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Energy Density Waves
The vibrational energy per unit length along the string, or wave
energy density [320] 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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