Total Energy in a Rigidly Terminated String

The total energy ${\cal E}$ in a length $L$, rigidly terminated, freely vibrating string can be computed as

$${\cal E}(t) \isdef \int_{0}^L W(t,x)dx = \int_{t_0}^{t_0+2L/c}{\cal P}(\tau,x) d\tau$$ (G.56)

$$\approx \sum_{m=0}^{N/2-1} W(t_n,x_m)X = \sum_{n=1}^{N}{\cal P}(t_0 + t_n,x_m) T$$ (G.57)

for any $x\in[0,L]$. Since the energy never decays, $t$ and $t_0$ are also arbitrary. Thus, because free vibrations of a doubly terminated string must be periodic in time, the total energy equals the integral of power over any period at any point along the string.