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