Approximate representations of the scattering amplitude

This is the second in a series of articles about the theory of scattering from a rough surface. A symmetric representation of the scattering amplitude and a formal statement of the composite model, both derived in the previous article, are used to approximate the scattering amplitude in terms of the known results for a reference surface. When the reference surface is a plane, the small slope approximation follows, while the traditional composite model is obtained if the curvature of the reference surface is neglected. The medium is assumed to be homogeneous, and the calculation is performed for scalar waves obeying Dirichlet or Neumann boundary conditions, electromagnetic waves obeying perfect conductor boundary conditions and the interface between two media for both types of waves. Finally, the requirement that the formal composite model must reproduce the traditional composite model in the appropriate limit is used to obtain a new approximation to the scattering amplitude. This is done for the Dirichlet, Neumann, and perfect conductor problems. The error goes as the square of the curvature.