Problems to Chapter 9

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### Problems to Chapter 9

Problem 1.

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

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".