A New Concept of the Surface Wave of the Interference Type for the Smooth Convex Surfaces Embedded in ℝ3

A new concept of surface waves of an interference nature is described in details for the case of a whispering gallery waves propagating along a smooth strictly concave surface embedded in 3D Euclidean space. Numerous papers consider whispering gallery and creeping waves in 2D case. However, in general, surface waves propagate along geodesic lines on the surface, and typically form caustics. The geodesic lines themselves are not plane curves in 3D and, therefore, their torsion must be taken into account. Our approach enables one to adequately describe surface waves propagation along smooth surfaces embedded in 3D Euclidean space. It is based on the consideration of the geodesic on a surface, which is associated with the surface wave generated by a source. For each geodesic line, we present an asymptotic solution of the Helmholtz equation, localized in its tubular and having no singularities on caustics. The surface wave under consideration is then presented as a superposition (integral) of the localized solutions.