Axially Symmetric Cavities

Suppose that the direction of symmetry is along the -axis, and that the length of the cavity in this direction is . The boundary conditions at and demand that the dependence of wave quantities be either or , where . In other words, all wave quantities satisfy

as well as

(1328) |

where stands for any component of or . The field equations

(1329) | ||

(1330) |

must also be satisfied.

Let us write each vector and each operator in the above equations as the sum of a transverse part, designated by the subscript , and a component along . We find that for the transverse fields

When one of Equations (1333)-(1334) is used to substitute for the transverse field on the right-hand side of the other, and use is made of Equation (1329), we obtain

Thus, all transverse fields can be expressed in terms of the components of the fields, each of which satisfies the differential equation

where stands for either or , and is the two-dimensional Laplacian operator in the transverse plane.

The conditions on and at the boundary (in the transverse plane) are quite different: must vanish on the boundary, whereas the normal derivative of must vanish to ensure that in Equation (1336) satisfies the appropriate boundary condition. If the cross-section is a rectangle then these two conditions lead to the same eigenvalues of , as we have seen. Otherwise, they correspond to two different sets of eigenvalues, one for which is permitted but , and the other where the opposite is true. In every case, it is possible to classify the modes as transverse magnetic or transverse electric. Thus, the field components and play the role of independent potentials, from which the other field components of the TE and TM modes, respectively, can be derived using Equations (1335)-(1336).

The mode frequencies are determined by the eigenvalues of Equations (1329) and (1337). If we denote the functional dependence of or on the plane cross-section coordinates by then we can write Equation (1337) as

Let us first show that , and, hence, that . Now,

(1337) |

It follows that

(1338) |

where the integration is over the transverse cross-section, . If either or its normal derivative is to vanish on the conducting surface, , then

(1339) |

We have already seen that . The allowed values of depend both on the geometry of the cross-section, and the nature of the mode.

For TM modes, , and the dependence of is given by . Equation (1338) must be solved subject to the condition that vanish on the boundaries of the plane cross-section, thus completing the determination of and . The transverse fields are then given by special cases of Equations (1335)-(1336):

For TE modes, in which , the condition that vanish at the ends of the cylinder demands a dependence on , and a which is such that the normal derivative of is zero at the walls. Equations (1335)-(1336), for the transverse fields, then become

and the mode determination is complete.