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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 420 \\
    Homework \#2, Rev. A, due Monday, 2015-09-21 }
\end{center}

\vspace{.3 in}

{\bf 1.} Calculate and plot the {\bf Fourier transform} $\psi(x) =
\int_{-\infty}^\infty dk\, A(k) e^{ikx}$ of a square frequency distribution
$A(k)=A_0$ if $k_1< k < k_2$ and 0 otherwise.  Calculate the uncertainty
$\Delta k$ in frequency.  Show that the Heisenberg uncertainty principle holds
for this wavefunction, but very badly!

{\bf 2. Surface waves}---According to the
\href{https://en.wikipedia.org/wiki/Airy_wave_theory}{Airy theory} of gravity
waves, the dispersion relation for water waves is $\omega^2 = g k \tanh(kh)$,
where $h$ is the water depth and $g$ is acceleration due to gravity.
Calculate the phase velocity $v_\phi=\omega/k$, group velocity
$v_g=d\omega/dk$, and rate of dispersion $\beta=d^2\omega/dk^2$ for water wave
packet centered about frequency $k_0$ in the shallow water $(kh<<1)$ and deep
water ($kh>>1$) limits.  Do individual crests move forward or backward within
the wave packet?  What if anything is the significance of negative dispersion
$\beta$?  Calculate the same coefficients at short wavelengths, where surface
tension displaces gravity as the restoring force.  Now the dispersion relation
becomes $\omega = \sqrt{k^3\gamma/\rho}$, where $\gamma\approx 73~\q{mN/m}$
and $\rho \approx 1~\q{g/cm}^3$ are the suface tension and density of water,
respectively.  Do surface ripples go forward or backward in their wave packet?

% {\bf 3. [bonus] Beats}---Resolve the wavefunction
% $\Psi(x)=Ae^{ik_0}+Be^{i(k_0-\Delta k)x} + Be^{-i(k_0-\Delta k)x}$ into the
% product of a phase wave and a repeating wave packet, and plot it for
% different values of $0 \leq B \leq 2$.  Show by direct calculation that if
% $\omega(k) = a + bk$, where $a>0$, then $v_\phi > v_g$, but otherwise the
% wave propagates undistorted.

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