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Deadline Monday, October 1, 9 pm

APM 346 (2012) Home Assignment 2

Problem 1

Consider equation with the initial conditions \begin{align} & u_{tt}-4u_{xx}=0,\qquad &&t>0, x>vt, \label{eq-1}\\ &u|_{t=0}= e ^{-x}, \qquad &&x>0, \label{eq-2}\\ &u_t|_{t=0}= e^{-x}, \qquad &&x>0, \label{eq-3} \end{align}

1. Let $v=3$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-1})-(\ref{eq-3}) so that the resulting problem would have a unique solution:

1. None,
2. $u|_{x=vt}=0$ ($t>0$),
3. $u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).

Solve the problem you deemed as a good one.

2. Let $v=1$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-1})-(\ref{eq-3}) so that the resulting problem would have a unique solution:

1. None
2. $u|_{x=vt}=0$ ($t>0$),
3. $u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).

Solve the problem you deemed as a good one.

3. Let $v=-3$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-1})-(\ref{eq-3}) so that the resulting problem would have a unique solution:

1. None
2. $u|_{x=vt}=0$ ($t>0$),
3. $u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).

Solve the problem you deemed as a good one.

Problem 2

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 $$u_{tt} = c^2 \bigl(u_{rr}+\frac{2}{r}u_r\bigr) \qquad\text{(“spherical wave equation”).} \label{eq-4}$$

1. Change variables $v = ru$ to get the equation for $v$: $v_{tt} = c^2 v_{rr}$.

2. Solve for $v$ using $$v = f(x+ct)+g(x-ct) \label{eq-5}$$ and thereby solve the spherical wave equation.

3. Use $$v(r,t)=\frac{1}{2}\bigl[ \phi (r+ct)+\phi (r-ct)\bigr]+\frac{1}{2c}\int_{r-ct}^{r+ct}\psi (s)\,ds \label{eq-6}$$ with $\phi(r)=v(r,0)$, $\psi(r)=v_t(r,0)$ to solve it with initial conditions $u(r, 0) = \Phi (r)$, $u_t(r, 0) = \Psi(r)$.

4. Find the general form of solution continuous as $r=0$.

Problem 3

By method of continuation combined with D'Alembert formula solve each of the following four problems (a)--(d).

1. \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=1, \qquad &&x>0,\\ &u|_{x=0}=0, \qquad &&t>0. \end{aligned}\right.
2. \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=1, \qquad &&x>0,\\ &u_x|_{x=0}=0, \qquad &&t>0. \end{aligned}\right.

3. \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=x, \qquad &&x>0,\\ &u|_{x=0}=0, \qquad &&t>0. \end{aligned}\right.

4. \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=x, \qquad &&x>0,\\ &u_x|_{x=0}=0, \qquad &&t>0. \end{aligned}\right.

5. Problem 4

For a solution $u(x, t)$ of the wave equation with $\rho = T =1 \Longrightarrow c = 1$, the energy density is defined as $e=\frac{1}{2}\bigl(u_t^2+u_x^2\bigr)$ and the momentum density as $p = u_t u_x$.
1. Show that $$\frac{\partial e}{\partial t} = \frac{\partial p}{\partial x} \qquad \text{and} \qquad \frac{\partial p}{\partial t} = \frac{\partial e}{\partial x} . \label{eq-11}$$
2. Show that both $e(x, t)$ and $p(x,t)$ also satisfy the wave equation.
3. Note Notion of momentum density defies a naive physical interpretation.