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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#7, Rev. B, due Tuesday, 2020-10-20 }
\end{center}

\vspace{.3 in}

{\bf 1.} \href{https://physics.info/waves/animation-surface-wave.html}{\bf Gravity waves} propagate along the interface between a liquid and a gas (or any two fluids of different density, for example air and water), as gravity or {\it buoyancy} tries to restore equilibrium.  The particles in the liquid follow elliptical trajectories with their amplitude decaying exponentially in depth.  If the velocity field $\bs v(x,z)$ is irrotational ($\nabla\times\bs v=0$), it can be represented by the gradient of a scalar flow potential $\bs v=-\nabla\phi(x,z)$.  If the fluid is also incompressible ($\nabla\cdot\bs v=0$), its flow satisfies the Laplace equation $\nabla^2\phi=0$.  Let the gas-liquid interface be at height $z=\eta(x,t)$ above the equilibrium level at $z=0$, and $h$ be water depth (the sea bed is at $z=-h$).

In addition to gravity, surface tension $\gamma$, exerts a downward pressure $P=-\gamma \nabla^2_{\perp} \eta$ on the liquid, also propagating ripples on the surface.  This is the two-dimensional analog of the vertical force $dF=Tf''dx$ of a wave on an element of string under horizontal tension $T$.  Both of these effects can be described by \href{https://en.wikipedia.org/wiki/Airy_wave_theory}{Airy wave theory}, which we develop below.

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

{\bf \quad b)} Show this solution satisfies the boundary condition $v_z(x,-h)=0$, at the bottom of the liquid.  The boundary condition on the top surface is $v_z=\dot\eta$, evaluated at $z=0$ (the boundary to 0$^{th}$ order).  Find $A$ so that $\phi$ satisfies this boundary condition at the interface {$\eta(x,t) = a\sin(kx-\omega t)$.}

{\bf \quad c)} Integrating Newton's law over $z$ leads to Bernoulli's law $\rho\partial_t\phi = \rho g\eta -\gamma\partial_x^2\eta$, where $\rho$ is the mass density of the liquid.  Substitute $\phi$ and $\eta$ into Bernoulli's law at $z=0$ 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 d)} Calculate the wavelength $\lambda_c$ below which waves are dominated by surface tension, using $\gamma=72.8$~mN/m and $\rho=1.00$~g/cm$^3$ for water.  What is the dispersion relation in this limit?

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

{\bf \quad f)} 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 10~km deep [shallow!] ocean waters?  How long will it take one wavelength to pass?


{\bf 2.} [bonus] \href{https://en.wikipedia.org/wiki/Linear_elasticity#Elastodynamics_\%E2\%80\%93_the_wave_equation}{\bf Elastic waves} in solids, also known as body, bulk, \href{https://en.wikipedia.org/wiki/Seismic_wave}{seismic}, stress, or strain waves, are the three-dimensional analog of waves traveling along a Slinky.  They have three polarizations: one longitudinal polarized acoustic P-wave, (primary or pressure), and two transverse polarized {S-waves} (secondary or shear).  [Note there are three completely different and inconsistent definitions of S- and P-waves for seismic, optical, and quantum mechanical angular momentum waves!]

Pressure and shear waves are both described in terms of elastic deformation.  Similar to the displacement $f(x,t)$ of coils in a Slinky, the displacement field $\bs u(\bs r,t)$ describes the shift of the particle at position $\bs r$ in equilibrium to the new position $\bs r+\bs u$.  Elastic strain (deformation) $\bs\epsilon$ indicates the change in displacement $\bs{du}=\bs{\epsilon}\cdot\bs{dr}$ between neighboring particles, analogous to $f'(x,t)$ for the Slinky.  This symmetric {\it strain  tensor} $\epsilon_{ij}\equiv \frac12(\partial_{i}u_{j}+\partial_{j}u_{i})$ is the matrix of all nine possible linear deformations.  On the other hand, the {\it stress tensor} $\tau_{ij}=dF_{i}/da_{j}$ describes all three components of force $dF_{i}$ per area along all three independent directions of surface area $da_{j}$ of the interface between two neighboring elements at $\bs r$ and $\bs r+\bs{dr}$ at equilibrium in the bulk, so that the equal and opposite force between these elements is $\bs{dF}=\pm\bs\tau\cdot\bs{da}$.  The normal component of stress to the interface ($i=j$) is called {\it pressure}, while the tangential components ($i\neq j$) are called {\it shear}, each propagating their own polarization.  Stress transfers energy and momentum across the interface, propagating the wave according to Newton's second law $\bs{dF}=dm\,\bs{A}$, which is written $\partial_{j}\tau_{ij}=\rho \ddot u_{i}$, analogous to $dF=\mu\,dx \ddot f$ for the Slinky.

The generalization of Hooke's law, $dF=\kappa f''dx$, from the Slinky to the bulk of an elastic material, $\tau_{ij}=\lambda\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij}$, relates the stress (force) $\bs\tau$ to the strain (stretch) $\bs\epsilon$.  This is the most general isotropic and rotationally invariant relation, which contains two independent ``spring constant'' {\it Lam\'e parameters}: $\lambda$ (pressure), and $\mu$ (sheer).  Other combinations of these constants, each with their own physical interpretation are: Poisson's ratio $\nu=\lambda/2(\lambda+\mu)$, the P-wave (longitudinal, pressure) modulus $M=\lambda+2\mu$, the S-wave (transverse, shear) modulus $G=\mu$, the {\it bulk modulus} (compressibility) $K=M-\frac43 G$, and {\it Young's modulus} (stretchiness) $E=2G(1+\nu)=3K(1-2\nu)$.  Nonviscous fluids do not support sheer strain ($\mu=0$) and only accommodate P-waves.  For an ideal gas, the isentropic bulk modulus $K_{S}=\gamma P$ is used, where the {\it adiabatic index} $\gamma=C_{P}/C_{V}$ is the ratio of heat capacities at constant pressure and volume, respectively, and $P$ is the equilibrium pressure.  In air at STP, $\gamma=1.41$ and $P=1~\q{bar}=100~\q{kPa}$.
%https://en.wikipedia.org/wiki/Bulk_modulus

{\bf \quad a)} Combine Hooke's and Newton's laws in the form given above to obtain the wave equation $\rho\, \ddot{\bs u} = M\, \nabla\nabla{\cdot} \bs u - G\, \nabla{\times}\nabla{\times} \bs u$.  Note that the two operators $\nabla^{2}_{\parallel}\equiv \nabla\nabla{\cdot}$ and $\nabla^{2}_{\perp}\equiv -\nabla{\times}\nabla{\times}$ are the longitudinal and transverse projections of the {\it Laplacian} $\nabla^{2} \equiv \nabla{\cdot}\nabla = \nabla^{2}_{\parallel} + \nabla^{2}_{\perp}$ (the fundamental second vector derivative) that describe the curvature of longitudinal and transverse waves, respectively.

{\bf \quad b)} For a P-wave, $\nabla\times\bs u=0$.  Take the divergence of both sides of the wave equation to show that $\rho\ddot n=M\nabla^{2}n$, where $n\equiv\nabla\cdot\bs u$ is the compression of the medium.  Show that the wave $n=a e^{i(\bs k\cdot \bs x-\omega t)}$ is a solution with amplitude $a$ traveling in the direction $\bh k$, and obtain the dispersion relation $\omega(\bs k)$.  Calculate the velocity of a seismic P-wave ($K=$ 200~GPa, $\rho=$ 1700~kg/m$^3$ for rock), and of sound in air ($\rho=$ 1.225~kg/m$^3$), water ($K=$ 2.2~GPa), and steel ($K=$ 160~Gpa, $\rho=$ 8050~kg/m$^3$).

{\bf \quad c)} For an S-wave, $\nabla\cdot\bs u=0$.  Take the curl of both sides of the wave equation to show that $\rho\ddot{\bs m} = G \nabla^{2}\bs m$, where $\bs m\equiv\nabla\times\bs u$ is the sheer strain which is perpendicular to the direction of propagation.  Show that the wave $\bs m= \bs a e^{i(\bs k\cdot \bs x-\omega t)}$ is a solution with transverse amplitude $\bs a$ traveling perpendicular to $\bs k$, and obtain the dispersion relation $\omega(\bs k)$.  Calculate the velocity of a seismic S-wave ($M=$ 100~GPa for rock), and of a sheer wave in steel ($M=$79.3~GPa).  There are no S-waves in water, air, or in the molten \href{https://en.wikipedia.org/wiki/Seismic_wave#/media/File:Speeds_of_seismic_waves.PNG}{\it outer core} of the earth, which is how we know it exists.

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