A solution for the diffraction by a semi-infinite impedance cone

Analytical mathematical methods elaborated to find the exact expression of the field scattered by a semi-infinite object with geometrical singularities and mixed boundary conditions have principally progressed for 2D objects such as wedges. Thus, for a field satisfying the scalar wave equation (Helmholtz) and scattered by a classical semi-infinite conical vertex of any angle with a circular section, there exist some rigorous analytical approaches for Neumann and Dirichlet boundary conditions on the scatterer but no analytical method concerning the general solution for the case of mixed boundary conditions of constant impedance type. In fact, D.S. Jones indicates that there does not even exist a theorem for the conditions that would ensure that this problem with constant impedance boundary conditions is well-posed. By a particular use of complex analysis and integral transforms, with the assumption of plane wave illumination in a harmonic state and some conditions of analysis of the scattered field, we derive a reduction of the mathematical problem of the field to a well-posed non-oscillatory integral equation, independent of wave number.