Synopsis

Chapter 5 summarizes the basic propagation processes that are encountered when studying radiation or edge diffraction. Three problems of progressive difficulty are studied. We begin by calculating the transient, antiplane radiation excited by a line source at the surface of a half-space. The Cagniard–deHoop method is used to invert the integral transforms. We then return to considering how plane waves and a knowledge of their interactions can be used to construct more general wavefields. We calculate the time harmonic, inplane radiation, from a two-dimensional center of compression buried in a half-space. Plane-wave spectral techniques are used and the resulting integrals are approximated by the method of steepest descents. This method is discussed in detail. Lastly, we extend our knowledge of plane-wave interactions by calculating the diffraction of a time harmonic, plane, antiplane shear wave by a semi-infinite slit or crack. This problem is solved exactly by using the Wiener–Hopf method and approximately by using matched asymptotic expansions. An Appendix describing the reduction of the diffraction integral to Fresnel integrals is included.

Antiplane Radiation into a Half-Space

We consider an elastic half-space. The x1 coordinate stretches along its surface and the positive x2 coordinate extends into the interior. At the origin a line load is applied to an otherwise traction-free surface. The line load is a tangentially acting force very localized in x1 and directed from −∞ to ∞ in the x3 direction.