A waveguide with lossless dielectric inside and perfectly conducting walls has a cross-sectional contour ?! that departs slightly from a comparison contour Co whose fields are known. The difference in boundaries is described by d(x, y), the length measured from C0 to ?! along the normal to C0 at the boundary point (x, y). The derivative dd/ds along the boundary is higher order in small quantities.
(a) If the eigenvalue parameters and solutions for ?! and C0 are (?32, I^) and (?320, I^0), respectively, without degeneracy, show that to first order in d
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where only the first (second) term in the numerator occurs for TM (?c??) modes.
(b) Determine the perturbed value of ?32 for the lowest ?c?? and TM modes (TE1.0 and TM1.1) in a rectangular guide if the change in shape is as shown in the accompanyingsketch.
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(a) If the eigenvalue parameters and solutions for ?! and C0 are (?32, I^) and (?320, I^0), respectively, without degeneracy, show that to first order in d
where only the first (second) term in the numerator occurs for TM (?c??) modes.
(b) Determine the perturbed value of ?32 for the lowest ?c?? and TM modes (TE1.0 and TM1.1) in a rectangular guide if the change in shape is as shown in the accompanyingsketch.
