Consider an evacuated waveguide of rectangular cross-section that runs along the -direction, and is enclosed by perfectly conducting (i.e., infinite conductivity) metal walls located at , , , and . Suppose that an electromagnetic wave propagates along the waveguide in the -direction. For the sake of simplicity, let there be no -variation of the wave electric or magnetic fields. The wave propagation inside the waveguide is governed by the two-dimensional wave equation [cf., Equation (537)]

where represents the electric component of the wave, which is assumed to be everywhere parallel to the -axis, and is the velocity of light in vacuum. The appropriate boundary conditions are

because the electric field inside a perfect conductor is zero (otherwise, an infinite current would flow), and, according to standard electromagnetic theory (see Appendix C), there cannot be a tangential discontinuity in the electric field at a conductor/vacuum boundary. (There can, however, be a normal discontinuity. This permits to be non-zero at and .)

Let us search for a separable solution of Equation (883) of the form

where represents the -component of the wavevector (rather than its magnitude), and is the effective wavenumber for propagation along the waveguide. The previous solution automatically satisfies the boundary condition (884). The second boundary condition (885) is satisfied provided

(887) |

where is a positive integer. Suppose that takes its smallest possible value . ( cannot be zero, because, in this case, everywhere.) Substitution of expression (886) into the wave equation (883) yields the dispersion relation

where

(889) |

This dispersion relation is analogous in form to the dispersion relation (797) for an electromagnetic wave propagating through a plasma, with the

(890) |

is superluminal. This is not a problem, however, because the group velocity,

(891) |

which is the true signal velocity, remains sub-luminal. (Recall, from Section 10.2, that a high frequency electromagnetic wave propagating through a plasma exhibits similar behavior.) Not surprisingly, the signal velocity goes to zero as , because the signal ceases to propagate at all when .

It turns out that waveguides support many distinct modes of propagation. The type of mode discussed previously is termed a TE (for transverse electric-field) mode, because the electric field is transverse to the direction of propagation. (See Exercise 12.) There are many different sorts of TE mode, corresponding, for instance, to different choices of the mode number, (Fitzpatrick 2008). However, the mode has the lowest cut-off frequency. There are also TM (for transverse magnetic-field) modes (see Exercise 13), and TEM (for transverse electric- and magnetic-field) modes (ibid.). TM modes also only propagate when the wave frequency exceeds a cut-off frequency. On the other hand, TEM modes (which are the same type of mode as that supported by a conventional transmission line) propagate at all frequencies. However, TEM modes are only possible when the waveguide possesses an internal conductor running along its length (ibid.).