Aperiodic wave diffraction by a half-plane

Since A. Sommerfeld founded the mathematical theory of diffraction or scattering theory, a lot of mathematicians have tried to attack boundary value problems for time-harmonic and aperiodic wave fields. Different methods have been developed for canonical obstacles, like half-planes, wedges, cones, etc. relying on the Wiener-Hopf technique, involving the factorization of scalar and matricial Fourier symbols, or treating functional equations in strips of the complex plane, the so-called Maliuzhinets method. In the present paper we give an overview of the recent achievements in the subject by the Darmstadt research group. Some sucess in tackling time-aperiodic boundary-transmission problems allowed us to close the gap between the two above-mentioned methods. Very general representations of the time-dependent wave-fields in acoustics, electrodynamics and elastodynamics can be obtained in the 2D case using conformal mapping techniques. The explicit factorization of parameter-dependent Fourier symbols relies strongly on the time-harmonic case, but it can be extended to further situations including, for example, the integrodifferential equations of linear viscoelastodynamics.