Q21 of 36 Page 109

The displacement of an elastic wave is given by the function y = 3 sin ωt + 4 cos ωt. where y is in cm and t is in second. Calculate the resultant amplitude.

Given:


Displacement equation of an elastic wave y = 3sinωt + 4cosωt


Let 3 = Acosθ and 4 = Asinθ, then the equation for displacement becomes:



by using the property



we have ,




Which is the equation for a simple harmonic wave with amplitude A and phase difference θ. For θ we have





For amplitude A, squaring and adding 3 = Acosθ and 4 = Asinθ, we have





And


Therefore, the equation for the wave becomes



with its amplitude being 5 m.


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