A boundary layer approach to the whispering gallery phenomenon

We consider the eigenvalue problem for the Laplace operator in a two dimensional domain, bounded by a smooth, closed convex curve $C$, on which the eigenfunctions vanish. Waves whose wavelengths $\lambda$ are very small may be viewed as particles traveling along specific paths termed rays, along which the waves propagate. Small wavelengths $\lambda$ correspond to large wave numbers $k=\frac {2 \pi }{\lambda }$, which are the eigenvalues. Therefore, we restrict attention to the consideration of large eigenvalues. Keller and Rubinow considered the problem from the point of view of geometrical optics. They constructed eigenvalues and eigenfunctions by means of rays for special geometries, and qualitatively discussed general geometries. They noted the existence of caustics, i.e., envelopes of rays in the domain, with the property that the amplitude of the eigenfunctions is appreciable only in the region between the boundary and the caustic. Inside the caustic the solution is negligibly small. The caustic is the locus of the turning points on the rays, at which the behavior changes from oscillatory (wavelike) to exponentially decaying. We are interested in those eigenfunctions for which the caustic is “near” the boundary. They correspond, in acoustics, to the whispering gallery modes, by means of which a person speaking near the wall of a convex room can be heard across the room, but not in the interior of the room. We consider domains bounded by a general smooth convex curve $C$, and employ a boundary layer approach to the problem, to determine these eigenvalues and eigenfunctions. Our results provide improvements to those of Keller and Rubinow.