Consider a vacuum-filled wave-guide which runs parallel to the -axis.
An electromagnetic wave trapped inside the wave-guide satisfies Maxwell's equations for free space:

(1280) | |||

(1281) | |||

(1282) | |||

(1283) |

Let , and , where is the wave frequency, and the wave-number parallel to the axis of the wave-guide. It follows that

Equations (1287) and (1289) yield

(1292) |

(1293) |

(1294) |

(1295) |

Here, and are the

Substitution of Eqs. (1296) and (1297) into Eqs. (1288) and (1291) yields the equations satisfied by
the longitudinal fields:

The remaining equations, (1284) and (1285), are automatically satisfied provided Eqs. (1296)-(1299) are satisfied.

We expect
inside the walls of the wave-guide,
assuming that they are perfectly conducting. Hence, the appropriate
boundary conditions at the walls are

(1300) | |||

(1301) |

It follows, by inspection of Eqs. (1296) and (1297), that these boundary conditions are satisfied provided

at the walls. Here, is the normal vector to the walls. Hence, the electromagnetic fields inside the wave-guide are fully specified by solving Eqs. (1298) and (1299), subject to the boundary conditions (1302) and (1303), respectively.

Equations (1298) and (1299) support two independent types
of solution. The first type has , and is consequently called a *transverse electric*, or TE, mode. Conversely, the
second type has , and is called a *transverse
magnetic*, or TM, mode.

Consider the specific example of a *rectangular* wave-guide, with conducting walls
at , and . For a TE mode, the longitudinal
magnetic field can be written

(1304) |

(1305) | |||

(1306) |

where , and . Clearly, there are many different kinds of TE mode, corresponding to the many different choices of and . Let us refer to a mode corresponding to a particular choice of as a mode. Note, however, that there is no mode, since is uniform in this case. According to Eq. (1299), the dispersion relation for the mode is given by

where

According to the dispersion relation (1307), is imaginary for
. In other words, for
wave frequencies below , the mode
fails to propagate down the wave-guide, and is instead attenuated. Hence,
is termed the *cut-off frequency* for the mode.
Assuming that , the TE mode with the lowest cut-off frequency is
the mode, where

(1309) |

For frequencies above the cut-off frequency, the phase-velocity of the
mode is given by

which is always less than . Of course, energy is transmitted down the wave-guide at the group-velocity, rather than the phase-velocity. Note that the group-velocity goes to zero as the wave frequency approaches the cut-off frequency.

For a TM mode, the longitudinal electric field can be written

(1312) |

(1313) | |||

(1314) |

where , and . The dispersion relation for the mode is also given by Eq. (1307). Hence, Eqs. (1310) and (1311) also apply to TM modes. However, the TM mode with the lowest cut-off frequency is the mode, where

(1315) |

There is, in principle, a third type of mode which can propagate down
a wave-guide. This third mode type is characterized by ,
and is consequently called a *transverse electromagnetic*, or
TEM, mode. It is easily seen, from an inspection of
Eqs. (1286)-(1291), that a TEM mode satisfies

(1317) | |||

(1318) |

where satisfies

The boundary conditions (1302) and (1303) imply that

at the walls. However, there is no non-trivial solution of Eqs. (1319) and (1320) for a conventional wave-guide. In other words, conventional wave-guides

(1321) |

(1322) |