Energy Density Waves

Energy density = potential + kinetic energy densities:

$$W(t,x) \mathrel{\stackrel{\mathrm{\Delta}}{=}} \underbrace{\frac{1}{2} Ky'^2(t,x)}_{\hbox{potential}} + \underbrace{\frac{1}{2} \epsilon {\dot y}^2(t,x)}_{\hbox{kinetic}}$$

Sampled wave energy density can be expressed as

$$W(t_n,x_m) \mathrel{\stackrel{\mathrm{\Delta}}{=}}W^{+}(n-m) + W^{-}(n+m)$$

where

$$\begin{eqnarray*} W^{+}(n) &=& \frac{{\cal P}^{+}(n)}{c} = \frac{f^{{+}}(n)v^{+}(n)}{c} = \epsilon \left[v^{+}(n)\right]^2 = \frac{\left[f^{{+}}(n)\right]^2}{K} \\ W^{-}(n) &=& \frac{{\cal P}^{-}(n)}{c} = -\frac{f^{{-}}(n)v^{-}(n)}{c} = \epsilon \left[v^{-}(n)\right]^2 = \frac{\left[f^{{-}}(n)\right]^2}{K} \nonumber \end{eqnarray*}$$

Total wave energy in string of length $L$ :

$${\cal E}(t) = \int_{x=0}^L W(t,x)dx \approx \sum_{m = 0}^{L/X-1} W(t,x_m)X$$