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.

# Introduction

During the Christmas holidays in 2019, I wanted to escape from the topic of the
everyday 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

## 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.