A Grazing Gaussian Beam

TL;DR

This paper analyzes the behavior of Gaussian beams tangent to a boundary in Friedlander's wave equation, showing that the reflected wave amplitude is approximately half of the incident beam for large wave numbers.

Contribution

It provides a leading-order calculation of reflected Gaussian beams tangent to a boundary, extending boundary interaction analysis to grazing incidence.

Findings

Reflected wave amplitude is nearly half of the incident beam for large k.

The analysis is done to leading order in k on the tangent ray path.

Boundary interaction results differ from non-tangential cases.

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.