Collocation discretization for an integral equation in ocean acoustics with depth‐dependent speed of sound. (12th July 2016)

• Record Type: Journal Article
• Authors: Lechleiter, Armin; Rienmüller, Tobias
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 5(2017)
• Page Start: 1608
• Page End: 1624
• Publication Date: 2016-07-12
• Subjects: scattering -- Lippmann‐Schwinger integral equation -- Helmholtz equation -- waveguide scattering -- collocation method; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4082
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We analyze two collocation schemes for the Helmholtz equation with depth‐dependent sonic wave velocity, modeling time‐harmonic acoustic wave propagation in a three‐dimensional inhomogeneous ocean of finite height. Both discretization schemes are derived from a periodized version of the Lippmann‐Schwinger integral equation that equivalently describes the sound wave. The eigenfunctions of the corresponding periodized integral operator consist of trigonometric polynomials in the horizontal variables and eigenfunctions to some Sturm‐Liouville operator linked to the background profile of the sonic wave velocity in the vertical variable. Applying an interpolation projection onto a space spanned by finitely many of these eigenfunctions to either the unknown periodized wave field or the integral operator yields two different collocation schemes. A convergence estimate of Sloan [J. Approx. Theory, 39:97–117, 1983] on non‐polynomial interpolation allows to show converge of both schemes, together with algebraic convergence rates depending on the smoothness of the inhomogeneity and the source. Copyright © 2016 John Wiley & Sons, Ltd.