Relation between the solutions of the Helmhotz and parabolic equations for sound propagation

In this paper we establish an integral transform relation between the solutions of the Helmholtz and parabolic equations for sound propagation in an arbitrary two‐dimensional waveguide. The sound speed is a function of both depth and range. For range‐independent sound speeds, the integral transform is proved to be exact by using a normal‐mode expansion. For range‐dependent sound speeds the stationary phase approximation of the transform is, in lowest order, equivalent to the usual parabolic approximation. Corrections to the parabolic approximation are also calculated using the stationary phase method.