(1354) |

The

where

(1356) |

is the cutoff frequency. There is an absolute cutoff frequency associated with the mode of lowest frequency: that is, the mode with the lowest value of .

For real (i.e., ), it is clear from Equation (1357) that the wave propagates along the guide at the phase velocity

(1357) |

It is evident that this velocity is greater than that of electromagnetic waves in free space. The velocity is not constant, however, but depends on the frequency. The waveguide thus behaves as a dispersive medium. The group velocity of a wave pulse propagated along the guide is given by

(1358) |

It can be seen that is always smaller than , and also that

(1359) |

For a TM mode ( ), Equations (1342)-(1343) yield

(1360) | ||

(1361) |

where use has been made of . For TE modes ( ), Equations (1344)-(1345) give

(1362) | ||

(1363) |

The time-average component of the Poynting vector, , is given by

It follows that

for TE modes, and

(1366) |

for TM modes. The subscript 0 denotes the peak value of a wave quantity.

For a given mode, waveguide losses can be estimated by integrating Equation (1319) over the wall of the guide. The energy flow of a propagating wave attenuates as , where

(1367) |

Thus,

(1368) |

where the numerator is integrated over unit length of the wall, and the denominator is integrated over the transverse cross-section of the guide. It is customary to define the

(1369) |

Here, both integrals are over the transverse cross-section of the guide. It follows from Equations (1366) and (1367) that

(1370) |

for TE modes, and

(1371) |

for TM modes. For both types of mode, .