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1. Vibrations of a uniform string are described by the wave equation au at2 02 a2u ar2 = 0, where v > 0 is a constant. Consider a string clamped at the ends; this corresponds to boundary conditions u(0,t) = u(L,t) for all t. Suppose at the initial moment t=0, the string is displaced as follows: A sin 2 0<x<L/2 ul 0, L/2 < x <L. L; u(2,0) = { = where the amplitude A is a parameter. The initial velocity is zero. (a) Expand u(x, 0) in the complete orthogonal system X;(x) = sin(kjx), where kj TjL, j = 1, 2, .... That is, find the coefficients C; in = == ? u(x,0) = c;X;(x). j=1 Ecx You may find useful the identity = 1 2 sin(k22) sin(k; 2) = 3 (cos(k2 – k;)– cos(k2 + kj)a). x x x]. Please distinguish between even and odd ;. The coefficient C2, corresponding to j = 2, is special. Please compute it separately. (b) Search for u(x, t) as a superposition of the normal modes: u(x, t) = T;(t)X;(x). ,),α). = j=1 Obtain an ODE for each of functions T;(t) and initial conditions for it, in terms of the coefficients C; from part (a). (c) Solve the ODEs from part (b), thereby finding T;(t) and u(x, t) for all t > 0.