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# 8soln8 - Physics 315 Oscillations and Waves Homework 8...

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Physics 315: Oscillations and Waves Homework 8: Solutions 1. Let x measure vertical distance upward, such that the bottom of the rope is at x = 0, and the top at x = L . Let the rope be of uniform mass per unit length ρ . The weight of that section of the rope suspended below some point situated at coordinate x is ρ x g . Thus, if the rope is in vertical equilibrium then the tension in the rope at this point is T ( x ) = ρ x g . Consider a wave pulse centered at coordinate x . Suppose that the spatial extent of the pulse is much less than the length of the rope. In this case, the variation of the rope tension across the pulse is negligible. Hence, the dispersion relation for the sinusoidal waves which make up the pulse can be calculated assuming that the rope tension takes the constant value T ( x ). In other words, the dispersion relation becomes ω bracketleftbigg T ( x ) ρ bracketrightbigg 1 / 2 k = g 1 / 2 x 1 / 2 k. (1) Thus, the group velocity of the pulse is v g = dk g 1 / 2 x 1 / 2 . (2) It follows that the pulse equation of motion takes the form dx dt = v g g 1 / 2 x 1 / 2 . (3) Hence, the time required for the pulse to travel from x = 0 to x = L is T = integraldisplay L 0 dx v g g - 1 / 2 integraldisplay L 0 dx x 1 / 2 = g - 1 / 2 bracketleftBig 2 x 1 / 2 bracketrightBig L 0 = 2 radicalbig L/g. (4) 2. From Eq. (5.13) in the lecture notes, the phase velocity of a transverse wave on a beaded string of finite length is ω = 2 parenleftbigg T m a parenrightbigg 1 / 2 sin( k a/ 2) , (5)

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where T is the string tension, a the bead spacing, and
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