A Grazing Gaussian Beam
Analysis of PDEs
2017-07-13 v1
Abstract
We consider Friedlander's wave equation in two space dimensions in the half-space x > 0 with the boundary condition u(x,y,t)=0 when x=0. For a Gaussian beam w(x,y,t;k) concentrated on a ray path that is tangent to x=0 at (x,y,t)=(0,0,0) we calculate the "reflected" wave z(x,y,t;k) in t > 0 such that w(x,y,t;k)+z(x,y,t;k) satisfies Friedlander's wave equation and vanishes on x=0. These computations are done to leading order in k on the ray path. The interaction of beams with boundaries has been studied for non-tangential beams and for beams gliding along the boundary. We find that the amplitude of the solution on the central ray for large k after leaving the boundary is very nearly one half of that of the incoming beam.
Cite
@article{arxiv.1707.03477,
title = {A Grazing Gaussian Beam},
author = {James Ralston and Neelesh Tiruviluamala},
journal= {arXiv preprint arXiv:1707.03477},
year = {2017}
}