Degenerate caustic in a waveguide

For a standard acoustic problem of two-dimensional oscillations in a region bounded by perfectly reflecting branches of a hyperbola, the case in which the congruence of the waves directed into the foci gives the geometric-optic interpretation of one of the eigenfunctions is considered. The straight line y = 0, which is a singular line of this congruence, is designated a degenerate caustic. The reflection coefficient and transmission coefficient for a degenerate caustic can be obtained by reduction of the problem to the quantum mechanical problem of the scattering of particles by a potential barrier with a height exactly equal to the energy of the particles. This problem is solved asymptotically using the equations of the parabolic cylinder. The reflection coefficient of eigenoscillations with a nearly degenerate caustic, corresponding to hyperbolas or ellipses with very large or very small eccentricities, respectively, is nonexponentially small.