Getting to Grips with Aircraft Noise

9 - A BIT OF THEORY
Using trigonometric relationship for the product of two cosines leads to:
1 1
2 2
1
2
1 2 1 2
1 2
,
Then the integration over T seconds of the product of these two cosines is null.
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( ωωωω ( t − r / c)
) . cos(
ωωωω ( t − r / c)
) = [ cos(
( ωωωω + ωωωω ) t − ( r + r ) / c)
+ cos(
( ωωωω − ωωωω ) t + ( r − r ) / c)
]
cos 2 1
The average value of each squared cosine being equal to ½, we finally have:
2
2
2 1max
2max
1 2 2 2 ⎥ ⎥ ⎡ p ⎤ ⎡ p ⎤
pe = +
(9.3.3-4)
⎢⎢⎣ r ⎥⎥⎦ ⎢⎢⎣ r ⎦
The SPL corresponding to the combination of the two sound sources at the chosen
point is then:
2 2 ⎡ p ⎤
e1
+ pe2
SPL = 10log
(9.3.3-5)
2
⎢⎢
⎣ pe0
⎥⎥
⎦
• It can then be deduced from the above that if the two sources have got the same
effective pressure, the SPL is increased by only 3 dB compared to a single
source emission.
• If the source number 2 has got an effective pressure that is half the one of the
source number 1, then:
2
2
2
⎡ 5 p ⎤ ⎡ ⎤ 5 ⎡ ⎤
e1
pe1
⎛ ⎞ pe1
SPL = 10log = 10log
+ 10log⎜⎝
⎟⎠
= 10log
+ 0.
97
2
2
2
⎢⎢
⎣4
p
4
e0
⎥⎥
⎦ ⎢⎢
⎣ pe0
⎥⎥
⎦
⎢⎢
⎣ pe0
⎥⎥
⎦
(9.3.3-6)
It shows that if one source is less powerful than the other, the increase in terms
of SPL is rather small. The most powerful source masks the other one,
highlighting the masking effect.
9.3.4. COMPLEX SOUND SIGNALS
• We already mentioned that the simplest form of an acoustic wave is the pure
tone.
( ωωωω + ϕϕϕϕ )
p( t)
= Acos
t
(9.3.4-1)
Where: A is the amplitude of the wave (in Pa)
ϕϕϕϕ is the phase of the wave (in rad)
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Getting to grips with aircraft noise