Halliday, Resnick & Walker — Waves–I: Problem Problem_92

Two waves, \[ y_1 = (2.50 \text{ mm}) \sin[(25.1 \text{ rad/m})x - (440 \text{ rad/s})t] \] and \[ y_2 = (1.50 \text{ mm}) \sin[(25.1 \text{ rad/m})x + (440 \text{ rad/s})t] \], travel along a stretched string. (a) Plot the resultant wave as a function of \( t \) for \( x = 0, \lambda/8, \lambda/4, 3\lambda/8, \) and \( \lambda/2 \), where \( \lambda \) is the wavelength. The graphs should extend from \( t = 0 \) to a little over one period. (b) The resultant wave is the superposition of a standing wave and a traveling wave. In which direction does the traveling wave move? (c) How can you change the original waves so the resultant wave is the superposition of standing and traveling waves with the same amplitudes as before but with the traveling wave moving in the opposite direction? Next, use your graphs to find the place at which the oscillation amplitude is (d) maximum and (e) minimum. (f) How is the maximum amplitude related to the amplitudes of the original two waves? (g) How is the minimum amplitude related to the amplitudes of the original two waves?