$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
Problem 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; \\ 4u_{tt}-9u_{xx}=0. \end{gather}
Problem 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. 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 $c_1< c_2$.
Problem 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}
Problem 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. 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 $0$.
Problem 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. 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}
Problem 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 $p =c u_t u_x$.
Problem 10.
Problem 11.
Generalize Problem 4:
A spherical wave is a solution of the $n$-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{n-1}{r}u_r\bigr) \qquad\text{("spherical wave equation").} \label{eq-4b} \end{equation}
Remark 1. For even $n$ spherical waves do not exist