Find the total energy of vibration of a string of length L, fixed at both ends, oscillating in its n-th characteristic mode with an amplitude A. The tension in the string is T and its total mass is M. (Hint: consider the integrated kinetic energy at the instant when the string is straight so that it has no stored potential energy over and above what it would have when not vibrating at all.) (b) Calculate the total energy of vibration of the same string if it is vibrating in the following superposition of normal modes: x(z,t) = A1sin( π z L ) cos ω1 t + A3sin( 3 z L ) cos(ω3t - π 4 ). Show explicitly that it is the sum of the energies of the two modes taken separately.