High frequency solutions to partial differential equations (PDEs) are notoriously
difficult to simulate numerically due to the large number of grid points required to resolve the wave
oscillations. In applications, one often must rely on approximate solution methods to describe the
wave field in this regime. Gaussian beams are asymptotically valid high frequency solutions concen-
trated on a single curve through the domain. We show that one can form integral superpositions of
such Gaussian beams to generate more general high frequency solutions to PDEs.
¶ As a particular example, we look at high frequency solutions to the constant coefficient wave
equation and construct Gaussian beam solutions with Taylor expansions of several orders. Since
this PDE can be solved via a Fourier transform, we use the Fourier transform solution to gauge the
error of the constructed Gaussian beam superposition solutions. Furthermore, we look at an example
for which the solution exhibits a cusp caustic and investigate the order of magnitude of the wave
amplitude as a function of frequency at the tip of the cusp. We show that the observed behavior is
in agreement with the predictions of Maslov theory.
Publié le : 2008-06-15
Classification:
high frequency waves,
superpositions,
Gaussian beams,
caustics,
ray methods,
wave equation,
caustics,
35L05,
41A60,
65M25
@article{1214949931,
author = {Tanushev, N. M.},
title = {Superpositions and higher order Gaussian beams},
journal = {Commun. Math. Sci.},
volume = {6},
number = {1},
year = {2008},
pages = { 449-475},
language = {en},
url = {http://dml.mathdoc.fr/item/1214949931}
}
Tanushev, N. M. Superpositions and higher order Gaussian beams. Commun. Math. Sci., Tome 6 (2008) no. 1, pp. 449-475. http://gdmltest.u-ga.fr/item/1214949931/