$\newcommand{\const}{\mathrm{const}}$

Deadline January 29

APM346 (2015S) Home Assignment 2

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

Problem 1

Consider equation with the initial conditions \begin{align} & u_{tt}-9u_{xx}=0,\qquad &&t>0, x>vt, \label{eq-HA2.1}\\ &u|_{t=0}= \cos (x), \qquad &&x>0, \label{eq-HA2.2}\\ &u_t|_{t=0}= \sin(x), \qquad &&x>0, \label{eq-HA2.3} \end{align}

  1. Let $v=4$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-HA2.1})-(\ref{eq-HA2.3}) 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.

  1. Let $v=2$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-HA2.1})-(\ref{eq-HA2.3}) 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.

  1. Let $v=-4$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-HA2.1})-(\ref{eq-HA2.3}) so that the resulting problem would have a unique solution:

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

  2. Let $v=-3$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-HA2.1})-(\ref{eq-HA2.3}) 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

\begin{equation} u_{tt} = c^2 \bigl(u_{rr}+\frac{2}{r}u_r\bigr) \qquad\text{(“spherical wave equation”).} \label{eq-HA2.4} \end{equation}

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

  2. Solve for $v$ using \begin{equation} v = f(r+ct)+g(r-ct) \label{eq-HA2.5} \end{equation} and thereby solve the spherical wave equation.

  3. Use \begin{equation} 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-HA2.6} \end{equation} 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 $u$ to (\ref{eq-HA2.4}) which is continuous as $r=0$.

Problem 3

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

\begin{equation} \left\{\begin{aligned} &u_{tt}-9u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=\cos (x), \qquad &&x>0,\\ &u|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \label{eq-HA2.7} \end{equation}

\begin{equation} \left\{\begin{aligned} &u_{tt}-9u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=\cos (x), \qquad &&x>0,\\ &u_x|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \label{eq-HA2.8} \end{equation}

\begin{equation} \left\{\begin{aligned} &u_{tt}-9u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=\sin(x), \qquad &&x>0,\\ &u|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \label{eq-HA2.9} \end{equation}

\begin{equation} \left\{\begin{aligned} &u_{tt}-9u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=\sin(x), \qquad &&x>0,\\ &u_x|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \label{eq-HA2.10} \end{equation}

Problem 4

For a solution $u(x, t)$ of the wave equation $u_{tt}=c^2u_{xx}$, the energy density is defined as $e=\frac{1}{2}\bigl(u_t^2+c^2 u_x^2\bigr)$ and the momentum density as $p =c u_t u_x$.

  1. Show that \begin{equation} \frac{\partial e}{\partial t} = c\frac{\partial p}{\partial x} \qquad \text{and} \qquad \frac{\partial p}{\partial t} = c\frac{\partial e}{\partial x}. \label{eq-HA2.11} \end{equation}
  2. Show that both $e(x, t)$ and $p(x,t)$ also satisfy the same wave equation.