Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 79 / 1 / a / Solution

Assume small-amplitude, nonrotating, inviscid Boussinesq approximation motion about a fluid at rest, with stable background stratification. In either uniform layer let $b=-g{\rho }^{\prime }/{\rho }_{\ast }$ be the buoyancy perturbation and $N^2=-(g/{\rho }_{\ast }){\rho }_0^{\prime }(z)>0$. The linear equations areFor a plane internal gravity wave proportional to $e^{i(kx+mz-\omega t)}$, continuity gives $u=-mw/k$, and horizontal momentum gives $p={\rho }_{\ast }\omega u/k$. Substituting these into vertical momentum and buoyancy evolution givesChoose $k>0$, $\omega >0$. Its group velocity isThus downward energy propagation has $m>0$. If $\theta$ is the acute angle between the ray and the vertical in its direction of propagation, then $\tan \theta =|m|/k$, soThe phase velocity is parallel to $(k,m)$, while ${\mathbf{c}}_g\cdot (k,m)=0$: phase and group velocities are perpendicular, and their vertical components have opposite signs. Propagating incident waves require $0<\omega <N_1$; both layers support propagating waves only when $0<\omega <\min (N_1,N_2)$. At or above $N_2$, the lower layer cannot carry a propagating downward internal wave.

For the amplitude convention used below, $\eta$ multiplies the unit particle-displacement vector along the ray, with positive horizontal component. That vector is $(\sin \theta ,-\cos \theta )$ for a downward wave and $(\sin \theta ,+\cos \theta )$ for an upward wave. Stating this polarization convention is important when assigning a sign to a reflection coefficient.