Three waves of equal frequency having amplitudes 13 μ m , 6 μ m , 5 μ m arrive at a given point with successive phase difference of π /2 . The amplitude of the resultant wave is (in μ m ) ( cos 9 0 ∘

Question

Three waves of equal frequency having amplitudes 13 μ m , 6 μ m , 5 μ m arrive at a given point with successive phase difference of π /2 . The amplitude of the resultant wave is (in μ m ) ( cos 9 0 ∘ = 0 , cos π = − 1 )

TestHub · Accepted Answer

Let the three waves be represented by phasors $A_{1}$ , $A_{2}$ , and $A_{3}$ .

Taking the first wave along the x-axis, the phase angles of the three waves are $0$ , $π /2$ , and $π$ respectively.

The x-component of the resultant amplitude is:
$R_{x} = A_{1} cos (0) + A_{2} cos (π /2) + A_{3} cos (π)$
$R_{x} = 13 (1) + 6 (0) + 5 (- 1) = 13 - 5 = 8 μ m$

The y-component of the resultant amplitude is:
$R_{y} = A_{1} sin (0) + A_{2} sin (π /2) + A_{3} sin (π)$
$R_{y} = 13 (0) + 6 (1) + 5 (0) = 6 μ m$

The resultant amplitude $R$ is given by:
$R = R_{x}^{2} + R_{y}^{2}$
$R = 8^{2} + 6^{2} = 64 + 36 = 100 = 10 μ m$

Answer: $10$

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