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

Andrey V. Shanin, Valentin D. Kunz, Raphael C. Assier

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.

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

Abstract

We propose an extension of the plane-wave representation for wave fields defined on the real sphere . 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 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.

Paper Structure

This paper contains 18 sections, 6 theorems, 104 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Laplace--Beltrami equation on a sphere
  3. Laplace--Beltrami equation on a real sphere. The Green's function
  4. The complex sphere and its compactification
  5. Characteristic lines on the complex sphere
  6. Complexification of the Laplace--Beltrami equation
  7. A plane wave analogue for the sphere
  8. Asymptotics of Green's function
  9. Singularities of plane waves
  10. Plane-wave representation of a field
  11. Structure of a plane-wave representation
  12. A first plane-wave representation of the Green's function
  13. Extension of the plane-wave representation by sliding contours
  14. On projective spaces and the complex sphere
  15. Some properties of the characteristic lines
  16. ...and 3 more sections

Key Result

Lemma 2.5

The following statements are true:

Figures (7)

  • Figure 1: $(\theta,\varphi)$ coordinate system on $\mathcal{S}^2$ (left), and $\mathcal{S}^2 \subset \mathbb{R}^3$ with a point source located at $\boldsymbol{x}_0=\boldsymbol{x}_{\textsc{np}}$ (right).
  • Figure 2: $\Xi$ as a strip (left) and as a sphere (right)
  • Figure 3: Illustration of the Maps $\Psi$ and $\Phi$
  • Figure 4: Contours $C^{(m)}$ on $\mathcal{S}^2$ (left); Contours $\gamma^{(1)}$ and $\gamma^{(2)}$ on $\Xi$ (right)
  • Figure 5: Domain $X^{(1)}\cap X^{(2)}$ (left); deformation of $\gamma^{(1)}$ into $\gamma^{(2)}$ on $\Xi$ (right)
  • ...and 2 more figures

Theorems & Definitions (23)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 13 more