EXACT SOLUTION TO THE GREEN'S FUNCTION OF THE HELMHOLTZ EQUATION IN A LINEAR GRADED MEDIUM USING GEODESIC POLAR COORDINATES

摘要

Studies of acoustic propagation in underwater and aero environments are frequently reduced to that of solving the Helmhotz equation with a depth dependent sound speed profile. The Helmholtz equation becomes separable in Cartesian coordinate yielding normal mode solutions for several well known model sound speed profiles. Green's functions may then be constructed by normal mode expansion, a technique which must be terminated with a finite number of modes when applied numerically. In this work the Helmholtz equation is expressed in a coordinate system in which the bicharacteristic curves are chosen as one of the coordinates (equivalently the characteristic surfaces are chosen as one of the coordinate surfaces). The resulting equation yields closed form solutions for the acoustic field due to a point source for several well known sound speed profiles. The general technique will be presented and the solutions compared to other well known approximate solutions; such as ray trace solutions, truncated mode expansions and known Green's functions for the Parabolic Equation.