Connection between the solution of the Helmholtz and parabolic equations for sound propagation

Abstract

Using a conformal mapping technique in a rectangular waveguide, we present an exact integral relation between the solutions of the Helmholtz equation whose sound speed c(x,y) varies as a function of both depth y and range x and the solutions of a parabolic equation whose sound speed varies in the mapped depth coordinate. The relation of the corresponding boundary value problems is also discussed, as well as the use of the . parabolic approximation in underwater sound propagation problems. The conformal transformation interrelates the sound speeds of the two equations. Several examples are discussed. When c(x,y) = c(y) is only a function of depth we get the recent result of Polyanskii. Other examples for a general conformal transformation are functions c(x»y) which are sinusoidal in depth and exponentially decrease to a constant in range. Several alternative methods of using these results are also discussed.