Energy Density Waves

The vibrational energy per unit length along the string, or wave energy density [297] is given by the sum of potential and kinetic energy densities:

$$W(t,x) \isdef \underbrace{\frac{1}{2} Ky'^2(t,x)}_{\mbox{potential}} + \underbrace{\frac{1}{2} \epsilon {\dot y}^2(t,x)}_{\mbox{kinetic}}$$ (G.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

$$W(t_n,x_m) \isdef W^{+}(n-m) + W^{-}(n+m)$$ (G.51)

where

$$W^{+}(n) = \frac{{\cal P}^{+}(n)}{c} = \frac{f^{{+}}(n)v^{+}(n)}{c} = \epsilon \left[v^{+}(n)\right]^2 = \frac{\left[f^{{+}}(n)\right]^2}{K}$$ (G.52)

$$W^{-}(n) = \frac{{\cal P}^{-}(n)}{c} = -\frac{f^{{-}}(n)v^{-}(n)}{c} = \epsilon \left[v^{-}(n)\right]^2 = \frac{\left[f^{{-}}(n)\right]^2}{K}$$

Thus, traveling power waves (energy per unit time) can be converted to energy density waves (energy per unit length) by simply dividing by $c$, 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:

$${\cal E}(t) = \int_{x=-\infty}^\infty W(t,x)dx \approx \sum_{m = -\infty}^\infty W(t,x_m)X$$ (G.53)

In practice, of course, the string length is finite, and the limits of integration are from the $x$ coordinate of the left endpoint to that of the right endpoint, e.g., 0 to $L$.