Scattering by the Edge
Here we will derive a uniform formula of the edge-waveamplitude, valid throughout the entire domain of the existence of this wave. The idea of such a description is based on the merging formulas of the boundary layer and geometric theory of diffraction. According to the mentioned theory (see Section 1.7), the edge-wave amplitude can be obtained by its continuation along diffracted rays, if it is known in a small vicinity of the edge. The edge-wave amplitude near the edge can be found by solving some standard diffraction problem, which produces a similar congruence of diffracted rays in the vicinity of the edge (Keller, 1962). As a standard, we use the problem of diffraction of a plane wave on a 3-D system of wedge-shaped regions. We will continue the edge-wave amplitude, obtained from the solution of this problem, outside the neighborhood of the edge in accordance with the geometric theory of diffraction. The obtained formulas fail in the neighborhoods of the shadow boundaries of the reflected/transmitted waves, i.e., in the boundary layers. In the boundary layers we can use the formulas of Chapter VI. However, the formulas are invalid outside the boundary layers. To get the uniform description of the edge-wave amplitude, we generalize the boundary layer formulas in such a way that they correctly behave outside the boundary layers.
We give a physical interpretation of obtained formulas in the form of a concept of scattering by the edge. This concept can be expressed in the simplest form not in the usual Euclidean space but on a Riemann multifold surface (Frank and Mises, 1935). The other, less illustrative, way of interpretation is the generalization of the plan-waves theory to the complex directions of propagation. Here we prefer the latter, because it allows us to express all ideas in more or less ordinary terms.