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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

$\displaystyle {\cal E}(t)$ $\displaystyle \isdef$ $\displaystyle \int_{0}^L W(t,x)dx = \int_{t_0}^{t_0+2L/c}{\cal P}(\tau,x) d\tau$ (C.54)
  $\displaystyle \approx$ $\displaystyle \sum_{m=0}^{N/2-1} W(t_n,x_m)X = \sum_{n=1}^{N}{\cal P}(t_0 + t_n,x_m) T$ (C.55)

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.
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