Singular integral equation for antiplane-wave scattering by a semi-infinite crack

The problem of antiplane-wave scattering by a semi-infinite crack is reduced to a singular integral equation of the Cauchy type. This equation is obtained by treating the problem as the limiting case of a sequence of problems for which the crack-opening displacements decay exponentially at infinity, and by using real-variable (as opposed to complex-variable) Fourier-transform methods. An integral identity is used to obtain the solution of the singular integral equation. The solution is shown to coincide with the classical Wiener-Hopf solution of the problem.