Connection Between the Solutions of the Helmholtz and Parabolic Equations for Sound Propagation

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 cx,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 cx,y cy is only a function of depth we get the recent result of Polyanskii. Other examples for a general conformal transformation are functions cx,y which are sinusoidal in depth and exponentially decrease to a constant in range. Several alternative methods of using these results are also discussed.