Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 1 / c / Solution

Write $\mu =\tan \theta =m/k$. The incident down-right ray from $(0,s)$ has slope $-\cot \theta$ and meets the parabolic bottom $H(x)=2x^2$ at $x=x_0\in (0,1/2)$ providedAt the impact point the bottom slope is $S=H^{\prime }(x_0)=4x_0$. A stationary reflection preserves frequency and tangential wavenumber $k+Sm$. For the incident wavevector $(k,m)=a(1,\mu )$ and a reflected up-left ray $(k_r,m_r)=a_r(-1,\mu )$, tangential matching givesA positive reflected magnitude $a_r$ therefore exists exactly when $S\mu >1$. This is the supercritical-slope condition for reflection of an internal-wave ray and produces a group velocity in the second quadrant.

Since $x_0<1/2$, such a point exists whenFor any such $\theta$, chooseThe initial ray then hits the parabola at a supercritical point and its reflected ray travels up and left, as required.