Title: A Grazing Gaussian Beam,
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.