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## Problem

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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