Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 74 / 1 / c / Solution

Use the spatial internal-wave ray tracing model, with $dx/dz=\pm 16z$, and reflect the rays so that the frequency and tangential wave number are preserved. At a vertical wall the horizontal energy direction reverses; at a horizontal wall the vertical energy direction reverses. On the sloping wall the reflection depends on internal-wave slope criticality. The three-reflection orbit below encounters that wall at a height above $3/16$, so the wall reverses the horizontal energy direction there.

First take the packet to leave the bottom upward and to the left, with $0<x_0<6$. The four segments of the loop areThe left-wall reflection is at $z=\sqrt{x_0/8}$ and the top reflection at $x=8-x_0$. The slope intersection solves $8z_s^2+3z_s+x_0-10=0$, giving the internal-wave ray return mapThus the next formal bottom reflection is at $(R(x_0),0)$. On $0<x_0<6$, the slope heights are between approximately $0.544$ and $0.946$, consistent with the assumed reflection type.

An internal-wave attractor is a stable periodic ray, so solve $R(x_{\ast })=x_{\ast }$. This gives $z_s=2/3$, and thereforeThe successive vertices areThis is clockwise circulation in the $(x,z)$ plane. Differentiating the internal-wave ray return map showsNeighbouring bottom intersections converge to the periodic orbit, geometrically focusing the energy beam.

For completeness, the opposite upward launch direction has the reversed three-reflection itinerary: slope, top, left. In the interval where this itinerary is valid, putThe three-reflection reverse map is $R^{-1}$, with domain $R(0)<x_0<R(6)$. Its fixed point is the same, but $R_{\mathrm{rev}}^{\prime }(x_{\ast })=41/23>1$. Counterclockwise packets defocus from that orbit. Except for the exactly periodic reverse ray, they eventually leave this itinerary. In the full geometric reflection construction, omitted-side or additional-slope reflections redirect generic rays into the clockwise attracting branch. Tangency and exactly critical reflections are singular exceptions requiring finite-wavelength or dissipative treatment.

The bottom reflection is a formal spatial-ray idealization: part (b) shows that the WKB approximation itself fails near $z=0$, so the diagram does not imply arrival and reflection there in finite WKB travel time. In a real fluid, viscous attenuation of an internal-wave beam, finite wavelength, internal-wave breaking and ensuing turbulence and mixing limit the beam width and energy density. Unlimited focusing is a property of the ideal geometric model, not a prediction for the resolved physical wave.