PHIL 130 Lecture 13: PHIL 130 - Lecture 13 (4)

Consider the evolution of a string of the length L, the ends of which are fixed. The positions of the ends are x = -L/2 and x = L/2. The motion of the string in the xy-plane is given by the wave equation Uu = C2 Uxx, for the amplitude u = u(x, t), x [-L/2, L/2], with the boundary conditions u(-L/2, t) = u(L/2, t) = 0. The initial condition is u(x, 0) = 0, ut(x, 0) = V0[1 - (2x/L)2]. [That is the initial amplitude is zero, but the initial velocity is finite, except for the end points.] Re-scale x, t, and u to get rid of letters in the equation, boundary conditions, and the initial condition. If you find it more convenient, you may also shift x. Make sure that the boundary conditions are canonical. Construct the orthonormal basis of the eigenfunctions of the Laplace operator in the space of functions obeying the boundary conditions (2). Find the solution u(x, t) of the problem (1)-(3) in the form of the Fourier series in terms of the constructed ONB. Restore the original variables x, t, and u.