In underwater acoustics, the parabolic wave equation is often solved by employing techniques based on finite difference techniques. A major drawback of the finite difference treatment of interfaces to date is that mesh points have been required to lie exactly on the interface. With regularly spaced, predetermined meshes this can be a serious computational restriction. Previous treatments of the parabolic wave equation using finite differences have assumed an equispaced mesh and that at interfaces a mesh point will lie precisely on the interface. In this paper we remove both of these restrictions. This approach also makes it apparent that mesh points can be added or deleted during the solution process, and so presents a more general model to work with.