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
![\begin{eqnarray*}
W^{+}(n) &=& \frac{{\cal P}^{+}(n)}{c} \,\mathrel{\mathop=}\,\...
...ht]^2 \,\mathrel{\mathop=}\,\frac{\left[f^{{-}}(n)\right]^2}{K}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img3508.png)
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


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Root-Power Waves
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Power Waves