Abstract

One dimensional complex analysis and Fourier transforms are very successful tools in diffraction theory. They led to many innovative mathematical methods such as the Wiener-Hopf technique. Important canonical problems such as the diffraction by a half-plane can be dealt with efficiently and elegantly with those. The resulting wave fields are often written as an inverse Fourier transform integral over the real axis of the complex Fourier space. It is often convenient to assume that the wavenumber has a small positive imaginary part so that this integral is actually well defined, and one then has to let the imaginary part of the wavenumber tend to zero somehow. This process is not straightforward and necessitates an indentation of the integration contour. However, even if they are very successful, such techniques are mainly applicable to “simple” two-dimensional problems. For intrinsically three-dimensional problems (e.g. quarter-plane) or complicated two-dimensional problems (e.g. penetrable wedge), it is not sufficient. One approach is to use double Fourier transforms together with complex analysis in two complex variables. In this talk, we will consider physical fields defined as double inverse Fourier integrals of a special class of spectral functions: those with the so-called real-property. We will generalise the concept of contour indentation to higher dimensions, and provide a concise notation, the bridge and arrow notation, that specifies the relative position of the integration surface and the singularities of the spectral functions. Using this we will aim to shed some light on the far-field asymptotics of such physical-field and show that it can be reduced to the local study of a finite set of very specific points in the double Fourier space. We will illustrate this method by applying it to the long-standing problem of diffraction by a quarter-plane. This is some ongoing joint work with A.V. Shanin and A.I. Korolkov.