## 1. Introduction

[2] The study of wave scattering by sections of perfect (perfectly conducting, rigid or soft) conical surfaces has a fundamental role in diffraction theory. Such problems are models for study of different physical phenomena. Analysis of the scattering and diffraction by the conical sections as well as radiation from them has been of great interest recently due to prediction and reduction of the radar cross section (RCS) of different targets, wide-band antennas as well as horn and phased antennas design. This problem serves as a simple model of duct structures such as jet engine intakes of aircrafts and defects' occurring on surfaces of complicated bodies.

[3] Some of the conical scatterers have been analyzed thus so far using a variety of different analytical asymptotical and numerical methods by *Ufimtsev* [1962], *Northover* [1965a, 1965b], *Pridmore-Brown* [1968], *Senior and Uslenghi* [1971], *Bevensee* [1973], *Syed* [1981], *Eremin and Sveshnikov* [1987], *Goshin* [1987], and *Davis and Scharstein* [1994]. It is known that fields of perfect conical scatterers can be completely described within discrete modal representation. So, the most adequate mathematical approach, which could be used for studying the corresponding diffraction processes, is the mode matching technique. Such an approach is widely used in field analysis of canonical structures, but its formal application is known to become substantially complicated for wave diffraction analysis in resonance and quasioptical frequency bands. At the same time, there are well known approaches, which allow the substantial improvement of the possibilities of the mode matching in the case of internal boundary diffraction problems for bifurcations of infinitely extended (rectangular or cylindrical) waveguides [*Mittra and Lee*, 1971; *Shestopalov et al.*, 1984]. Accounting the importance of conical structures for solution of a wide range of problems in technical physics, and the complexity of rigorous analysis of the corresponding diffraction problems the analytical regularisation technique was developed by *Kuryliak* [2000a, 2000b] and *Kuryliak and Nazarchuk* [2000, 2006] for obtaining of the solutions. The proposed method includes four principal steps: (a) application of the mode matching technique and derivation of the rule for correct receiving of the infinite system of linear algebraic equations (ISLAE); (b) separation of the matrix operator of convolution type: *A*:{*a*_{pn} = (*ξ*_{p} − *z*_{n})^{−1}} from the static part of ISLAE, where *ξ*_{p}, *z*_{n} are the functions which are linear depended from the indexes at *p*, *n* → ∞; (c) finding of the inverse operator *A*^{−1} in analytical form; (d) application of the pair of operators *A* and *A*^{−1} for the development of analytical regularisation technique, which includes the separation of the convolution type operator *A* from ISLAE, its analytical inversion by the operator *A*^{−1} and transition to the ISLAE of the second kind.

[4] The main reason for application of this technique is that operators of the convolution type consist of the principal part of the asymptotic of ISLAE for initial diffraction problem. It means that if the indexes of the matrix elements of ISLAE tend to infinity, the matrix elements of this system tend to the elements of the matrix operators of the convolution type for any geometrical parameters and frequency. The effectiveness of the proposed technique for solution of wave diffraction problem is stipulated by the analytical inversion of the principal part of its asymptotic. This leads to improvement of the convergence of the reduction procedure and allows obtaining the solution with prespecified accuracy for any parameters of the problem. In this paper the main principles of such an approach are demonstrated. This approach can be used to obtain the rigorous solutions for a wide range of problems, particularly, for the structure with coaxial sections with the same efficiency as it is made for waveguides. The time factor is assumed to be *e*^{−iωt} and suppressed throughout this paper.