Uniform theory of diffraction

Definition

Consider time-harmonic scattering of a wave of wave number $k$ by a perfectly conducting body with edges. Assume that $k L \gg 1$ for every length $L$ of the geometry. Use the time factor $e^{j\omega t}$ of the engineering literature. The uniform theory of diffraction approximates the total field at a point $P$ by the sum of these ray contributions:

1. the incident and reflected fields of geometrical optics, each present only where its ray reaches $P$;
2. the diffracted fields. Let a diffracted ray from an edge point $Q$ reach $P$ at distance $s$. It contributes $$E^d(P) = E^i(Q) \cdot \mathbf{D} \, \sqrt{\frac{\rho}{s (\rho + s)}} \, e^{-j k s},$$ where $\rho$ is the caustic distance of the diffracted ray and $\mathbf{D}$ is a dyadic diffraction coefficient.

The coefficients $\mathbf{D}$ are those of Keller's geometrical theory of diffraction, with one change. Each term that is singular at a shadow or reflection boundary is multiplied by a transition function. That function is $$F(X) = 2 j \sqrt{X} \, e^{j X} \int_{\sqrt{X}}^{\infty} e^{-j \tau^2} \, d\tau, \qquad X > 0,$$ and its argument $X$ tends to $0$ at that boundary.