Confluent tip singularity of the electromagnetic field at the apex of a material cone

Abstract

The tip singularity of the electromagnetic field at the apex of a cone (conical sheet) is investigated in its most general framework. To this end one considers, without loss of generality, a circularly symmetric cone which separates two simple media having different constitutive parameters, and tries to reveal the asymptotic behaviour of the electromagnetic field created near the apex of the cone by any rotationally symmetric source distribution. To cover various boundary conditions which are extensively used in actual investigations, the cone is supposed to be formed by an infinitely thin material sheet having its own constitutive parameters. The results show that the type and order of the singularity depend, in general, on various parameters such as (i) the apex angle of the cone, (ii) the constitutive parameters of the mediums separated by the cone, (iii) the constitutive parameters of the material cone itself and (iv) the topology of the conical surface. The problem of determining the order in question gives rise to a transcendental algebraic equation involving the Legendre functions of the first kind with complex orders. If the order is a simple root of this equation, then the singularity is always of the algebraic type whereas a multiple root gives rise also to logarithmic singularities. A numerical method suitable to find a good approximate solution to this equation is also established. Since the general expressions of the boundary conditions on the material cone, which are compatible with both the Maxwell equations and the topology of the cone, are not known, an attempt has also been made to derive these expressions. Some examples concerning the boundary conditions which are extensively considered in actual investigations are given.