A trial to develop a formulation of Helmholtz equation in helical coordinates
I tried to develop in a non-standard way the mathematical formulation for Helmholtz in helical coordinates. Here are the results of several days trial.
2020-01-05 2020-01-06

Introduction

During the Christmas holidays in 2019, I wanted to escape from the topic of the every day work, instead, I would enjoy any kind of hobby research to extend my knowledge base. In the year of 2019, helical waveguides with metallic wall aroused my interest. However, I had no time and no chance to study it among the working days. Hence, during the wet and cold Christmas holidays, I did some basic derivation of its equations from scratch, starting with a try to write the Helmholtz equation regarding the helical transformation.

Results

The derivation for Cartesian coordinates can be downloaded here. The description of the used symbols can also be found in that document. Only the final equations are shown here.

Cartesian Coordinates

Equation in Cartesian coordinates

Cylindrical Coordinates

Equations in cylindrical coordinates

Discussion

These equations, however, are not easily analytically solvable. If one implements the numerical solution in FEM or BEM, these are not the standard Helmholtz eigenvalue equations; thus, the functions have to be re-derived. Falling back to the guess of β is a stupid way. Maybe if one day I have to calculate the dispersion numerically, I would choose a more mature approach such as this one.