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[VW30] Zero slope, y(x,t) ok an ห= 0 2L Consider a taut string with equilibrium length L. The string's left end is tied down while its right end is attached to a massless ring which slides vertically along a frictionless pole. Since the ring is massless it moves with the string without the string having to exert a vertical force on it. As a result of this the slope of the string is always zero at x = L. (a) Formulate the two boundary conditions on the solution of the wave equation de- scribing the string. (b) Start from the standing wave solutions we derived in class and determine the normal modes of the string, as well as their wavenumbers kn, wavelengths in and frequencies Wn. Take the speed of the waves in the string to be v. You should be able to label the modes with an integer n = 0,1,2,... (C) Draw the first three (lowest frequency) modes.

(d) Write down the general form of the solution of y(x,t) in terms of the quantities you defined above and the mode amplitudes An. Assume that the string starts moving from rest. To determine the An coefficients we can use Fourier analysis, since the mode functions again form a complete orthogonal set. It should be clear by now that the integrals required to prove the orthogonality of these functions and to calculate the An coefficients are usually quite simple, but also tedious to do by hand. We will therefore let Mathematica perform these integrals for us for a particular choice of initial condition. (e) Complete the Mathematica code in the provided file. From the final animation, comment on how the wave evolves at points away from the two ends, and also on what happens to the wave at the two ends. Are these results consistent with what you know about the general solution of the wave equation and about wave reflection? (f) [Bonus] Suppose the ring had a non-zero mass of m. (i) Formulate the boundary conditions on the string, keeping in mind the assump- tion yr <1 entering in the derivation of the wave equation. (ii) If the mass of the ring is one fifth of that of the string, determine, in terms of L, the wavelengths of the first two modes.