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.

One-way or parabolic wave equations for time-harmonic propagation in two-dimensional elastic waveguides are considered. It is shown that the direct application of a rational linear approximation with real coefficients to the elastic wave propagation case results in exponential growth in the numerical solutions. Elementary analysis demonstrates that this kind of approximation does not treat properly the modes with complex wavenumber which can exist in elastic waveguides. A new bilinear square-root approximation with complex coefficients is introduced that accommodates all mode types and leads to stable numerical solutions. In the case of thick elastic layers (such as sea-bottom sediments), this new approximation gives accurate total field prediction. When thin elastic layers (such as ice on the sea surface) are present, however, the method introduces excessive damping to modes with wavenumbers significantly different from a reference wavenumber.

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PECan is a Canadian N× 2D/3D parabolic equation (PE) underwater sound propagation model that was developed for matched-field processing applications. It is based on standard square-root operator and/or propagator approximations that lead to an alternating direction solution of the 3-D problem. A 2-D split-step Padé approximation is employed for propagation in range. The 3-D azimuthal corrections are computed using either a split-step Fourier method or a Crank–Nicolson finite-difference approximation. It features a heterogeneous formulation of the differential operators on an offset vertical grid, energy conservation, a choice of initial field including self-starter, and both absorbing and nonlocal boundary conditions. Losses due to shear wave conversion in an elastic bottom are handled in the context of a complex density approximation. In this paper, PECan is described and validated against some standard benchmark solutions to underwater acoustics problems. Subsequently, PECan is applied to several single-frequency test cases that were offered for numerical consideration at the SWAM'99 Shallow Water Acoustic Modeling workshop.

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