Wave energy density, energy flux and total energy

Let $n\ge 1$, $c>0$, let $U\subseteq {\mathbb{R}}^n$ be open, let $I\subseteq \mathbb{R}$ be an open interval and let $u:U\times I\to \mathbb{R}$ be $C^2$ ($C^k$ maps and multi-index derivative notation in Euclidean space). Write $u_t:={\mathrm{\partial }}_{t}u$ and let

$Du:=({\mathrm{\partial }}_{0}u,\dots ,{\mathrm{\partial }}_{n-1}u)$

be the spatial gradient of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, a $C^1$ field $U\times I\to {\mathbb{R}}^n$ whose Euclidean norm is $\mathrm{\mid }Du\mathrm{\mid }$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

The kinetic density, potential density and energy density of $u$ are the continuous functions

$e_{\mathrm{k}\mathrm{i}\mathrm{n}}:=\frac{1}{2}{u}_t^2,e_{\mathrm{p}\mathrm{o}\mathrm{t}}:=\frac{c^2}{2}\mathrm{\mid }Du{\mathrm{\mid }}^2,e:=e_{\mathrm{k}\mathrm{i}\mathrm{n}}+e_{\mathrm{p}\mathrm{o}\mathrm{t}}=\frac{1}{2}(u_t^2+c^2\mathrm{\mid }Du{\mathrm{\mid }}^2),$

and the energy flux is the $C^1$ vector field

$q:=-c^2{u}_{t}Du$

(Divergence and curl of a $C^1$ vector field). For a Lebesgue-measurable $\mathrm{\Omega }\subseteq U$ and a time $t\in I$ with ${\int }_{\mathrm{\Omega }}e(x,t)dx<\mathrm{\infty }$, the total energy in $\mathrm{\Omega }$ is the real number

$E_{\mathrm{\Omega }}(t):={\int }_{\mathrm{\Omega }}e(x,t)dx$

(The nonnegative Lebesgue integral); when the integral is infinite the total energy is $+\mathrm{\infty }$ and no real value is assigned.

Sign convention. If $\mathrm{\Omega }$ is admissible for the divergence theorem with outward unit normal $\nu$, then ${\int }_{\mathrm{\partial }\mathrm{\Omega }}q\cdot \nu dS$ is the energy leaving $\mathrm{\Omega }$ across $\mathrm{\partial }\mathrm{\Omega }$ per unit time. This is the convention under which the pointwise identity ${\mathrm{\partial }}_{t}e+\mathrm{div}q=u_t{\mathrm{\square }}_{c}u$ of The local wave-energy conservation law holds with ${\mathrm{\square }}_c={\mathrm{\partial }}_t^2-c^2\mathrm{\Delta }$ (Wave equation, Cauchy data and wave speed): the flux vector $q=-c^2{u}_{t}Du$ points in the direction of energy transport, and $\mathrm{div}q$ is the local rate at which energy leaves a point.

Speed convention. The gradient term carries the exact factor $c^2$, so at unit speed the energy density is the familiar $\frac{1}{2}{u}_t^2+\frac{1}{2}\mathrm{\mid }Du{\mathrm{\mid }}^2$ and the flux is $-u_{t}Du$. The general-speed statements of this page are rendered at $c>0$ throughout.

No equation for $u$, no finiteness of a particular integral and no regularity of any boundary are asserted here: each statement that uses these objects states its own hypotheses, and sufficient settings in which $E_{\mathrm{\Omega }}$ is finite and constant are supplied by Conservation of total wave energy in three admissible settings.