Parabolic Equation Techniques

Parabolic equation techniques are based on rational approximations of the square root and other functions. The split-step Pade solution is based on an exponential of the square root [1], which takes into account the range numerics and allows large range steps. An initial condition may be obtained using the self-starter [2], which is based on a far-field approximation of a Hankel function of the square root. Accurate solutions to range-dependent problems may be obtained with the energy-conserving parabolic equation [3], which is based on the fourth root. Three-dimensional parabolic equations are derived by factoring the wave equation without making the uncoupled azimuth approximation [4–6]. When horizontal variations in the environment are sufficiently gradual so that energy coupling between modes may be neglected, three-dimensional calculations can be avoided by solving horizontal wave equations for the mode coefficients [7, 8].