\(\renewcommand{\Re}{\operatorname{Re}}\) \(\renewcommand{\Im}{\operatorname{Im}}\) \(\newcommand{\erf}{\operatorname{erf}}\) \(\newcommand{\dag}{\dagger}\) \(\newcommand{\const}{\mathrm{const}}\) \(\newcommand{\arcsinh}{\operatorname{arcsinh}}\)

Problems to Chapter 9

1. Problem 1
2. Problem 2
3. Problem 3
4. Problem 4

Problem 1.

Consider wave equation : \begin{equation} u_{tt}-\Delta u=0 \label{eq-9.P.1} \end{equation} in the disk \(\{r\le a,\ 0\le \theta \le 2\pi\}\).

We consider solution in the form \(u=v(r,\theta)e^{i\omega t}\). Then \(v\) satisfies Helmholtz equation \begin{equation} (\Delta +\omega^2)u=0 \label{eq-9.P.2} \end{equation}

Separate variables \(v=R(r)\Theta(\theta)\).

1. Write down ODE which should satisfy \(\Theta\) and solve it (using periodicity).
2. Write down ODE which should satisfy \(R\).

Problem 2.

Consider Laplace equation \(\Delta u=0\) in the cylinder \(\{r\le a,\ 0<z<b,\ 0\le \theta \le 2\pi\}\).

Separate variables \(u=R(r)Z(z)\Theta(\theta)\).

1. Write down ODE which should satisfy \(\Theta\) and solve it (using periodicity).
2. Write down ODE which should satisfy \(Z\) and solve it using \(Z(0)=Z(b)=0\).
3. Write down ODE which should satisfy \(R\).

Problem 3.

Consider wave equation (\ref{eq-9.P.1}) in the cylinder \(\{r\le a,\ 0< z <b,\ 0\le \theta \le 2\pi\}\).

We consider solution in the form \(u=v(r,z,\theta)e^{i\omega t}\). Then \(v\) satisfies Helmholtz equation (\ref{eq-9.P.2})

Separate variables \(v=R(r)Z(z)\Theta(\theta)\).

1. Write down ODE which should satisfy \(\Theta\) and solve it (using periodicity).
2. Write down ODE which should satisfy \(Z\) and solve it using \(Z(0)=Z(b)=0\).
3. Write down ODE which should satisfy \(R\).

Problem 4.

Consider wave equation (\ref{eq-9.P.1}) in the ball \(\{\rho\le a,\ 0<\phi <\pi,\ 0\le \theta \le 2\pi\}\).

We consider solution in the form \(u=v(\rho,\phi,\theta)e^{i\omega t}\). Then \(v\) satisfies Helmholtz equation (\ref{eq-9.P.2})

Separate variables \(v=P(\rho)\Phi(\phi)\Theta(\theta)\).

1. Write down ODE which should satisfy \(\Theta\) and solve it (using periodicity).
2. Write down ODE which should satisfy \(\Phi\).
3. Write down ODE which should satisfy \(P\).

Hint. In the spherical coordinates \begin{equation*} \Delta u= u_{\rho\rho}+2\rho^{-1}u_\rho + \rho^{-2}\bigl(\Phi'' +\cot (\phi)\Phi'\bigr) + \rho^{-2}\sin^{-2}(\phi)u_{\theta\theta} \end{equation*}

Remark. "Solve" everywhere means "write down solution without justification".

\(\Leftarrow\)  \(\Uparrow\)  \(\uparrow\)  \(\Rightarrow\)