Premium
A particle limit for the wave equation with a variable wave speed
Author(s) -
Hagedorn George A.
Publication year - 1984
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160370106
Subject(s) - gravitational singularity , mathematics , limit (mathematics) , wavelength , wave equation , fourier transform , geometrical optics , mathematical analysis , zero (linguistics) , energy (signal processing) , wave propagation , variable (mathematics) , physics , optics , linguistics , philosophy , statistics
We study short wavelength solutions to the n ‐dimensional wave equation u tt =(c(x)) 2 Δu. We prove that the propagation of certain spatially localized pulses is determined by a time dependent analogue of geometrical optics up to an error whose energy tends to zero in the zero wavelength limit. Our method is very explicit, and no difficulties are incurred as the pulses propagate through caustics. Our goal is to study the physical question of where the energy propagates, rather than the more mathematical question of the propagation of singularities which has been studied in Fourier integral operators.