Consider small amplitude waves on a string of length 1 (in dim'less units) lying on 0 x 1, pinned at both ends such that the waves travel at a speed of 2 (in dim'less units). At t=0, the shape of the string is given by the profile H(x) = A x(x - 1), with A < < 1 (i.e. small amplitude), and string is momentarily at rest. Determine the complete wavefunction y(x, t) in terms of... a) a Fourier eigenfunction superposition (expressed so that all terms in the superposition are nonzero); the d'Alembert solution (expressed in terms of H and/or its periodic extensions). b) c) As a multiple of A, what is the exact (i.e. rational) displacement and (transverse) velocity of the particle at x= 3/8 when t = 7/16? (Note: the derivative of an even/odd function is odd/even respectively.)