$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###Problems > 1. [Problem 1](#problem-2.7.P.1) > 2. [Problem 2](#problem-2.7.P.2) > 3. [Problem 3](#problem-2.7.P.3) > 4. [Problem 4](#problem-2.7.P.4) > 5. [Problem 5](#problem-2.7.P.5) **Problem 1.** For equation \begin{equation} u\_{tt}-c^2 u\_{xx}+ f(u)=0, \qquad x\> 0 \end{equation} prove energy conservation law \begin{equation} E(t)=\frac{1}{2}\int\_0^\infty \bigl(u\_t^2+ c^2 u\_{x}^2+F(u)\bigr)\,dx \end{equation} under Dirichlet ($u|\_{x=0}=0$) or Neumann ($u\_x|\_{x=0}=0$) boundary condition; here $F$ is a primitive of $f$. **Problem 2.** For *beam equation* \begin{equation} u\_{tt}+K u\_{xxxx}=0, \qquad x\> 0,\\\\ \end{equation} prove energy conservation law \begin{equation} E(t)=\frac{1}{2}\int\_0^\infty (u\_t^2+ K u\_{xx}^2)\,dx \end{equation} under each of the pair of the boundary conditions: \begin{align} &u|\_{x=0}=u\_x |\_{x=0}=0;\\\\ &u|\_{x=0}=u\_{xxx} |\_{x=0}=0;\\\\ &u\_{x}|\_{x=0}=u\_{xx} |\_{x=0}=0. \end{align} **Problem 3.** a. For problem \begin{align} &u\_{tt}-c^2 u\_{xx}=0, &&x \>0,\\\\ &(\alpha\_0 u\_x+\alpha\_1 u)|\_{x=0}=0 \end{align} find energy conservation law \begin{equation} E(t)=\frac{1}{2}\int\_0^\infty (u\_t^2+ c^2 u\_{x}^2)\,dx + \frac{1}{2}a u(0)^2 \end{equation} (you need to calculate $a$). b. For problem \begin{align} &u\_{tt}-c^2 u\_{xx}=0, &&0\< x \< l,\\\\ &(\alpha\_0 u\_x+\alpha\_1 u)|\_{x=0}=0,\\\\ &(\beta\_0 u\_x-\beta\_1 u)|\_{x=l}=0 \end{align} find energy conservation law \begin{equation} E(t)=\frac{1}{2}\int\_0^\infty (u\_t^2+ c^2 u\_{x}^2)\,dx + \frac{1}{2}a u(0)^2+ \frac{1}{2}b u(l)^2 \end{equation} (you need to calculate $a$, $b$). **Problem 4.** For problem \begin{align} &u\_{tt}-c^2 u\_{xx}=0, &&x \>0,\\\\ &(u\_x-i\alpha u\_t)|\_{x=0}=0 \end{align} with real $\alpha$ prove energy conservation law \begin{equation} E(t)=\frac{1}{2}\int\_0^\infty (|u\_t|^2+ c^2 |u\_{x}|^2)\,dx \end{equation} **Problem 5.** Consider [Problem 2.6.2](./S2.6.P.html#problem-2.6.2). \begin{align\*} & u\_{tt}-c\_1^2u\_{xx}=0, &&&t\>0, x\>0, \\\\ & u\_{tt}-c\_2^2u\_{xx}=0, &&&t\>0, x\<0, \\\\ &u|\_{t=0}= \phi (x), &&u\_t|\_{t=0}= c\_1\phi'(x) &x\>0, \\\\ &u|\_{t=0}=0, &&u\_t|\_{t=0}=0, &x\<0,\\\\ &u|\_{x=+0}=\alpha u|\_{x=-0}, &&u\_x|\_{x=+0}=\beta u\_x|\_{x=-0} &t\>0 \end{align\*} Let $E\_1(t)=\frac{m\_1}{2}\int\_0^\infty (u\_t^2 + c\_1^2 u\_x^2)\,dx$, $E\_2(t)=\frac{m\_2}{2}\int\_{-\infty}^0 (u\_t^2 + c\_2^2 u\_x^2)\,dx$. a. Find $m\_1:m\_2$ such that $E(t)=E\_1(t)+E\_2(t)$ is conserved. b. In this case prove that if $\phi (x)=0$ for $x\> L$ then for $t \> L/c\_1$ $E\_1(t)= k\_1 E(0)$, $E\_2(t)= k\_2 E(0)$. Calculate $k\_1,k\_2$ and prove that $k\_1+k\_2=1$. ______________ [$\Leftarrow$](./S2.6.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](./S2.8.html)