scholarly journals Outgoing solutions and radiation boundary conditions for the ideal atmospheric scalar wave equation in helioseismology

2020 ◽  
Vol 54 (4) ◽  
pp. 1111-1138 ◽  
Author(s):  
Hélène Barucq ◽  
Florian Faucher ◽  
Ha Pham
Keyword(s):  
Boundary Conditions ◽  
Acoustic Waves ◽  
Whittaker Functions ◽  
Radiation Boundary Conditions ◽  
Time Harmonic ◽  
Whole Space ◽  
Radiation Boundary ◽  
Dirichlet To Neumann Map ◽  
Radiation Conditions ◽  
The Ideal

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.

scholarly journals Atmospheric radiation boundary conditions for the Helmholtz equation

2018 ◽  
Vol 52 (3) ◽  
pp. 945-964 ◽  
Author(s):  
Hélène Barucq ◽  
Juliette Chabassier ◽  
Marc Duruflé ◽  
Laurent Gizon ◽  
Michael Leguèbe
Keyword(s):  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Acoustic Waves ◽  
Numerical Study ◽  
Outgoing Wave ◽  
Atmospheric Radiation ◽  
Angle Of Incidence ◽  
Radiation Boundary Conditions ◽  
Radiation Boundary ◽  
Dirichlet To Neumann Map

This work offers some contributions to the numerical study of acoustic waves propagating in the Sun and its atmosphere. The main goal is to provide boundary conditions for outgoing waves in the solar atmosphere where it is assumed that the sound speed is constant and the density decays exponentially with radius. Outgoing waves are governed by a Dirichlet-to-Neumann map which is obtained from the factorization of the Helmholtz equation expressed in spherical coordinates. For the purpose of extending the outgoing wave equation to axisymmetric or 3D cases, different approximations are implemented by using the frequency and/or the angle of incidence as parameters of interest. This results in boundary conditions called atmospheric radiation boundary conditions (ARBC) which are tested in ideal and realistic configurations. These ARBCs deliver accurate results and reduce the computational burden by a factor of two in helioseismology applications.

RADIATION BOUNDARY CONDITIONS FOR VERTICALLY HETEROGENEOUS ACOUSTIC MEDIA

1993 ◽  
Vol 01 (03) ◽  
pp. 321-333 ◽  
Author(s):  
GONGQIN LI ◽  
JOSEPH E. MURPHY ◽  
STANLEY A. CHIN-BING
Keyword(s):  
Boundary Conditions ◽  
Heterogeneous Media ◽  
Plane Waves ◽  
Reflection Coefficients ◽  
Radiation Boundary Conditions ◽  
Media Comparison ◽  
Acoustic Media ◽  
Radiation Boundary ◽  
Radiation Conditions ◽  
Effective Reflection Coefficient

Several radiation boundary conditions for inhomogeneous acoustic media are investigated. Previous investigators have developed various approximate radiation conditions and have studied their accuracy by calculating an effective reflection coefficient for plane waves incident on such radiating boundaries. In this paper, it is shown that effective reflection coefficients can be calculated for a class of parabolic approximations to the Helmholtz equation. These results are valid for vertically heterogeneous media. Comparison of these radiation conditions is given through numerical examples.

HIGH-ORDER RADIATION BOUNDARY CONDITIONS FOR STRATIFIED MEDIA AND CURVILINEAR COORDINATES

2012 ◽  
Vol 20 (02) ◽  
pp. 1240002 ◽  
Author(s):  
THOMAS HAGSTROM
Keyword(s):  
Boundary Conditions ◽  
Acoustic Waves ◽  
Curvilinear Coordinates ◽  
Stratified Media ◽  
Computational Domain ◽  
Progressive Wave ◽  
Radiation Boundary Conditions ◽  
Local Radiation ◽  
Radiation Boundary ◽  
Homogeneous Media

Optimized local radiation boundary conditions to truncate the computational domain by a rectangular boundary have been constructed for acoustic waves propagating into a homogeneous, isotropic far field. Here we try to achieve comparable efficiencies in stratified media and cylindrical coordinates. We find that conditions constructed for homogeneous media are highly effective in the stratified case. On the circle we derive boundary conditions by optimizing a semidiscretized perfectly matched layer. Though we are unsuccessful in matching the accuracies of the Cartesian case, our experiments show that older sequences based on the progressive wave expansion are surprisingly efficient.

UNIFORM RADIATION CONDITIONS FOR A SOUND PROPAGATION MODEL

2009 ◽  
Vol 17 (02) ◽  
pp. 135-157
Author(s):  
A. T. PEPLOW ◽  
S. FINNVEDEN
Keyword(s):  
Boundary Conditions ◽  
Sound Propagation ◽  
Fundamental Problem ◽  
Propagation Model ◽  
Radiation Boundary Conditions ◽  
Spectral Finite Element Method ◽  
Radiation Boundary ◽  
Spectral Finite Element ◽  
Radiation Conditions ◽  
Grass Strip

In this work, a new set of uniform radiation boundary conditions for a half-space model are derived and applied to a fundamental problem in outdoor sound propagation. This original approach derived here relies upon a high-order family of local radiation boundary conditions related to plane wave reflection coefficients. Validity of the approximation is carried out by examining sound propagation above an impedance ground using a spectral finite element method. This is followed by computational results verified against an analytic solution for sound propagation over hard ground. Finally, the case of sound propagation above a grass-strip surrounded by rigid ground and a rigid-strip in a grassland environment with atmospheric profiles are studied.

RADIATION AND OUTFLOW BOUNDARY CONDITIONS FOR DIRECT COMPUTATION OF ACOUSTIC AND FLOW DISTURBANCES IN A NONUNIFORM MEAN FLOW

1996 ◽  
Vol 04 (02) ◽  
pp. 175-201 ◽  
Author(s):  
CHRISTOPHER K.W. TAM ◽  
ZHONG DONG
Keyword(s):  
Boundary Conditions ◽  
Acoustic Waves ◽  
Maximum Velocity ◽  
Mean Flow ◽  
Direct Computation ◽  
Radiation Boundary Conditions ◽  
Outflow Boundary Conditions ◽  
The Mean ◽  
Radiation Boundary ◽  
Outflow Boundary

It is well known that Euler equations support small amplitude acoustic, vorticity and entropy waves. To perform high quality direct numerical simulations of flow generated noise problems, acoustic radiation boundary conditions are required along inflow boundaries. Along boundaries where the mean flow leaves the computation domain, outflow boundary conditions are needed to allow the acoustic, vorticity and entropy disturbances to exit the computation domain without significant reflection. A set of radiation and outflow boundary conditions for problems with nonuniform mean flows are developed in this work. Flow generated acoustic disturbances are usually many orders of magnitude smaller than that of the mean flow. To capture weak acoustic waves by direct computation (without first separating out the mean flow), the intensity of numerical noise generated by the numerical algorithm and the radiation and outflow boundary conditions (and the computer) must be extremely low. It is demonstrated by a test problem involving sound generation by an oscillatory source that weak acoustic waves with maximum velocity fluctuation of the order of 10−9 of the mean flow velocity can be computed accurately using the proposed radiation boundary conditions. The intensity of such acoustic waves is much smaller than the numerical error of the mean flow solution.

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