A theoretical foundation for the use of the parabolic wave equation/Fourier split‐step method for modeling electromagnetic tropospheric propagation is presented. New procedures are used to derive a scalar Helmholtz equation and to subsequently transform to a rectangular coordinate system without requiring approximations. The assumptions associated with reducing the resulting exact Helmholtz equation to the parabolic wave equation that is used for computations are then described. A similar discussion of the error sources associated with the Fourier split‐step solution technique is provided as well. These discussions provide an important indication of the applicability of the parabolic equation/split‐step method to electromagnetic tropospheric propagation problems. A rigorous method of incorporating an impedance boundary at the Earth's surface in the split‐step algorithm is also presented for the first time. Finally, a few example calculations which demonstrate agreement with other propagation models are provided.

The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current interest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary interest is with the computational methods, there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in themselves. These results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some of the more powerful techniques available are illustrated by applying them to a model problem. 3. Elementary solutions. We quote Rayleigh 128 who says, "The theory of the free vibrations of a membrane was first successfully considered by Poisson 116]. His theory in the case of the rectangle left little to be desired." For the rectangle 0 _< x _< a, 0 <_ y <_ b, the eigenfunctions are

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