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Question
Find the amplitude of the resultant wave produced due to interference of two waves given as, y1 = A1 sin ωt, y2 = A2 sin (ωt + ϕ)
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Solution
Expression for the amplitude of resultant wave:
- Consider two waves having the same frequency but different amplitudes A1 and A2. Let these waves differ in phase by Φ.
- The displacement of each wave at x = 0 is given as y1 = A1sinωt and y2 = A2sin (ωt + Φ)
- According to the principle of superposition of waves, the resultant displacement at x = 0 is y = y1 + y2
∴ y = A1sinωt + A2 sin (ωt+ Φ)
y = A1sinωt + A2sinωt cosΦ + A2cosωt sinΦ
y = (A1 + A2cosΦ)sinωt + A2sinΦ cosωt
Let, A1 + A2 cosΦ = A cosθ ….(1)
A2 sinΦ = Asinθ ….(2)
∴ y = Acosθ sinωt + A sinθ cos ωt
∴ y = Asin(ωt + θ) - This is the equation for the displacement of the resultant wave. It has the same frequency as that of the interfering waves.
- The resultant amplitude A is given by squaring and adding equations (1) and (2).
A2 cos2θ + A2sin2θ = (A1 + A2cosΦ)2 + `"A"_2^2` sin2Φ
A2 = `"A"_2^2` + 2A1A2cosΦ + `"A"_2^2` cos2Φ + `"A"_2^2` sin2Φ
∴ A = `sqrt("A"_1^2 + 2"A"_1"A"_2"cos"phi + "A"_2^2)`
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