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# Superposition of travelling waves (Beats)

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To see how two traveling waves of nearly the same frequency can create beats and to interpret the superposition as a "walking" wave, consider two similar traveling transverse waves, which might be traveling for example along a string:

y1(x,t) = Asin(k1 x - w1 t) and y2(x,t) = Asin(k2 x - w2 t). They are similar because we assume that k1 and k2 are nearly equal and also that w1 and w2 are nearly equal.

a). Find C, y_envelope (x,t), and y_carrier (x,t). Express your answer in terms of A, k1, k2, x, t, w1 and w2. Separate the three terms with commas. Recall that y_envelope (the second term) varies slowly whereas y_(rm carrier) (the third term) varies quickly. Both y_envelope and y_carrier should be trigonometric functions of unit amplitude.

b). The envelope function can be written simply in terms of delta_k = k_1 - k_2 and delta_w= w1 - w2. If you do so, what is v_group, the velocity of propagation of the envelope? Express your result in terms of delta_k and delta_omega.