\documentclass{article}

\usepackage{fullpage}
\usepackage{amsmath}
\usepackage{color}
\newcommand{\gp}[1]{{\left({#1}\right)}}
\newcommand{\px}[2]{\frac{\partial #1}{\partial #2}}

\begin{document}

\title{Math 142-2, Homework 6}
\author{Your name here}
\date{\today}
\maketitle

\section*{Problem 64.1}
\textbf{Consider the linear car-following model, equation 64.2, with a response
  time $T$ (a delay).}

\textbf{(a) Ignore this part.}

\textbf{(b) Assume the lead driver's velocity varies periodically
$$ v_0 = \mbox{Re}(1+f_0 e^{i \omega t}). $$
Also assume the $n$-th driver's velocity varies periodically
$$ v_n = \mbox{Re}(1+f_n e^{i \omega t}), $$
where $f_n$ measures the amplification or decay which occurs. Show that
$$ f_n = \gp{1 + \frac{i \omega}{\lambda} e^{i \omega T}}^{-n} f_0, $$
where $0 < f_0 < 1$.  Note: $f_n$ will be complex, $f_0$ is real.  Unlike the
problem as stated in the book, I am assuming $f_0 < 1$, since otherwise the
model predicts that even the first car stops frequently.  If $f_0 \approx 0$,
then the first car drives quite smoothly, and the effects on later cars are more
meaningful.  This alteration does not make the problem more difficult.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Show the magnitude of the amplification factor $f_n$ decreases with $n$ if
$$ \frac{\sin \omega T}{\omega} < \frac{1}{2\lambda}. $$}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) Show that the above inequality holds for all $\omega$ only if
  $\lambda T < \frac{1}{2}$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(e) Conclude that if the product of the sensitivity and the time lag is greater
than $\frac{1}{2}$, it is possible for following cars to drive much more erratically
than the leader. In this case we say the model predicts instability if
$\lambda T > \frac{1}{2}$ (i.e., with a sufficiently long time lag). (This
conclusion can be reached more expeditiously through the use of Laplace transforms.)}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 65.1}
\textbf{Determine the solution of $\px{\rho}{t} = (\sin\, x) \rho$ which satisfies $\rho(x, 0) = \cos\, x$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 67.3}
\textbf{Suppose initially ($t = 0$) that the traffic density is $\rho = \rho_0 +
  \epsilon \sin\,x$, where $|\epsilon| \ll \rho_0$. Determine $\rho(x, t)$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 67.5}
\textbf{Based on a linear analysis, would you say $\rho = \rho_0$, a constant, is a stable or
unstable equilibrium solution of equation 66.1?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 68.1}
\textbf{Explain why a density wave moves forward for light traffic. Consider
  both cases in which the traffic is getting heavier down the road and lighter.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 71.1}
\textbf{Experiments in the Lincoln Tunnel (combined with the theoretical work
  discussed in exercise 63.7) suggest that the traffic flow is approximately
$$q(\rho) = a \rho [\ln(\rho_{\max})-\ln(\rho)] $$
  (where $a$ and $\rho_{\max}$ are known constants). Suppose the initial density
  $\rho(x, 0)$ varies linearly from bumper-to-bumper traffic (behind $x = -x_0$)
  to no traffic (ahead of $x = 0$) as sketched in Fig. 71-6. Two hours later,
  where does $\rho = \rho_{\max}/2$?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 71.9}
\textbf{Show that $\rho = f(x - q'(\rho) t)$ satisfies equation 71.1 for any
  function $f$. Note that initially $\rho = f(x)$. Briefly explain how this
  solution was obtained.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Additional Problem}

\textbf{The equations that describe compressible flow in 1D are:
\begin{align*}
\px{\rho}{t} + \px{}{x}(\rho u) & = 0 \\
\px{(\rho u)}{t} + \px{}{x}(\rho u^2 + p) & = 0 \\
\px{E}{t} + \px{}{x}(E u + p u) & = 0
\end{align*}
There are four variables here that must be determined ($\rho$, $u$, $E$, and
$p$), so one additional equation is required.  This takes the form of an
\textit{equation of state}, which is some equation expressing $p$ in terms of
the other quantities.  An example of such an equation of state is the ideal gas
law $p = \rho e (\gamma - 1)$, where $e$ is related to $E$ and the temperature
$T$ by
$$ E = \rho e + \frac{1}{2} \rho u^2 \qquad e = k T. $$}
\begin{center}
\begin{tabular}{cl}
\hline
var & meaning \\
\hline
$\rho$ & density \\
$u$ & velocity \\
$e$ & internal energy \\
$E$ & total energy \\
$p$ & pressure \\
$\gamma$ & gas-specific constant \\
$T$ & temperature \\
$k$ & specific heat capacity \\
\hline
\end{tabular}
\end{center}

\vspace{1cm}
\textbf{(a) Deduce the SI units ($kg$, $m$, $s$, $K$) of $u$, $e$, $E$, $p$,
  $\gamma$, and $k$ from the equations above, given that $\rho \to kg\,m^{-1}$
  and $T \to K$.  Note that we are assuming 1D, so some of these units will not
  be what you may be used to seeing in 3D.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) A conservation law is said to be in conservation form when written
  as
$$ \px{U}{t} + \px{F}{x} = 0, $$
where $U$ is a vector of conserved quantities and $F$ is a vector of quantities
called \textit{fluxes}.  Show that if a differential equation of this form holds
for a vector of quantities $U$, then for each component of $U$ there is some
corresponding conserved quantity.  Identify the conserved quantity for each of
the three components of $U$.  You may assume periodic boundary conditions.
(That is, consider the system to live in some interval $[a,b]$, where $U(a,t) =
U(b,t)$, $p(a,t) = p(b,t)$, $F(a,t) = F(b,t)$, etc.)}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Consider again (b) on the interval $[a,b]$, but this time do not
  assume periodicity.  Provide a physical interpretation for the role that the
  fluxes $F$ play.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\end{document}
