Two waves having same amplitude A , angular frequency ω and propagation constant k , traveling in opposite directions, have a phase difference of 6 0 ∘ . If they interfere with each other, the resulta

Question

Two waves having same amplitude A , angular frequency ω and propagation constant k , traveling in opposite directions, have a phase difference of 6 0 ∘ . If they interfere with each other, the resultant displacement y of the wave is

TestHub · Accepted Answer

Let the equations of the two waves traveling in opposite directions be $y_{1} = A sin (ω t - k x)$ and $y_{2} = A sin (ω t + k x + ϕ)$ .

Given that the phase difference is $ϕ = 6 0^{\circ}$ .

The resultant displacement is given by the principle of superposition:

$y = y_{1} + y_{2}$

$y = A sin (ω t - k x) + A sin (ω t + k x + 6 0^{\circ})$

Using the trigonometric identity $sin C + sin D = 2 sin (\frac{C + D}{2}) cos (\frac{C - D}{2})$ , we get:

$y = 2 A sin (\frac{ω t - k x + ω t + k x + 6 0 ^{\circ}}{2}) cos (\frac{ω t + k x + 6 0 ^{\circ} - ( ω t - k x )}{2})$

$y = 2 A sin (ω t + 3 0^{\circ}) cos (k x + 3 0^{\circ})$

Answer: $2 A sin (ω t + 3 0^{\circ}) cos (k x + 3 0^{\circ})$

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