Geometrization of the Leading Term in Acoustic Gaussian Beams

DOI

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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TY  - JOUR
AU  - Matias F. Dahl
PY  - 2021
DA  - 2021/01/07
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  - 10.1142/S1402925109000042
ID  - Dahl2021
ER  -