The following Research Report is available on Hal:
H. Barucq, F. Faucher and H. Pham
Outgoing solutions to the scalar wave equation in helioseismology,
Inria Research Report, July 2019, 130 pages, hal-02168467, https://hal.archives-ouvertes.fr/hal-02168467.
Abstract
This report studies the construction and uniqueness of physical solutions for the time-harmonic scalar wave equation arising in helioseismology. Intuitively speaking, physical solutions are characterized by their $L^2 (R^3)$-boundedness in the presence of absorption, while without, by their profile at infinity approximated by outgoing spherical waves (or retarded). For brevity, we unite these two families (with and without absorption) under the label ‘outgoing’ or ‘physical’. The definition of outgoing solutions to the equation in consideration or their construction and uniqueness has not been discussed before in the context of helieoseismology. In our work, we use the Liouville transform to conjugate the original equation to a potential scattering problem for Schrödinger operator, with the new problem containing a Coulomb-type potential. Under assumptions (in terms of density and background sound speed) generalizing ideal atmospheric behavior, for γ not 0, we obtain existence and uniqueness of variational solutions using only basic techniques in analysis. For γ = 0, under the same assumptions, the theory of long-range scattering with singular potentials is employed to construct the resolvent by means of Limiting Absorption Principle (LAP). Solutions obtained in this manner are characterized uniquely by a Sommerfeld- type radiation condition at a new wavenumber denoted by k. The appearance of this wavenumber is only clear after applying the Liouville transform. Another advantage of the conjugated form is that it makes appear the Whittaker functions, when ideal atmospheric behavior is extended to the whole domain R^3 or outside of a sphere. This allows for the explicit construction of the outgoing Green kernel and the exact Dirichlet-to-Neumann map and hence reference solutions and radiation boundary condition. In addition, the role played by k in radiation condition and asymptotic expansion of the solution suggests that k should be the more natural choice to use as gauge function in approximating the exact nonlocal radiation condition. This perspective gives rise to a simpler family of radiation boundary conditions. To supplement the theoretical discussion, some preliminary numerical tests are carried out to investigate the robustness of this new family, compared to those already existent in literature which were obtained in terms of the original complex frequency ω.
