New concept of surface waves of interference nature on smooth, strictly convex surfaces embedded in $\mathbb R^3$

M. M. Popov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 50, Tome 493 (2020), pp. 301-313

Résumé

The new concept of surface waves of interference nature is described in detail for the case of whispering gallery waves propagating along a smooth strictly concave surface embedded in 3D Euclidean space. In a numerous articles devoted to surface waves of whispering gallery and creeping waves it is assumed that they propagate along boundaries formed by a smooth plane curves. However, the process of surface waves propagation along smooth surfaces is mush more complicated then along plane curves. Indeed, the surface waves slide along geodesic lines on the surface where they normally form numerous caustics. Besides, the geodesic lines itself are not plane curves in 3D and therefore their torsion has to be taken into account. Our approach enables resolving both these peculiar problems of waves propagation along smooth surfaces imbedded in 3D Euclidian space. It is based on consideration of geodesic flow on the surface which is associated with the surface wave generated by a source. For each geodesic line we construct an asymptotic solution of the Helmholtz equation localized in a tube vicinity of the geodesic line and having no singularities on caustics. The surface wave under consideration is then presented as a superposition (integral) of the localized solutions.