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Problem

Waventerference

Consider two travelling waves

$ y_1(t) = 0.3\sin(4x - 3t + \pi) $
$ y_2(t) = 0.3\sin(4x - 3t - \pi) $

Determine the sum of the two waves.

Hint: Use the identity

$ \sin(u) + \sin(v) = 2\sin \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right) $

Solution

$ y_1(t) + y_2 (t) = 0.3\sin(4x - 3t + \pi) + 0.3\sin(4x - 3t - \pi) $
$ y_1(t) + y_2 (t) = 0.3 \left[\sin(4x - 3t + \pi) + \sin(4x - 3t - \pi) \right] $

Use the identity

$ \sin(u) + \sin(v) = 2\sin \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right) $.
$ y_1(t) + y_2 (t) = 0.3 \left[2\sin(4x - 3t) \cos(\pi) \right] $
$ y_1(t) + y_2 (t) = -0.6\sin(4x - 3t) $
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