Problems to Chapter 9

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

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".


$\Leftarrow$  $\Uparrow$  $\uparrow$  $\Rightarrow$