Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 2 / c

The maximum boundary slope is $s_{\max }=h_0{k}_T$, so subcritical internal-wave reflection requiresor equivalently

Take $k,m>0$ without loss of generality, putand write the incident field as the imaginary part of ${\tilde{w}}_i{e}^{i(kx+mz-\omega t)}$. The kinematic boundary condition on $z=h(x)$ isFor a flat boundary the reflected wave is $-{\tilde{w}}_i\sin (kx-mz-\omega t)$. Expanding the boundary condition in a Taylor expansion about $z=0$ creates the topographic sidebands of an internal gravity wave. With upward-radiating vertical wavenumber $-\mu |p_j|$, their complex amplitudes through second order areThus the general compact result isFor the convenient nondegenerate case $k>2k_T$, all displayed sideband wavenumbers are positive. Defining ${\Theta }_j=p_{j}x-\mu p_{j}z-\omega t$, the same answer is the explicitly real formulaValidity requires a linear incident wave, an inviscid uniformly stratified bulk, an outgoing-radiation condition, strict separation from critical slopes, and small boundary excursions for every retained mode, in particular $h_0{k}_T\ll 1$ and $\mu h_0{\max }_{|j|\le 2}|p_j|\ll 1$. A vanishing $p_j$ is a degenerate zero-horizontal-wavenumber case and must be treated by taking the corresponding zero-amplitude limit rather than dividing by $p_j$.

If $h_0{k}_T>\tan \theta$, each period contains two critical points satisfying $|h_x|=\tan \theta$, with supercritical intervals around the steepest parts. Internal-wave ray tracing sends neighbouring reflected rays towards caustics attached to those critical points; rays can reverse horizontal direction in the supercritical intervals and intersect rays reflected elsewhere on the sinusoid. At equality the inviscid reflected wavelength collapses locally. The physical pattern is therefore a set of intense finite-width beams and mixing regions once viscosity, scalar diffusion, and wave breaking regularize the ray caustics.