Canadian Applied Mathematics Society abstract

This paper was presented by the first author, Matthew Yedlin, who is currently cross-appointed between the departments of Electrical Engineering and Earth & Ocean Sciences at UBC, at the 1988 Canadian Applied Mathematics Society meeting at SFU, Burnaby, BC.

Diffraction Coefficient Formulation of Exact Time-Domain Solutions
for Acoustic Diffraction by a Half Plane

by Matthew J. Yedlin and David R. Dalton


Young's (1802) hypothesis that obstacle edges act to diffract an incoming wave field is now well-accepted. For the simple canonical problems of edge diffraction of incoming acoustic, elastodynamic or electromagnetic (EM) radiation, exact solutions for the total field may be obtained. By asymptotically expanding the resultant integrals, the total wave field can be partitioned into a geometrical optics (G.O.) component and a diffraction component. Alternatively, the total-field integral solution may be partitioned into G.O. terms, which have step function discontinuities across the shadow boundaries, and diffracted terms, which have signum function discontinuities such that the total field is continuous.

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.


Bowman, J.J., Senior, T.B.A. and Uslenghi, P.L.E., 1969. Electromagnetic and Acoustical Scattering by Simple Shapes, North Holland, Amsterdam.

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.