### 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

*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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