Volume 54, Number 4, July-August 2020
|Page(s)||1111 - 1138|
|Published online||18 May 2020|
Outgoing solutions and radiation boundary conditions for the ideal atmospheric scalar wave equation in helioseismology
Inria Project-Team Magique 3D, E2S–UPPA, CNRS, Pau, France
2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
* Corresponding author: firstname.lastname@example.org
Accepted: 6 December 2019
In this paper, we study the time-harmonic scalar equation describing the propagation of acoustic waves in the Sun’s atmosphere under ideal atmospheric assumptions. We use the Liouville change of unknown to conjugate the original problem to a Schrödinger equation with a Coulomb-type potential. This transformation makes appear a new wavenumber, k, and the link with the Whittaker’s equation. We consider two different problems: in the first one, with the ideal atmospheric assumptions extended to the whole space, we construct explicitly the Schwartz kernel of the resolvent, starting from a solution given by Hostler and Pratt in punctured domains, and use this to construct outgoing solutions and radiation conditions. In the second problem, we construct exact Dirichlet-to-Neumann map using Whittaker functions, and new radiation boundary conditions (RBC), using gauge functions in terms of k. The new approach gives rise to simpler RBC for the same precision compared to existing ones. The robustness of our new RBC is corroborated by numerical experiments.
Mathematics Subject Classification: 00A71 / 33C15 / 35J10 / 35L05 / 35A08 / 35B40 / 33C55 / 85A20
Key words: Helioseismology / Whittaker functions / Coulomb potential / outgoing fundamental solution / exact Dirichlet-to-Neumann map / Schrödinger equation / Liouville transform / radiation conditions
© The authors. Published by EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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