Problem 1. Find the general solutions of \begin{gather} u_{tt}-u_{xx}=0; \\[5pt] u_{tt}-4u_{xx}=0; \\[5pt] u_{tt}-9u_{xx}=0; \\[5pt] 4u_{tt}-u_{xx}=0; \\[5pt] 4u_{tt}-9u_{xx}=0. \end{gather}
Problem 2.
Solve IVP
\begin{equation}
\left\{\begin{aligned}
&u_{tt}-c^2u_{xx}=0,
\\
&u|_{t=0}=g(x),\quad u_t|_{t=0}=h(x)
\end{aligned}\right.
\end{equation}
with
\begin{align}
&g(x)=\left\{\begin{aligned}
&0 &&x < 0,\\
&1 &&x \ge 0,
\end{aligned}\right.
&&h(x)=0;
\\[5pt]
&g(x)=\left\{\begin{aligned}
&1 &&|x| < 1,\\
&0 &&|x| \ge 1,
\end{aligned}\right.
&&h(x)=0;
\\[5pt]
&g(x)=\left\{\begin{aligned}
&1-|x| &&|x| < 1,\\
&0 &&|x| \ge 1,
\end{aligned}\right.
&&h(x)=0;
\\[5pt]
&g(x)=\left\{\begin{aligned}
&1-x^2 &&|x| < 1,\\
&0 &&|x| \ge 1,
\end{aligned}\right.
&&h(x)=0;
\\[5pt]
&g(x)=\left\{\begin{aligned}
&\cos (x) &&|x| < \pi/2,\\
&0 &&|x| \ge \pi/2,
\end{aligned}\right.
&&h(x)=0;
\\[5pt]
&g(x)=\left\{\begin{aligned}
&\cos ^2(x) &&|x| < \pi/2,\\
&0 &&|x| \ge \pi/2,
\end{aligned}\right.
&&h(x)=0;
\\[5pt]
&g(x)=\left\{\begin{aligned}
&\sin (x) &&|x| < \pi,\\
&0 &&|x| \ge \pi,
\end{aligned}\right.
&&h(x)=0;
\\[5pt]
&g(x)=\left\{\begin{aligned}
&\sin^2 (x) &&|x| < \pi,\\
&0 &&|x| \ge \pi,
\end{aligned}\right.
&&h(x)=0;
\end{align}
\begin{align}
&g(x)=0, &&h(x)=\left\{\begin{aligned}
&0 &&x < 0,\\
&1 &&x \ge 0;
\end{aligned}\right.
\\[5pt]
&g(x)=0, &&h(x)=\left\{\begin{aligned}
&1-x^2 &&|x| < 1,\\
&0 &&|x| \ge 1;
\end{aligned}\right.
\\[5pt]
&g(x)=0, &&h(x)=\left\{\begin{aligned}
&1 &&|x| < 1,\\
&0 &&|x| \ge 1;
\end{aligned}\right.
\\[5pt]
&g(x)=0, &&h(x)=\left\{\begin{aligned}
&\cos (x) &&|x| < \pi/2,\\
&0 &&|x| \ge \pi/2;
\end{aligned}\right.
\\[5pt]
&g(x)=0, &&h(x)=\left\{\begin{aligned}
&\sin (x) &&|x| < \pi,\\
&0 &&|x| \ge \pi.
\end{aligned}\right.
\end{align}
Problem 3. Find solution $u=u(x,t)$ and describe domain, where it is uniquely defined \begin{align} &u_{tt}-u_{xx}=0; \label{A}\\[5pt] &u|_{t=x^2/2}= x^3; \label{B}\\[5pt] &u_t|_{t=x^2/2}= 2x. \label{C} \end{align}
Problem 4.
Problem 5. 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 6.
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 7. Find formula for solution of the Goursat problem \begin{align} &u_{tt} - c^2 u_{xx}=0, && x > c|t|; \\[5pt] &u|_{x=-ct}=g(t), && t < 0; \\[5pt] &u|_{x=ct}=h(t), &&t > 0. \end{align} as long as $g(0)=h(0)$.
Problem 8. Find solution u=u(x,t) and describe domain, where it is uniquely defined \begin{align} &u_{tt}-u_{xx}=0, \\[2pt] &u|_{t=x^2/2}= x^3, &&|x|\le 1,\\[2pt] &u_t|_{t=x^2/2}= 2x &&|x|\le 1. \end{align} Explain, why we imposed restriction $|x|\le 1$?
Problem 9. 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 10. 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 11. 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 12. 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 13.
Problem 14.
Generalize Problem 6:
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.