The paper by Clayton and Engquist (1977) is often quoted in the literature, almost exclusively when problems involving finite difference methods are being discussed when dealing with acoustic wave propagation and coupled P-S V wave propagation in an elastic medium using finite differences or related methods. With the recent interest in perfectly matched layers (PML) methods it is often used as a bench mark with which to determine the numerical accuracy of this relatively new method (for example, Zhu and McMechan, 1991). This attention is for the most part based on one page of the 1977 paper. There are some cursory instructions on how to proceed to obtain paraxial approximations for the wave equations in the vicinity of finite, usually perfectly reflecting, boundaries and how to employ them. As the single page is followed by an appendix for its implementation, little thought has been paid to what has been said on that page. Here we would like to expand on that page for the information of others who wish to use this method for similar, usually more complex, problems and for its possible use in hybrid methods, where one or more of the spatial derivatives have been removed by integral transform methods. As others have questioned the authors on this topic, it was thought that this mild tutorial could be useful.
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