### 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

#### ABSTRACT

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.

#### REFERENCES

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.