3.7 Visualization of Acoustic Waves |
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j 2π=N |
k ∙ i |
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This is the discrete Fourier transform formula we will be using to calculate the weighted sound pressure level spectrum in Sect. 9.5.
3.7Visualization of Acoustic Waves
3.7.1Plotting Traveling Wave
The solutions of the acoustic wave equation expressed in both complex exponential functions and real trigonometric functions are derived in the previous chapter, and only Form 2 : REP is shown below for reference:
p(x, t) ¼ p (x, t) + p+(x, t)
¼ A cos (ωt + kx + ϕ ) + A+ cos (ωt kx + ϕ+) (Form 2 : REP)
where cos(ωt + kx + ϕ ) is the backward wave since ωt and kx have the same signs and cos (ωt kx + ϕ+) is the forward wave since ωt and kx have different signs.
For simplicity, let both phase shift angles ϕ and ϕ+ be zero and both amplitudes A and A+ be one, and we thus arrive at:
p(x, t) ¼p (x, t) + p+(x, t)
¼A cos (ωt + kx) + A+ cos (ωt kx) (Form 2 : REP)
For plotting a backward wave, assume a positive increment of negative decrement in space (left-hand side of the origin) x ¼ wave becomes:
p ðx, tÞ ¼ cos ðωt þ kxÞ ¼ cos |
ω t þ 8 |
þ k x þ |
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time t ¼ T8 and a8λ. The backward
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For plotting a forward wave, assume a positive increment of time t ¼ T8 and a |
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positive increment in space (right-hand side of the origin) |
x ¼ 8λ. The forward wave |
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then becomes: |
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cos |
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3 Solutions of Acoustic Wave Equation |
3.7.2Plotting Standing Wave
This section will demonstrate how standing waves work. The pressure function of a standing wave is separated into two pressure functions of a backward wave and a forward wave:
pðx, tÞ ¼ 2 cos ðωtÞ cos ðkxÞ ¼ p ðx, tÞ þ pþðx, tÞ
For a source at x ¼ 0, the pressure functions for the backward wave and the forward wave are:
Backward wave:
p ðx, tÞ ¼ cos ðωt þ kxÞ
Forward wave:
pþðx, tÞ ¼ cos ðωt kxÞ
3.7 Visualization of Acoustic Waves |
73 |
For a source at x ¼ 2λ, the pressure functions for the backward wave and the forward wave are as follows:
p |
¼ cos (ωt + k(x 2λ)) ¼ cos (ωt + kx 2kλ) |
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¼ cos (ωt + kx 4π) ¼ cos (ωt + kx) |
p+ |
¼ cos (ωt k(x 2λ)) ¼ cos (ωt kx + 2kλ) |
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¼ cos (ωt kx + 4π) ¼ cos (ωt kx) |
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3 Solutions of Acoustic Wave Equation |
Combining the waves induced by the sources at x ¼ 0 and x ¼ 2λ, we arrive at the standing wave between x ¼ (0, 2λ):
p ¼ p þ pþ ¼ cos ðωt þ kxÞ þ cos ðωt kxÞ ¼ 2 cos ðωtÞ cos ðkxÞ
In addition, the wave at the left of x ¼ 0 yields:
p ¼ p þ p ¼ 2 cos ðωt þ kxÞ
Moreover, the wave at the right of x ¼ 2λ thusly yields: p ¼ pþ þ pþ ¼ 2 cos ðωt kxÞ
The following figure shows the motion of a standing wave (shown in blue curves) as the result of the addition of a forward traveling wave (shown in black curves) and a backward traveling wave (shown in red curved) between x ¼ (0, 2λ):