levels and Octave Bands |
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pRMS2 = T Z0 |
T |
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p2ðtÞdt |
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1 |
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= T |
T |
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Po þ |
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k¼1Pk |
cos ð2π f kt þ θkÞ dt |
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Because Fourier series is an orthogonal basis which means that for i 6¼ j: |
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Z0 T |
cos ð2π f it þ θkÞ cos 2π f jt þ θk dt ¼ 0 |
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Apply the above orthogonal property for p2RMS results:
pRMS = T Z0 |
T |
Po þ |
k¼1Pk cos ð2π f kt þ θkÞ dt |
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=Po |
T Z0 |
T |
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T |
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cos |
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dt þ Xk¼1Pk |
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ð2π f kt þ θkÞdt |
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¼ Po2 þ |
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X |
Pk2 |
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k¼1 |
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Frequency contents of amplitude Pk and phase θk can be treated as a sound source radiating a pure tone in frequency fk with an amplitude Pk and phase θk.
Example 9.5: Sound Pressure Level Calculated in Frequency Domain
Given:
The single-frequency pressure time domain function p(t):
pðx, tÞ ¼ P1 cos ð2π f 1tÞ
where:
P1 ¼ 3 ½Pa&
f 1 ¼ 500 ½Hz&
Calculate:
(a)The RMS pressure in the frequency domain
(b)The unweighted sound pressure level