$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###[Problems](id:sect-2.3.P) > 1. [Problem 1](#problem-2.3.P.1) > 2. [Problem 2](#problem-2.3.P.2) > 3. [Problem 3](#problem-2.3.P.3) > 4. [Problem 4](#problem-2.3.P.4) > 5. [Problem 5](#problem-2.3.P.5) > 5. [Problem 6](#problem-2.3.P.6) > 1. [Problem 7](#problem-2.3.P.7) > 2. [Problem 8](#problem-2.3.P.8) > 3. [Problem 9](#problem-2.3.P.9) > 4. [Problem 10](#problem-2.3.P.10) **[Problem 1.](id:problem-2.3.P.1)** Find the general solutions of \begin{gather} u\_{tt}-u\_{xx}=0; \\\\ u\_{tt}-4u\_{xx}=0; \\\\ u\_{tt}-9u\_{xx}=0; \\\\ 4u\_{tt}-u\_{xx}=0; \\\\ u\_{tt}-9u\_{xx}=0. \end{gather} **[Problem 2.](id:problem-2.3.P.2)** Solve IVP \begin{align} &u\_{tt}-c^2u\_{xx}=0, \\\\ &u|\_{t=0}=g(x),\quad u\_t|\_{t=0}=h(x) \end{align} with \begin{align\*} &g(x)=0,\ && h(x)=1; \\\\ &g(x)=\left\\{\begin{aligned} &0 &&x \< 0,\\\\ &1 &&x \ge 0. \end{aligned}\right. &&h(x)=0; \\\\ &g(x)=\left\\{\begin{aligned} &1 &&|x| \< 1,\\\\ &0 &&|x| \ge 1. \end{aligned}\right. &&h(x)=0; \\\\ &g(x)=\left\\{\begin{aligned} &\cos (x) &&|x| \< \pi/2,\\\\ &0 &&|x| \ge \pi/2. \end{aligned}\right. &&h(x)=0; \\\\ &g(x)=0, &&h(x)=\left\\{\begin{aligned} &0 &&x \< 0,\\\\ &1 &&x \ge 0. \end{aligned}\right. \\\\ &g(x)=0, &&h(x)=\left\\{\begin{aligned} &1 &&|x| \< 1,\\\\ &0 &&|x| \ge 1. \end{aligned}\right. \\\\ &g(x)=0, &&h(x)=\left\\{\begin{aligned} &\cos (x) &&|x| \< \pi/2,\\\\ &0 &&|x| \ge \pi/2. \end{aligned}\right. \end{align\*} **[Problem 3.](id:problem-2.3.P.3)** Find solution to equation \begin{equation} Au\_{tt}+2Bu\_{tx}+Cu\_{xx}=0 \end{equation} as \begin{equation} u=f(x-c\_1 t)+ g(x-c\_2t) \end{equation} with arbitrary $f,g$ and real $c\_1< c_2$. a. What equation should satisfy $c\_1$ and $c\_2$? b. When this equation has such roots? **[Problem 4.](id:problem-2.3.P.4)** A spherical wave is a solution of the three-dimensional wave equation of the form $u(r, t)$, where r is the distance to the origin (the spherical coordinate). The wave equation takes the form \begin{equation} u\_{tt} = c^2 \bigl(u\_{rr}+\frac{2}{r}u\_r\bigr) \qquad\text{("spherical wave equation").} \label{eq-4a} \end{equation} a. Change variables $v = ru$ to get the equation for $v$: $v\_{tt} = c^2 v\_{rr}$. b. Solve for $v$ using \begin{equation} v = f(r+ct)+g(r-ct) \end{equation} and thereby solve the spherical wave equation. c. Use \begin{equation} v(r,t)=\frac{1}{2}\bigl[ \phi (r+ct)+\phi (r-ct)\bigr]+\frac{1}{2c}\int\_{r-ct}^{r+ct}\psi (s) \,ds \end{equation} with $\phi(r)=v(r,0)$, $\psi(r)=v\_t(r,0)$ to solve it with initial conditions $u(r, 0) = \Phi (r)$, $u\_t(r, 0) = \Psi(r)$. d. Find the general form of solution $u$ to (\ref{eq-4a}) which is continuous as $r=0$. **[Problem 5.](id:problem-2.3.P.5)** Find formula for solution of the *Goursat problem* \begin{align} &u\_{tt} - c^2 u\_{xx}=0, && x \> c|t|, \\\\ &u|\_{x=-ct}=g(t), && t\<0, \\\\ &u|\_{x=ct}=h(t), &&t \> 0 \end{align} as $g(0)=h(0)$. **[Problem 6.](id:problem-2.3.P.6)** Often solution in the form of *travelling wave* $u=\phi (x-vt)$ is sought for more general equations. Here we are interested in the bounded solutions, especially in those with $\phi(x)$ either tending to $0$ as $|x|\to \infty$ (solitons) or periodic (kinks). Plugging such solution to equation we get ODE for function $\phi$, which could be either solved or at least explored. Sure we are not interested in the trivial solution which is identically equal to $0$. a. Find such solutions for each of the following equations \begin{align} &u\_{tt}-c^2 u\_{xx}+m^2 u=0;\\\\ &u\_{tt}-c^2 u\_{xx}-m^2 u=0; \end{align} the former is *Klein-Gordon equation*. Describe all possible velocities $v$. b. Find such solutions for each of the following equations \begin{align} &u\_{t}-K u\_{xxx}=0;\\\\ &u\_{t} - iKu\_{xx} =0;\\\\ &u\_{tt}+Ku\_{xxxx}=0; \end{align} **[Problem 7.](id:problem-2.3.P.7)** Look for solutions in the form of travelling wave for sine-Gordon equation \begin{equation} u\_{tt}-c^2 u\_{xx}+\sin(u)=0. \end{equation} observe that resulting ODE is describing *mathematical pendulum* which could be explored. Describe all possible velocities $v$. **[Problem 8.](id:problem-2.3.P.8)** Look for solutions in the form of travelling wave for each of the following equations \begin{align} u\_{tt}-u\_{xx}+u -2u^3=0; \\\\ u\_{tt}-u\_{xx}-u +2u^3=0; \end{align} a. Describe kinks. Describe all possible velocities $v$. b. Find solitons. Describe all possible velocities $v$. **[Problem 9.](id:problem-2.3.P.9)** For a solution $u(x, t)$ of the wave equation $u\_{tt}=c^2u\_{xx}$, the *energy density* is defined as $e=\frac{1}{2}\bigl(u\_t^2+c^2 u\_x^2\bigr)$ and the *momentum density* as $p =c u\_t u\_x$. a. Show that \begin{equation} \frac{\partial e}{\partial t} = c\frac{\partial p}{\partial x} \qquad \text{and} \qquad \frac{\partial p}{\partial t} = c\frac{\partial e}{\partial x}. \label{eq-HA2.11} \end{equation} b. Show that both $e(x, t)$ and $p(x,t)$ also satisfy the same wave equation. **[Problem 10.](id:problem-2.3.P.10)** a. Consider wave equation $u\_{tt}-u\_{xx}=0$ in the rectangle $0\< x\< a$, $0\< t \< b$ and prove that if $a$ and $b$ are not *commensurable* (i.e. $a:b$ is not rational) then Dirichlet problem $u|\_{t=0}=u\_{t=b}=u|\_{x=0}=u|_{x=a}=0$ has only trivial solution. b. On the other hand, prove that if $a$ and $b$ are *commensurable* then there exists a nontrivial solution $u=\sin (px/a)\sin (qt/b)$. ______ [$\Uparrow$](../contents.html)  [$\uparrow$](./S2.3.html)  [$\Rightarrow$](./S2.4.html)