Plane wave representations for wave scattering on a sphere

Abstract

Motivated by three-dimensional wave diffraction by conical shapes, we consider the two-dimensional problem of wave scattering on the real sphere ${\mathcal{S}}^2$ resulting from a point-source incidence on a scatterer $\mathbb{S}$. Using the recently introduced notion of plane waves on ${\mathcal{S}}^2$ as well as recent results on the Laplace--Beltrami Green's function $G$ on ${\mathcal{S}}^2$, we derive three plane-wave representations for the scattered field $u$. We then explain how $u$ can be analytically continued to the complexified sphere ${\mathcal{S}}_c^2$. This allows us to formulate a complexified version of Green's third identity, and to provide a characterisation of complex sources. All our results are then used to draw strong analogies between planar and spherical scattering problems, notably through the introduction of directivities.