Past exam of the mathematical methods II course of the University of Oxford / 2013 / 11 / d / Solution

In a homogeneous medium and the short-wavelength ray approximation, a smooth reflecting elliptical boundary sends a ray from focus $A$ to focus $B$ after one specular reflection, by the equal-angle property just proved. Moreover, every such route has the same path length $AP+PB=s$, hence the same propagation time $s/c_s$ for constant sound speed $c_s$.

An ideal two-dimensional elliptical reflector, or the corresponding ellipsoidal geometry, therefore concentrates sound near the other focus; equal travel times also support coherent arrival for a monochromatic source with a consistent reflection phase. Finite wavelength, imperfect surfaces, absorption and three-dimensional room geometry limit the focusing. The result predicts a finite physical focal region, not an infinite intensity at a mathematical point.