(1344) |

where satisfies the equation

(1345) |

and , , are cylindrical coordinates. Let

(1346) |

It follows that

(1347) |

or

(1348) |

where . The above equation can be recognized as Bessel's equation. The independent solutions of this equation are denoted and . The are regular at , whereas the are singular. Moreover, both solutions are regular at large .

Because the axis ( ) lies within the cavity, the radial eigenfunction must be regular at . This immediately rules out the solutions. Thus, the most general solution for a TM mode is

(1349) |

The are the eigenvalues of , and are determined by the solution of

(1350) |

The above constraint ensures that the tangential electric field is zero on the conducting walls surrounding the cavity ( ).

The most general solution for a TE mode is

(1351) |

In this case, the are determined by the solution of

(1352) |

where denotes differentiation with respect to argument. The above constraint ensures that the normal magnetic field is zero on the conducting walls surrounding the cavity. The oscillation frequencies of both TM and TE modes are given by

(1353) |

If is the ordinal number of a zero of a particular Bessel function of order (defined such that increases with increasing values of the argument) then each mode is characterized by three integers, , , , as in the rectangular case. The th zero of is conventionally denoted [so, ]. Likewise, the th zero of is denoted . Table 2 shows the first few zeros of , , , and . It is clear that, for fixed and , the lowest frequency mode (i.e., the mode with the lowest value of ) is a TE mode. The mode with the next highest frequency is a TM mode. The next highest frequency mode is a TE mode, and so on.