Journal of Nonlinear Mathematical Physics

Volume 16, Issue 1, March 2013, Pages 35 - 45

Geometrization of the Leading Term in Acoustic Gaussian Beams

Authors
Matias F. Dahlmatias.dahl@tkk.fi
Institute of Mathematics, P. O. Box 1100, 02015 Helsinki, Finland, matias.dahl@tkk.fi
Received 10 January 2008, Accepted 15 April 2008, Available Online 7 January 2021.
DOI
https://doi.org/10.1142/S1402925109000042How to use a DOI?
Keywords
Gaussian beams, wave packets, asymptotic analysis, wave equation, conservation of energy, leading amplitude term
Abstract

We study Gaussian beams for the wave equation on a Riemannian manifold. For the transport equation we geometrize the leading term at the center of the Gaussian beam. More precisely, if

u(x,t)=eiPθ(x,t)(u0(x,t)+u1(x,t)iP+u2(x,t)(iP)2+)
is a Gaussian beam propagating along a geodesic c, then we show that
u0(c(t),t)=C1(detY(t))1/2
where C is a constant and Y is a complex Jacobi tensor. Using a constant of motion for the non-linear Riccati equation related to the Jacobi equation, we prove that asymptotically the leading term of the energy carries constant energy.

Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
16 - 1
Pages
35 - 45
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1142/S1402925109000042How to use a DOI?
Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Matias F. Dahl
PY  - 2021
DA  - 2021/01
TI  - Geometrization of the Leading Term in Acoustic Gaussian Beams
JO  - Journal of Nonlinear Mathematical Physics
SP  - 35
EP  - 45
VL  - 16
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925109000042
DO  - https://doi.org/10.1142/S1402925109000042
ID  - Dahl2021
ER  -