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Acceleration of Convergence of Double Series for the Green's Function of the Helmholtz Equation in Polar Coordinates

Igor M. Braver

Abstract

The well-known expressions for the Green's functions for the Helmholtz equation in polar coordinates with Dirichlet and Neumann boundary conditions are transformed. The slowly converging double series describing these Green's functions are reduced by means of successive subtraction of several auxiliary functions to series that converge much more rapidly. A method is given for constructing the auxiliary functions (the first one is identical with the Green's function of the Laplace equation) in the form of single series and in the form of closed expressions. Formulas are presented for summing series of the Fourier-Bessel and Dini type; they are required to implement the two steps of the procedure for accelerating convergence. The effectiveness of the functions constructed is illustrated by the numerical solution of the problem of the dispersion properties of a slotted line with a coaxial circular cylindrical screen.

Acceleration of Convergence of Double Series for the Green's Function of the Helmholtz Equation in Polar Coordinates

Abstract

The well-known expressions for the Green's functions for the Helmholtz equation in polar coordinates with Dirichlet and Neumann boundary conditions are transformed. The slowly converging double series describing these Green's functions are reduced by means of successive subtraction of several auxiliary functions to series that converge much more rapidly. A method is given for constructing the auxiliary functions (the first one is identical with the Green's function of the Laplace equation) in the form of single series and in the form of closed expressions. Formulas are presented for summing series of the Fourier-Bessel and Dini type; they are required to implement the two steps of the procedure for accelerating convergence. The effectiveness of the functions constructed is illustrated by the numerical solution of the problem of the dispersion properties of a slotted line with a coaxial circular cylindrical screen.
Paper Structure (4 sections, 31 equations, 2 figures, 1 table)

This paper contains 4 sections, 31 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. DERIVATION OF THE SUMMATION FORMULAS
  3. EXAMPLE OF NUMERICAL REALIZATION
  4. REFERENCES

Figures (2)

  • Figure 1: Cross section of a circular waveguide with a slotted line in the diametral plane.
  • Figure 2: Wave number of the dominant mode for various widths of the slot: 1 and 2 -- calculations using asymptotic formulas (27) and (30), 3 -- solution of the rigorous dispersion equation (21).