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

Let $\theta$ be the angle made by the group velocity ray with the horizontal. The internal gravity wave dispersion relation givesEach sawtooth face changes height by $2h_0$ over horizontal distance ${\lambda }_T/2$, so its slope magnitude isThe internal-wave slope criticality criterion is $s_T<\tan \theta$. Hence reflection is subcritical forIn the corresponding internal-wave ray tracing sketch, every incident ray meets one planar face and leaves it into the fluid at the same angle $\theta$ to the horizontal. The reflected ray is steeper than either face, so it clears the sawtooth rather than running into an adjacent corner.

For a face of signed slope $s$, conservation of frequency and tangential wavenumber gives, on the branch relevant to an incident downward-right ray,At $s\mu =1$, the reflected wavenumber diverges and the reflected group velocity becomes tangent to the face. On a supercritical face the denominator changes sign: horizontal propagation reverses and rays from the two faces are directed towards a sawtooth corner. Successive reflections therefore focus energy and shorten the wavelength.

The inviscid ray pattern cannot persist indefinitely. Near-critical focusing amplifies gradients until kinematic viscosity, mass diffusivity, nonlinear wave steepening, and wave breaking matter; a real corner is also rounded on some finite scale. These effects replace the singular ray construction by dissipative boundary layers, mixing, and a finite-width reflected beam.