Word Count : 557

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We are given two waves: \[ y_1 (x, t) = 0.2 \sin (\pi (t - 2x)) \] \[ y_2 (x, t) = 0.2 \sin (\pi (t + 2x)) \] These waves travel in opposite directions, producing a standing wave. ### (a) Obtain the resultant displacement of the particle at x at time t. Using the principle of superposition: \[ y (x, t) = y_1 (x, t) + y_2 (x, t) \] Expanding the sine terms: \[ y (x, t) = 0.2 \sin (\pi (t - 2x)) + 0.2 \sin (\pi (t + 2x)) \] Using the trigonometric identity: \[ \sin A + \sin B = 2 \sin \left(\frac{A+B}{2} \right) \cos \left(\frac{A-B}{2} \right) \] Let \( A = \pi (t - 2x) \) and \( B = \pi (t + 2x) \): \[ \frac{A + B}{2} = \frac{\pi (t - __________ __________ _______ ____ _______ ______ ____ ________ _______.
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