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 𝒮2 resulting from a point-source incidence on a scatterer 𝕊. Using the recently introduced notion of plane waves on 𝒮2 as well as recent results on the Laplace--Beltrami Green's function G on 𝒮2, we derive three plane-wave representations for the scattered field u. We then explain how u can be analytically continued to the complexified sphere 𝒮2c. 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.
Related
- Stability of least-energy solutions of the harmonic map flow from the plane into the 2-sphere
- Scattering of plane waves and a modified Pompeiu criterion
- Prime ideals and representations of the bosonization of the super Jordan plane
- Semi-analytical representation of the Gaunt-modified Landau-Lifshitz equation in a plane wave
- Solving the scattering problem for open wave-guide networks, II: Outgoing estimates
BibTeX
@misc{kunz2026plane,
title = {{Plane wave representations for wave scattering on a sphere}},
author = {Valentin D. Kunz and Andrey V. Shanin and Raphael C. Assier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15121},
eprint = {2610.15121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15121},
}
RIS
TY - UNPB
TI - Plane wave representations for wave scattering on a sphere
AU - Valentin D. Kunz
AU - Andrey V. Shanin
AU - Raphael C. Assier
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.15121
UR - https://arxiv.org/abs/2610.15121
AB - Motivated by three-dimensional wave diffraction by conical shapes, we consider the two-dimensional problem of wave scattering on the real sphere 𝒮^2 resulting from a point-source incidence on a scatterer 𝕊. Using the recently introduced notion of plane waves on 𝒮^2 as well as recent results on the Laplace--Beltrami Green's function G on 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 𝒮^2_c. 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.
ER -