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Problems to Chapter 9

• Problem 1
• Problem 2
• Problem 3
• Problem 4

Problem 1.

Consider wave equation in dimension $2$: $$\begin{equation} u_{tt}-\Delta u=0 \label{eq-9.P.1} \end{equation}$$ 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 $$\begin{equation} (\Delta +\omega^2)v=0. \label{eq-9.P.2} \end{equation}$$

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 Dirichlet boundary conditions on the top and bottom of the cylinder $Z(0)=Z(b)=0$.
3. Write down ODE which should satisfy $R$.

Problem 3.

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

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 homogeneous Dirichlet or Neumann conditions at $z=0$ and $z=h$.
3. Write down ODE which should satisfy $R$.
4. Be ready to consider other related domains: cut from the cylinder by restriction $0<\theta < \alpha$ and homogeneous Dirichlet or Neumann conditions at $\theta=0$ and $\theta=\alpha$.

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".

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