A correction to the parabolic approximation

In a previous paper [J. A. DeSanto, J. Acoust. Soc. Am. 62, 295 (1977)] we showed that the solutions of the Helmholtz and parabolic equations were related by an integral transform. The stationary phase approximation of the transform led to corrections of the parabolic approximation (PA). Based on this analysis, we define in this paper a corrected parabolic approximation (CPA) involving the second range derivative of the parabolic field, numerically compare the results to a normal mode example, and show how the CPA reduces the error of the PA. The error reduction is most marked in the phase.