### Energy Density Waves

The vibrational energy per unit length along the string, or wave energy density [317] is given by the sum of potential and kinetic energy densities:

 (C.50)

Sampling across time and space, and substituting traveling wave components, one can show in a few lines of algebra that the sampled wave energy density is given by

 (C.51)

where

Thus, traveling power waves (energy per unit time) can be converted to energy density waves (energy per unit length) by simply dividing by , the speed of propagation. Quite naturally, the total wave energy in the string is given by the integral along the string of the energy density:

 (C.52)

In practice, of course, the string length is finite, and the limits of integration are from the coordinate of the left endpoint to that of the right endpoint, e.g., 0 to .

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