The total energy \(E(t)\) of the vibrating string is given as a function of time
by
$$
E(t)=\int_{0}^{L}\left[\frac{1}{2} \rho u_{t}^{2}(x, t)+\frac{1}{2} T
u_{x}^{2}(x, t)\right] d x ;
$$
the first term is the kinetic energy due to the motion of the string, and the
second term is the potential energy created by the displacement of the string
away from its equilibrium position.
For the displacement \(u(x, t)\) given by Eq. \((20),\) that is, for the solution
of the string problem with zero initial velocity, show that
$$
E(t)=\frac{\pi^{2} T}{4 L} \sum_{n=1}^{\infty} n^{2} c_{n}^{2}
$$
Note that the right side of Eq. (ii) does not depend on \(t .\) Thus the total
energy \(E\) is a constant, and therefore is conserved during the motion of the
string.
Hint: Use Parseval's equation (Problem 37 of Section 10.4 and Problem 17 of
Section \(10.3)\), and recall that \(a^{2}=T / \rho .\)