$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ##2.7. Energy integral ---------------------- > 3. [Energy integral: wave equation](#sect-2.7.1) > 4. [Energy integral: Maxwell equation](#sect-2.7.2) > 5. [Energy integral: Elasticity equations](#sect-2.7.3) ###Energy integral: wave equation Consider multidimensional wave equation \begin{equation} u\_{tt}-c^2 \Delta u=0. \label{eq-2.7.1} \end{equation} Recall that $\Delta =\nabla\cdot \nabla=\partial_x^2+\partial_y^2+\partial_z^2$ (the number of terms depends on dimension). Multiplying by $u\_t$ we get in the left-hand expression \begin{align\*} u\_t u\_{tt}-c^2 u\_t\nabla^2 u= &\partial\_t \bigl(\frac{1}{2}u\_t^2 \bigr)+ \nabla \cdot (-c^2 u\_t \nabla u )+ c^2 \nabla u\_t \cdot \nabla u \\\\ =&\partial\_t \bigl(\frac{1}{2}u\_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)+ \nabla \cdot \bigl(-c^2 u\_t \nabla u \bigr). \end{align\*} So we arrive to \begin{equation} \partial\_t \bigl(\frac{1}{2}u\_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)+ \nabla \cdot \bigl(-c^2 u\_t \nabla u \bigr). \label{eq-2.7.2} \end{equation} This is *energy conservation law in the differential form.* Here \begin{equation} e= \bigl(\frac{1}{2}u\_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr) \label{eq-2.7.3} \end{equation} is a *density of energy* and \begin{equation} \mathbf{S}=-c^2 u\_t \nabla u \label{eq-2.7.4} \end{equation} is a vector of *energy flow.* Then if we fix a volume (or an area in $2D$ case, or just an interval in $1D$ case) $V$ and introduce a full energy in $V$ at moment $t$ \begin{equation} E\_V(t)= \iiint \_V \bigl(\frac{1}{2}u\_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)\, dV \label{eq-2.7.5} \end{equation} then \begin{equation} E\_V(t\_2) - E\_V(t\_1) + \int\_{t\_1}^{t\_2}dt \iint\_\Sigma \mathbf{S}\cdot \mathbf{n}\,d\sigma =0 \label{eq-2.7.6} \end{equation} where $\Sigma$ is the surface bounding $V$, $d\sigma$ is an element of the surface area, and $\mathbf{n}$ is an unit exterior normal to $\Sigma$. ###Energy integral: Maxwell equation Similarly, *Maxwell equations* without charges and currents are \begin{align} &\varepsilon \mathbf{E}\_t = c\nabla \times \mathbf{H},\label{eq-2.7.7}\\\\ &\mu\mathbf{H}\_t = -c\nabla \times \mathbf{E},\label{eq-2.7.8}\\\\ &\nabla \cdot \varepsilon\mathbf{E}=\nabla \cdot \mu\mathbf{H}=0. \label{eq-2.7.9} \end{align} Here $\mathbf{E}, \mathbf{H}$ are intensities of electric and magnetic field respectively, $c$ is the speed of light in the vacuum, $\varepsilon$ and $\mu$ are dialectric and magnetic characteristics of the media ($\varepsilon\ge 1$, $\mu \ge 1$ and $\varepsilon=\mu=1$ in the vacuum). Multiplying (taking an inner product) (\ref{eq-2.7.7}) by $\mathbf{E}$ and (\ref{eq-2.7.8}) by $\mathbf{H}$ and adding we arrive to \begin{equation\*} \partial\_t \bigl(\frac{1}{2}\varepsilon|\mathbf{E}|^2 + \frac{1}{2}\mu |\mathbf{H}|^2\bigr)= c\bigl(\mathbf{E}\cdot (\nabla \times \mathbf{H}) - \mathbf{H}\cdot (\nabla \times \mathbf{E})\bigr)= -c\nabla \cdot \bigl(\mathbf{E}\times \mathbf{H}\bigr) \end{equation\*} where the last equality follows from vector calculus. Then \begin{equation} \partial\_t \bigl(\frac{1}{2}\varepsilon |\mathbf{E}|^2 + \frac{1}{2}\mu |\mathbf{H}|^2\bigr)+ \nabla \cdot \bigl(c\mathbf{E}\times \mathbf{H}\bigr)=0. \label{eq-2.7.10} \end{equation} In the theory of electromagnetism \begin{equation} e=\frac{1}{2}\bigl(\varepsilon |\mathbf{E}|^2 + \mu |\mathbf{H}|^2\bigr) \label{eq-2.7.11} \end{equation} is again *density of energy* and \begin{equation} \mathbf{S}=c \mathbf{E}\times \mathbf{H} \label{eq-2.7.12} \end{equation} is a vector of *energy flow* (aka *Poynting vector*). **Remark 1.** $\frac{c}{\sqrt{\mu \varepsilon}}$ is the speed of light in the given media. **Remark 2.** In inhomogeneous media $\varepsilon$ and $\mu$ depend on $(x,y,z)$; in anisotropic media (crystals) $\varepsilon$ and $\mu$ are matrices and then $e=\frac{1}{2}\bigl(\varepsilon \mathbf{E} \cdot \mathbf{E} + \mu \mathbf{H}\cdot \mathbf{E}\bigr)$. ###Elasticity equations Elasticity equations in homogeneous isotropic media are \begin{equation} \mathbf{u}\_{tt}= 2\mu \Delta \mathbf{u} + \lambda \nabla (\nabla \cdot \mathbf{u}) \label{eq-2.7.13} \end{equation} where $\mathbf{u}$ is a *displacement* and $\lambda>0$, $\mu>0$ are *Lamé parameters*. **Problem 1.** Multiplying (taking an inner product) (\ref{eq-2.7.13}) by $\mathbf{u}_t$ write conservation law in the differential form. What are $e$ and $\mathbf{S}$? ______________ [$\Uparrow$](../contents.html) [$\uparrow$](./S2.7.html) [$\downarrow$](./S2.7.P.html) [$\Rightarrow$](./S2.8.html)