 Solution

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.