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[VW29] φ(x) A Consider a taut string with equilibrium length L which is held fixed at y = 0 at x = 0 and x = L. At t = 0 we pluck the string, thereby starting it off from rest in the shape given by the triangular function (1) with height A shown in the figure below. х L/2 0 The first step to solving the string's wave equation is to express the initial condition (2) as a linear combination of the system's normal modes as $(x) = An sin(nac/L) n=1 where the mode amplitudes are given by An 1. = 1 $* ®lar173. 2 L 0:2) sin(n+x/L) dr. 0 пл = (a) Explain why you expect all the An amplitudes for even n to be zero. Hint: No calculations should be necessary here. Just think about the shapes of the functions appearing in the integral for An. As always, drawing some figures will help. (b) Prove that 8A An sin (6) n272 2 Show your steps carefully. You can save some work by using the result of part (a). (c) Write down the final form of the solution for y(x, t). Take the speed of waves in the string to be v. (d) Insert your expression for y(x, t) into the provided Mathematica file. You should find very good agreement between your result and the slow-motion video of a plucked rubber band.