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\begin{center}
{\Large Homework \#3, due October 17 at the beginning of class.}
\end{center}

\begin{enumerate}

\item Please do problem 3 in Section 2.3.

\item Please do problems 3, 4, and 6 in Section 3.1.  

For the problems from the book --- there are hints in the back of the
book. Please don't spend more than fifteen minutes on each problem
before looking at the hints!


\item Consider the
wave equation on the line with initial data
$$
\begin{cases}
u_{tt} - c^2 u_{xx} = 0 & \forall t > 0, x \in \R \\
u(x,0) = g(x) & \forall x \in \R \\
u_t(x,0) = h(x) & \forall x \in \R 
\end{cases}
$$ Assume the initial data satisfies $g \in C^2(\R)$ and $h \in
C^1(\R)$.  
\begin{enumerate}
\item Can you show that the solution depends continuously on the
initial data?  That is, if $u_1$ solves the IVP with $g_1$ and $h_1$
and $u_2$ solves the IVP with $g_2$ and $h_2$ then can you relate the
norm of their difference at time $t$, $u_1(\cdot,t)-u_2(\cdot,t)$, to
the difference of the norms of the initial data?  Use the $C^2(\R)$
norm:
$$
\| w \|_{C^2(\R)} :=
\max_{x \in \R} |w(x)| +
\max_{x \in \R} |w^\prime(x)| +
\max_{x \in \R} |w^{\prime \prime}(x)|
$$
\item If you fix the time
$t$ and take the initial data closer and closer to one another, does
$u_1(\cdot,t)-u_2(\cdot,t) \to 0$?  
\item What if you ask about {\it all} times?
\end{enumerate}

\item {\bf Transmitted and Reflected Waves in a Composite Bar}

The wave equation can describe displacements from rest in a solid material.  If 
$u(x,t)$ represents the displacement in the $x$ direction in a uniform material
that's axially loaded and if one assumes a linear relationship between stress and
strain (a reasonable assumption for small displacements) then the displacement
satisfies
$$
u_{tt} = \frac{E}{\rho} \; u_{xx}
$$
where $E$ is the Young's modulus and $\rho$ is the density of the material.

Consider an infinitely long composite bar.  For $x<0$ the density is
$\rho_1$ and the Young's modulus is $E_1$.  For $x>0$ the density is
$\rho_2$ and the Young's modulus is $E_2$.  The material changes type
at $x=0$.

Please search on ``wave reflection at a medium boundary'' at youtube.
The top video should come from ``bockphysics''.  Watch the video and
observe the incident (incoming), reflected (bounced back) and
transmitted (got through) waves.

\begin{enumerate}
\item Verify that $$u(x,t) = A e^{i \omega (t \pm \frac{x}{c_1})}$$
  solves the wave equation in material 1 if $c_1 = \sqrt{E_1/\rho_1}$.
  Using this observation, what is a corresponding {\it real-valued}
  solution of the wave equation?
\item You want to understand how a rightward moving incident wave produces a leftward moving reflected wave and a rightward moving transmitted wave at the interface at $x=0$.  Let
$$
u_1(x,t) = A_I e^{i \omega(t - \frac{x}{c_1})} + A_R e^{i \omega(t + \frac{x}{c_1})} 
$$
and 
$$
u_2(x,t) = A_T e^{i \omega(t - \frac{x}{c_2})} .
$$ ``$A_I$'' is the amplitude of the {\it incident} wave, ``$A_R$'' is
the amplitude of the {\it reflected} wave, and ``$A_T$'' the amplitude
of the {\it transmitted} wave.  Check that $u_1$ solves the wave
equation in the material on the left and $u_2$ solves the wave
equation in the material on the right.  Make sure that you understand
why the incident wave in $u_1$ is rightward moving and the reflected
wave is leftward moving.  Similarly, make sure you understand why the
transmitted wave ($u_2$) is rightward moving.
\item The displacement should be continuous at the interface between
the materials: $$u_1(0,t)= u_2(0,t).$$ Similarly, the stresses should
be continuous at the interface: $$E_1 \frac{\partial u_1}{\partial x}
(0,t) = E_2 \frac{\partial u_2}{\partial x} (0,t).$$ Use these to find
two equations involving $A_I$, $A_R$, and $A_T$.  Solve the equations
to find $A_T$ and $A_R$ in terms of $A_I$, $c_1$, $\rho_1$, $c_2$, and
$\rho_2$.
\item In terms of reflected and transmitted waves, the key quantities
are the ``mechanical impedances'' $\rho_1 c_1$ and $\rho_2 c_2$.  For
what types of materials will there be no reflected wave?  If the
materials have been chosen to have no reflected wave, how is the
magnitude of the transmitted wave ($A_T$) related to the magnitude of
the incident wave ($A_I$)?  Is it possible to create materials for
which there is {\it no} transmitted wave?
\end{enumerate}
If you're curious, google ``impedance matching''!

\item
Consider
the wave equation $u_{tt} - c^2 \Delta u = 0$ on 
$\R^n \times \R$.  You would like to find out what dimensions $n$
allow for distortionless radially symmetric wave propagation.

We say that distortionless radially symmetric wave propagation is
possible if there are functions $\alpha(r)>0$ and $\beta(r)>0$ with
$\alpha(1)=1$ and $\beta(0)=0$ such that
$$
u(\vec{x},t) := \alpha(r) \, \phi( t - \beta(r) )
$$
solves the wave equation for every ``reasonable'' $\phi$.  Let's say
$\phi(0)=\phi'(0)=0$ and $\phi \in C^2(\R)$.  Here,
$r := | \vec{x} |$.  
\begin{enumerate}
\item Find $\alpha(r)$ and $\beta(r)$
so that $u$ is a solution of the 
wave equation for every reasonable $\phi$.  (Hint: you'll succeed only in some dimensions.)
\item Choose $\phi$ so that initially $u(x,0)$ is compactly supported
on $1 \leq |x| \leq 2$ (for example).  Plot $u(\cdot,t)$ at some
subsequent times $t$.  (Note that these plots are pretty simple because
you can just plot things as a function of $r$.)
\item Consider a point $\vec{x}_0$ so that $|\vec{x}_0| = R > 2$.  At
what time would $u(\vec{x}_0,t)$ become nonzero?  At what time would
it become zero again for all perpetuity?  Define the cumulative energy at $\vec{x}_0$ to be
$$
\mathcal{E}(\vec{x}_0) = \frac{1}{2} \int_{t_0}^{t_1}
u_t(\vec{x}_0,t)^2 + c^2 | \nabla u(\vec{x}_0,t)| ^2 \; dt
$$
where $t_0$ and $t_1$ are these two times.  How does the cumulative
energy depend on $R$?  (For this one, if you like just choose the
initial $u(\vec{x},0)$ to be simple enough that you can do whatever
integrals you need.)
\item Are you happy that you live in the dimension you
  live in?  Why?
\end{enumerate}





\end{enumerate}

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