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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:
![$\displaystyle W(t,x) \isdefs \underbrace{\frac{1}{2} Ky'^2(t,x)}_{\mbox{potential}} + \underbrace{\frac{1}{2} \epsilon {\dot y}^2(t,x)}_{\mbox{kinetic}}$](img3508.png) |
(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
![$\displaystyle W(t_n,x_m) \isdefs W^{+}(n-m) + W^{-}(n+m),$](img3509.png) |
(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:
![$\displaystyle {\cal E}(t) \,\mathrel{\mathop=}\,\int_{x=-\infty}^\infty W(t,x)dx \approx \sum_{m = -\infty}^\infty W(t,x_m)X$](img3511.png) |
(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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