Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 39C / b

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.