Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 39C (Waves)

Assume the medium varies on scales much longer than the wavelength and period, so the leading-order WKB method gives the local eikonal equation. For the phase $\theta$ in the stated ansatz, defineThe local dispersion relation becomes the Hamilton--Jacobi equationDifferentiating it with respect to $x_i$, using equality of mixed derivatives, and choosing a ray whose velocity isgivesHere the dot, or $d/dt$, is the total derivative along a ray,and not merely a partial time derivative at fixed position.

Finally, differentiate $\omega =\Omega (\mathbf{k};\mathbf{x},t)$ along the same trajectory:because the last two terms cancel. The Hamiltonian ray-tracing equations are therefore

For the given internal gravity wave,The wavevector part of the Hamiltonian ray-tracing equations isApplying the initial data givesIn particular, $k(t)m(t)=k_0{m}_0$ is conserved.

The vertical group velocity is the $m$ derivative of the local dispersion relation:At release,Initial upward propagation is possible only when $m_0<0$. In that case $\dot{z}(0)>0$ precisely when

Assume the condition in part (ii), so $m_0<0$. Substituting the wavevector solution into the vertical ray equation gives the linear ordinary differential equationwhereThe integrating factor $e^{\gamma t}$ givesThe bracket is strictly positive, so $z(t)>0$ at every finite time. Moreover,so the improper integral converges to a finite positive value. It follows thatfor some $C>0$. The packet therefore approaches $z=0$ asymptotically and never reaches it at finite time. This is the vertical trapping of an internal gravity wave by planar strain.