Coaxial waveguides and the determination of their cutoff frequencies have been discussed by Marcuvitz . This involves finding zeros of a function that involve products of Bessel functions of 1st kind and 2nd kind for TM modes and products of derivatives of Bessel functions of 1st kind and 2nd kind for TE modes. These zeros pertain to a certain order and needs a number of iterations to be performed to obtain a set of cutoff wavenumbers. Finite difference methods in the conventional form have been applied t0 a variety of cross sections. However the same technique involving formation of rectangular meshes to a circular coaxial waveguide does not seem to be appealing in context with the selection of truncation boundaries and appropriate adjacent node points for the region between the inner and outer conductors. The increase in the number of spurious modes generated and the decrease in accuracy are also detrimental to this choice of meshing technique.
In my approach the curvilinear rectangular grid formation is used to mesh the region between the two conductors which enables us to encompass the cross sectional region completely thus eliminating the necessity of truncation of boundary. This also yields much accurate result than the conventional rectangular grid formation and is much simpler than the cumbersome method of finding roots from the closed form expressions. The Laplacian operator appearing in the Helmholtz equation in polar form is represented as a five-point difference operator along the radial and circumferential directions. The eigen values which give cutoff frequencies are evaluated from the characteristic equation expressed in terms of matrices obtained from a set of simultaneous equations satisfying the boundary conditions.