A Sketch of The Theoretical Analysis
Electromagnetic wave propagation along the axis of the waveguide is described by the wave equation, which is derived from Maxwell's equations, and where the wavelength depends upon the structure of the waveguide, and the material within it (air, plastic, vacuum, etc.), as well as on the frequency of the wave.
The spatial distribution of the time-varying electric fields and magnetic fields within the waveguide depends on boundary conditions imposed by the shape and materials of the waveguide. Let us assume that the waveguide is made of a metal that is such a good conductor that we can consider it to be a perfect conductor. Nearly all waveguides have copper interiors, but some of them are even plated with silver or gold on the inside - excellent conductors, and also resistant to corrosion. Now, the boundary conditions are these:
- Electromagnetic waves do not pass through conductors, but rather, they are reflected.
- Any electric field that touches a conductor must be perpendicular to it.
- Any magnetic field close to a conductor must be parallel to it.
These boundary conditions eliminate an infinite number of solutions to the wave equation, and the ones that remain are the possible solutions to the wave equation inside the waveguide. The rest of the analysis of the solutions of the electromagnetic waves inside a waveguide gets very mathematical.
All that remains that can be said without getting very mathematical is that commonly-used waveguides are only of a few categories. The most common kind of waveguide is one that has a rectangular cross-section, one that is usually not square. It is common for the long side of this cross-section to be twice as long as its short side. These are useful for carrying electromagnetic waves that have a horizontal or vertical polarization to them.
The second most commonly used kind of waveguide has a circular cross-section. These turn out to be quite useful when carrying electromagnetic waves with a rotating, circular polarization to them. Then, its electrical field traces out a helical pattern as a function of time.
The third kind of a waveguide - actually a seldom-used one - has an elliptical cross-section.
Read more about this topic: Waveguide
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