Propagation characteristics of equilateral triangular waveguides

Abstract Equilateral triangular waveguides happen to be one of the very few kinds of waveguides, whose field solutions can be constructed without necessarily solving the Maxwell’s equations. Solutions can be obtained simply by superposing some plane wave solutions and requiring the solutions to obey the necessary boundary conditions for each modes, as would be expected of any Maxwell equation’s solution or that of any Helmholtz equation resulting from the problem of heat flow, membrane vibrations, etc. In fact, this was how one of the earlier solutions or eigen functions found by Gabriel Lamé, were obtained. Julian Schwinger performed the same analysis, but used the superposition of complex exponential functions, in order to obtain the eigen functions of an equilateral triangular waveguide. The solutions exhibit other symmetric properties that leave their solutions invariant and these symmetries and applications are what we investigate in this work, particularly as it relates to their attenuation characteristics and quality factors. Finally, by employing the number theory of Eisenstein primes we have obtained more rigorous and well ordered eigenvalues of the equilateral triangular waveguides, which is definitely something missing in the literature.