Classical Electrodynamics · Problem 8.12

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · waveguide-boundary-perturbation

Problem

A waveguide with lossless dielectric inside and perfectly conducting walls has a cross-sectional contour

C

that departs slightly from a comparison contour

C_0

whose fields are known. The difference in boundaries is described by

\delta(x, y)

, the length measured from

C_0

C

along the normal to

C_0

at the boundary point

(x, y)

. The derivative

d\delta/ds

along the boundary is higher order in small quantities. (a) If the eigenvalue parameters and solutions for

C

and

C_0

are

(\gamma^2, \psi)

and

(\gamma_0^2, \psi_0)

, respectively, without degeneracy, show that to first order in

\delta

\gamma^2 - \gamma_0^2 = -\frac{\displaystyle\oint_{C_0} \delta(x,y)\left[\left|\frac{\partial\psi_0}{\partial n}\right|^2 - \psi_0^*\frac{\partial^2\psi_0}{\partial n^2}\right] dl}{\displaystyle\int_{S_0} |\psi_0|^2\, dx\, dy}

where only the first (second) term in the numerator occurs for TM (TE) modes. [Hint: Follow the same general approach as used in Section 8.6 for the effects of finite conductivity.] (b) Determine the perturbed value of

\gamma^2

for the lowest TE and TM modes (

\text{TE}_{1,0}

and

\text{TM}_{1,1}

) in a rectangular guide if the change in shape is as shown in the accompanying sketch.

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