Rational square-root approximations for parabolic equation algorithms

In this article, stable Padé approximations to the function 1+z are derived by choosing a branch cut in the negative half-plane. The Padé coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z<−1, I(z)=0 [I(z) denotes the imaginary part of z]. The characteristics of the corresponding square-root approximation are illustrated for various segments of the complex plane. In particular, for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case. An elastic parabolic equation algorithm is used to illustrate the application of the new Padé approximations to a realistic ocean environment, including elasticity in the ocean bottom.