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\title{MTH 352/652:
Homework \#3}
\vspace{-1in}
\date{Due Date: February 09, 2024}
\maketitle

\section{Problems for Everyone}
\begin{enumerate}
\item By changing variables to $\xi=x-ct$ and $\eta=x+ct$, show that the wave equation $u_{tt}=c^2u_{xx}$ is equivalent to the following PDE:
\begin{equation*}
u_{\xi \eta}=0.
\end{equation*}
\item Consider the following initial value problem for the wave equation:
\begin{align*}
u&_{tt}=c^2u_{xx},\,\, x\in \mathbb{R},\,\,t>0,\\
 u&(0,x)=\frac{4}{4+x^2} \text{ and } u_t(0,x)=0,
\end{align*}
where $c>0$ is a constant.
\begin{enumerate}[(a)]
\item What do the initial conditions model in this situation?
\item Calculate the exact solution $u(t,x)$ to this problem.
\item Assuming $c=1$, plot $u(t,x)$ at $t=0$, $t=1$, and $t=2$.
\item Calculate $\lim_{t\rightarrow \infty}u(t,x)$. 
\item Discuss the qualitative behavior of the solution.
\end{enumerate}

\item Consider the following initial value problem for the wave equation:
\begin{align*}
u&_{tt}=c^2u_{xx},\,\, x\in \mathbb{R},\,\,t>0,\\
 u&(0,x)=0 \text{ and } u_t(0,x)=\frac{4}{4+x^2},
\end{align*}
where $c>0$ is a constant.
\begin{enumerate}[(a)]
\item What do the initial conditions model in this situation?
\item Calculate the exact solution $u(t,x)$ to this problem.
\item Assuming $c=1$, plot $u(t,x)$ at $t=0$, $t=1$, and $t=2$.
\item Calculate $\lim_{t\rightarrow \infty}u(t,x)$. 
\item Contrast how the solution to this initial value problem is different from the solution to problem \#1.
\end{enumerate}

\item pg. 60-62, \#2.4.1-\#2.4.3

\item The \emph{Poisson-Darboux} equation is 
\begin{equation*}
u_{tt}-u_{xx}-\frac{2}{x}u_x=0.
\end{equation*}
By making the substitution, $w=xu$, solve the Poisson-Darboux equation with the initial data $u(0,x)=0$, $u_t(0,x)=g(x)$ where $g(x)=g(-x)$, i.e., $g$ is an even function.
\item For a classical solution $u(t,x)$ to of the unit speed wave equation $u_{tt}=u_{xx}$ the energy and momentum density are defined by
\begin{equation*}
E(t,x)=\frac{1}{2}\left(u_t^2+u_x^2\right) \text{ and } P(t,x)=u_tu_x,
\end{equation*}
respectively. 
\begin{enumerate}
\item Show that $E_t=P_x$ and $E_x=P_t$.
\item Show that both $E(t,x)$ and $P(t,x)$ satisfy the wave equation.
\item Show that 
\begin{equation*}
\int_{-\infty}^{\infty} u(t,x)dx
\end{equation*}
is constant in time.
\end{enumerate}

\item A spherical wave is a solution of the three-dimensional wave equation of the form $u(r,t)$, where $r=\sqrt{x^2+y^2+z^2}$ is the distance from the origin. The wave equation in this case takes the form:
\begin{align*}
u_{tt}&=c^2\left(u_{rr}+\frac{2}{r} u_{r}\right)\\
u(0,r)&= \phi(r),\\
u_t(0,r)&= \psi(r)
\end{align*}
where $\phi(r),\psi(r)$ are both even functions of $r$. By making the change of variables $v=ru$, solve the spherical wave equation. What does your solution tell you about the propagation of spherical waves in three dimensions?

\item Solve the equation $u_{xx}-3u_{xt}-4u_{tt}=0$ with the initial conditions $u(0,x)=e^{-x^2}$ and $u_t(0,x)=\frac{1}{1+x^2}$. \textbf{Hint:} Factor the operator as we did for the wave equation.
\end{enumerate}

\section{Choose One of These Problems to Complete}

\begin{enumerate}
\item In 1858 the first trans-Atlantic telegraph cable was laid down allowing direct communication between the United States and England. The first message sent by Queen Victoria to President James Buchanan contained 98 words and took 16 hours to communicate. This was an incredible improvement over the ten days it took to communicate by ships! However, within a month of its first operation Dr. Wildman Whitehouse (a medical doctor with no real training in electrical engineering) essentially melted the cable by applying too high of a voltage. The reaction from the media to this announcement was volatile with many writers hinting that the line was a hoax. Nevertheless, engineers persisted and a working cable was installed in 1866 that could transmit a blazing eight words per minute! 

 In this problem our goal is to mathematically understand how signals can be sent along wires. In particular, our goal is to tune specific electrical properties of the wire to arrive at the wave equation. As we saw in class, the wave equation is particularly nice for sending signals because the solutions retain the shape of the initial profile. That is, somebody receiving the signal would know precisely what the initial shape of the signal is. 

\begin{enumerate}
\item We will model the wire as an infinitely long straight line of resistance $R \geq 0$, inductance $L \geq 0$, grounding resistance $G \geq 0$ and capacitance $C\geq 0$. Letting $I(t,x)$ and $V(t,x)$ denote the current and voltage across the wire at position $x$ and time $t$, the partial differential equations satisfied by $I$ and $V$ are given by:
\begin{align}
V_x&=-RI-LI_t,\\
I_x&=-GV-CV_t.
\end{align}
Show that $I$ can be eliminated from these equations to yield the following equation:
\begin{equation}
LCV_{tt}+(LG+RC)V_t+RGV=V_{xx}.
\end{equation}
\item If we define $c=1/\sqrt{LC}$, $a=c^2(LG+RC)$, and $b=c^2RG$ then the above equation becomes:
\begin{equation}
V_{tt}+aV_{t}+bV=c^2V_{xx}.
\end{equation}
This equation is known as the \textbf{telegrapher's equation}. If $a=0$ and $b=0$ this equation becomes the wave equation. If $L>0$ and $C>0$ show that if $a=0$ and $b=0$ then both $R=0$ and $G=0$. Why is this an unrealistic case?
\item By making a substitution of the form $V(t,x)=e^{-\lambda t} u(t,x)$ and by appropriately picking $\lambda$ in terms of the constants $a,b$ and $c$, show that the telegraphers equation can be reduced to the following partial differential equation:
\begin{equation}
u_{tt}+ku=c^2u_{xx},
\end{equation}
where $k$ is a constant that depends on $a,b$ and $c$.
\item Show that $k=0$ only when $RC=LG$. 

\item In the case when $RC=LG$ with the initial conditions $u(0,x)=f(x)$ and $u_t(0,t)=0$, show that a solution to a solution to the telegrapher's equation is given by
\begin{equation}
V(t,x)=\frac{e^{-\lambda t}}{2}(f(x-ct)+f(x+ct)),
\end{equation}
What does this solution mean in practical terms? In particular, why is it useful for sending signals? \\
\\
\textbf{Remark:} This process of tuning the electrical parameters so the $RC=LG$ is called ``Pupinizing'' the cable after one of its discoverers Michael Pupin. 
\end{enumerate}
\item pg. 62, \#2.4.15 


\end{enumerate}
\end{document}