### 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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Root-Power Waves

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Power Waves