1. Consider the wave equation that describes the vibrating string with fixed ends J²u მ2 = J²u მ2 ди u(x, 0) = f(x); (x, 0) = g(x) Ət u(0,t) = u(L, t) = 0. Use its Fourier series solution u(x,t) u(x, t)=sin( to show that -Σsin ("7") [a, cos(x) + b, sin (TC)] n=1 u(x,t) = R(x-ct) + S(x + ct), where R and S are some functions. Hints: i. sin a cos b = 1 [sin(a + b) + sin(a − b)]. ii. sin a sin b [cos(a - b) - cos(a + b)]. L