Q.Derive an expression for displacement of a plane progressive wave.
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Start your 14-day free trial to unlock the full solution →Combining the SHM equation of the wave source with the time delay for the wave to travel a distance x gives the plane progressive wave equation y = A sin(wt - kx).
Consider a wave travelling in the positive x-direction through a medium with speed v. Suppose the particle at the origin (x = 0) executes simple harmonic motion:
y(0, t) = A sin(w t)
where A is the amplitude and w = 2pif is the angular frequency of oscillation.
A particle located at a distance x from the origin starts oscillating only after the disturbance (wave) reaches it, which takes a time x/v. So its displacement at time t is the same as the origin's displacement at the earlier time (t - x/v):
y(x, t) = A sin[w (t - x/v)]
Expanding:
y(x, t) = A sin(wt - wx/v)
Define the wave number k = w/v = 2pif/v = 2pi/lambda (since v = flambda). Then:
y(x, t) = A sin(wt - kx)
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