Waveguides

Waveguides

In the class we discussed plane electromagnetic waves traveling in infinite space. But when one wants to use electromagnetic waves to transmit information or power over long distances, one usually considers waves traveling in waveguides.

Waveguides are long tubes with conducting walls, such that the EM wave is confined to the inside of the tube. Such tubes may be hollow, or may have another conducting tube inside, in a coaxial configuration, such that the wave is confined to the space between the inner and outer tube.

At the surface of the bounding conducting walls of the waveguide, the field of the EM wave must satisfy the usual boundary conditions at the surface of a conductor:

tangential component of E = 0, and normal component of B = 0

The first follows from the usual condition that E = 0 inside a conductor, and that the tangential component of E is continuous at an interface. The second follows from the condition that the normal component of B is continuous at an interface, and we will assume that B is zero in the conductor.

Let's take the axis of the waveguide in the z-direction. Solving Maxwell's equations for a wave solution, traveling down the axis of the waveguide and obeying these boundary conditions, one finds that the electric and magnetic fields are not always transversely polarized, i.e., they have a component in the z-direction.

For a hollow waveguide, one finds solutions where either the E is transversely polarized (and B has a component in the z-direction), or B is transversely polarized (and E has a component in the z-direction). The first case are called TE waves (transerse electric), while the latter are called TM waves (transverse magnetic).

For a coaxial waveguide, there are also TEM solutions where both E and B are transversely polarized.

For a waveguide with a rectangular cross-section, find the solutions for TE and TM waves, and find the dispersion relation between frequency and wavenumber of these waves.

For a coaxial waveguide with cylindrical cross-section, find the solutions for the TEM waves, and find the dispresion relation for these waves.

Waveguides are a standard topic in many EM texts, for example you may find them discussed in Griffiths 3rd edition section 9.5. So you can look there for help.