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Interference of Sound Waves

When two waves of same frequency, same wavelength, and same velocity (nearly equal amplitude) move in the same direction, their superimposition results in the interference. Due to interference, the resultant intensity of sound at that point is different from the sum of intensities due to each wave separately. This modification of intensity due to superposition of two or more waves is called interference.
Let at a given point two waves arrive with phase difference φ and the equation of these waves is given by
y1 = a1 sin ωt, y2 = a2 sin (ω t + φ)
Then by the principle of superposition
32070.png y = A sin (ω t + θ)
where 32064.png
and tan θ = 32058.png
and since intensity ∝ A2,
So I = 32052.png

Some Important Points
Constructive interference: Intensity will be maximum when
φ = 0, 2π, 4π …, 2πn, where n = 0, 1, 2 …
x = 0, λ, 2λ, …, nλ, where n = 0, 1, ...
Imax = I1 I2 + 34987.png
It means the intensity will be maximum at those points where path difference is an integral multiple of wavelength λ. These points are called points of constructive interference or interference maxima.
Destructive interference: Intensity will be minimum when
φ = π, 3π, 5π, …, (2n – 1)π, where n = 1, 2, 3,…
x = λ/2, 3λ/2, …, (2n – 1)λ/2, where n = 1, 2, 3, ...
Imin = I1 I2 – 233504.png
⇒ Imin = 33498.png
  • All maxima are equally spaced and equally loud. Same is also true for minima. Also interference maxima and minima are alternate as for maximum Δx = 0, λ, 2λ, …, etc. and for minimum 33492.png.
  • 33538.png
  • If I1 = I2 = I0, then Imax = 4I0 and Imin = 0.

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