Keller (1962) advocated asymptotically expanding the diffracted wave field to obtain a diffraction coefficient. This coefficient, analogous to the reflection coefficient in wave scattering problems in inhomogeneous media, serves to describe the local effect of the edge on the incident wave field. For the canonical problem of diffraction of an acoustic, elastodynamic or electromagnetic incident wave field by a semi-infinite half-plane, the diffracted wave is a cylindrical wave multiplied by an angle-dependent diffraction coefficient. For the simple case of scalar diffraction by an edge, the diffraction coefficients are easily obtained by asymptotically expanding a Fresnel integral solution which, for local sources, is itself valid only in the far-field. These asymptotically-derived diffraction coefficients are not valid in paraboloidal transition regions centered on the direct and reflected shadow boundaries.
To extend the foregoing results, we use an exact time-domain solution for diffraction of a point-source acoustic wave field by a half-plane. This representation was developed by Dalton (1987), who computed the total field by analytically inverse Fourier transforming the classic frequency-domain solution obtained by Carslaw (1899) and simplified by Bowman et al. (1969). The diffraction coefficients are obtained in the time domain by removing the cylindrical wave form from the diffracted wave field. These coefficients, obtained for scattering by both hard and soft semi-infinite half-planes, are uniform across the geometric reflection and shadow boundaries and valid in the near field of the scattering edge.
Carslaw, H.S., 1899. Some multiform solutions of the partial differential equations of physical mathematics and their applications, Proc. London Math. Soc. 30, 121-161.
Dalton, D.R., 1987. Derivation and practical application of exact time-domain solutions for diffraction of acoustic waves by a half plane: M.Sc. thesis, Univ. of British Columbia.
Keller, J.B., 1962. Geometrical theory of diffraction, J. Acoust. Soc. Am. 52, 116-130. Young, T., 1802. {Phil. Trans. Roy. Soc. Lond. 12, 387.