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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#3, Rev. A, due Friday, 2015-09-16 }
\end{center}

\vspace{.3 in}

{\bf 1. Waves on a String} of linear mass density $\mu$ stretched horizontally
with tension $T$.

{\bf \quad a)} Derive the wave equation.  Hint: apply Newton's law, ignoring
gravity, to an segment of string of infinitesimal length $dx$ and mass $dm$.

{\bf \quad b)} Substitute the wave function $f(x,t)=A e^{i{kx-\omega t}}$ into
the wave equation to derive the dispersion relation $\omega(k)$.  Use this to
calculate the phase velocity of waves on the string.

{\bf \quad c)} A string of length $L$ is fixed at one end.  The other end is
tied to a massless ring freely slides up and down a rod.  Use boundary
conditions to find the spectrum of allowed frequencies.

{\bf \quad d)} In analogy with AC electrical impedance $Z=V/I=R + i\omega L +
1/i\omega C$, the mechanical impedance of an oscillating system (mass $M$,
spring constant $K$, drag coefficient $B$) is $Z=F/v= B + i\omega M +
K/i\omega$.  Calculate the characteristic impedance of the string, $Z=Tf'/\dot
f$ (ratio of vertical force to velocity) in terms of $T$ and $\mu$.  Show the
power transferred along the wave is $P=Z\dot f^2$.

{\bf \quad f)} Two strings of density $\mu_{1,2}$ and tension $T_{1,2}$ are
joined by a ring on a rod to support the difference in tension.  Justify the
boundary conditions $\Delta f=0$ and $\Delta(Tf')=\Delta(\pm Z\dot f)=0$.\\
Hint: apply Newton's law to vertical forces on the ring to obtain the second
condition.

{\bf \quad g)} An incident wave $A_Ie^{i(k_1x-\omega t)}$ from the left is
partially reflected at the ring due to the change in impedance.  The reflected
wave $A_Re^{i(-k_1x-\omega t)}$ is superimposed on the incident wave, and the
forward transmitted wave is $A_Te^{i(k_2x-\omega t)}$.  Apply the two boundary
conditions to obtain $A_R/A_I$ and $A_T/A_I$.  Calculate the coefficients of
reflected power $R = Z_1 A_R^2/Z_1 A_I^2$ and transmitted power $T = Z_2
A_T^2/Z_1 A_I^2$ in terms of $Z_{1,2}$.  Show they add up to 100\%.

{\bf \quad h) [bonus]} Repeat for a ring of impedance $Z$ (mass $M$, spring
constant $K$).

{\bf 2.} As {\bf surface waves} propagate along the interface between a liquid
and a gas, the liquid rotates around ellipses of exponentially decaying
amplitude with depth.  If the velocity field $\bv v(x,z)$ is irrotational
$\nabla\times\bv v=0$, we can represent it with a scalar flow potential
$\phi(x,z)$, define by $v=-\nabla\phi$.  If in addition the fluid
incompressible $\nabla\cdot\bv v=0$, the flow satisfies the Laplace equation
$\nabla^2\phi=0$.  Let the gas-liquid interface have height $z=\eta(x,t)$
above the equilibrium $z=0$.  Both gravity and surface tension $\gamma$, which
exerts pressure $P=\gamma \nabla^2 \eta$ on the liquid propagate the wave.

{\bf \quad a)} Show that the function
$\phi(x,z,t)=a\cosh(k(z+h))\sin(kx-\omega t)$ is a solution of
$\nabla^2\phi=0$.  Plot equipotentials of $\phi$ at $t=0$, with arrows showing
the direction of $\bv v$.

{\bf \quad b)} Show this solution satisfies the boundary condition $v_z(-h)=0$
at the \mbox{bottom of the liquid.}

{\bf \quad c)} The boundary condition on the top surface is $v_z=\dot\eta$,
evaluated at $z=0$ (approximately at the boundary).  Calculate the constant
$a$ which satisfies the \mbox{boundary condition $\eta(x,t) = A\cos(kx-\omega
  t)$.}

{\bf \quad d)} Integrating Newton's law over $z$ leads to Bernoulli's law
$\partial_t\phi = g\eta -\frac\gamma\rho \partial_x^2\eta$, where $\rho$ is
the mass density of the liquid.  Substitute $\phi$ and $\eta$ into Bernoulli's
law to obtain the dispersion relation $\omega^2=(gk+\frac\gamma\rho
k^3)\tanh(kh)$.  Plot $\omega(k)$, $v_\phi(k)=\omega/k$, and
$v_g(k)=d\omega/dk$.

{\bf \quad e)} Calculate the wavelength $\lambda_c$ below which waves are
dominated by surface tension ($\gamma=72.8$~mN/m and $\rho=1.00$~g/cm$^3$ for
water).  What is the dispersion relation in this limit?

{\bf \quad f)} Approximate $\phi(x,z,t)$ and $\omega(k)$ in the deep water
limit, where $kh>>1$.  Do individual crests move forward or backward within
the wave packet?

{\bf \quad g)} Approximate $\omega(k)$ in the shallow water limit, and show
that all frequencies have the same velocity.  What is the speed of a tsunami
($\lambda \approx 100$~km) in the 10~km deep [shallow!] ocean?  How long does
it take one wavelength to pass?

\end{document}