New Concept of Surface Waves of Interference Nature. Creeping waves

A new concept of surface waves of interference nature is described in detail for the case of creeping waves propagating along a smooth strictly concave surface embedded in 3D Euclidean space. In numerous papers devoted to whispering gallery and to creeping waves, it was assumed that they propagate along boundaries formed by smooth plane curves. However, the process of surface wave propagation along smooth surfaces is much more complicated than along plane curves. Indeed, the surface waves slide along geodesic lines on the surface where they typically form numerous caustics and that, in turn, generates singularities in the wave field asymptotics. In addition, the geodesic lines themselves are not plane curves in 3D and therefore their torsion has to be taken into account. Our approach allows us to resolve both of these specific problems of wave propagation along smooth surfaces embedded in ℝ3. It is based on the consideration of the 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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.