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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.

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.