Plane-wave representation for the Laplace--Beltrami equation on a sphere. Application to the Green's function

TL;DR

This paper extends the plane-wave representation to the sphere for wave fields, enabling better analysis of the Green's function on curved surfaces using complex analysis techniques.

Contribution

It introduces a novel plane-wave representation for the sphere, adapting methods from planar wave analysis to curved surfaces with complex geometry considerations.

Findings

Extended the validity region of the plane-wave representation on the sphere.

Developed a sliding-contours method for complexification of the sphere.

Applied the methodology to analyze the Green's function on the sphere.

Abstract

We propose an extension of the plane-wave representation for wave fields defined on the real sphere $\mathcal{S}^2$. This representation is well-known in the planar setting but has never been developed for curved surfaces. To achieve this, we need to carefully study the geometry of the complexification of $\mathcal{S}^2$ and the properties of the Laplace--Beltrami operator, while using concepts of multidimensional complex analysis. We extend the region of validity of such plane-wave representation by developing a sliding-contours method. Our methodology is illustrated through the study of the Green's function on the real sphere.