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 ?(x, y), the length measured from C₀ to ? along the normal to C₀ at the boundary point (x, y). The derivative d?/ds along the boundary is higher order in small quantities.

(a) If the eigenvalue parameters and solutions for ? and C₀ are (?², ?) and (?²₀, ?₀), respectively, without degeneracy, show that to first order in ?

where only the first (second) term in the numerator occurs for TM (??) modes.

(b) Determine the perturbed value of ?² for the lowest ?? and TM modes (TE_1.0?and TM_1.1) in a rectangular guide if the change in shape is as shown in the accompanying sketch.

8(x, y) dl n Co n Y - v = IWol dx dy So 2.