(a) Suppose that 2 p = p p and 2 q = q q in

(a) Suppose that ∇²_⊥ ψ_p = λ_pψ_p and ∇²_⊥ ψ_q = λ_qψ_q in a two dimensional domain A where either ψ|_C = 0 or ∂ψ/∂n|_C = 0 on the perimeter C of A. Use one of Green’s identities to show that

(b) Let E_p(r, t) = E_p(x, y)e^i(hpz−ωt) and B_p(r, t) = B_p(x, y)e^i(hpz−ωt) be the fields associated with the p^th mode of a cylindrical waveguide with cross sectional area A. When p ≠ q and both modes are TE, use
the results of part (a) to show that non-degenerate modes satisfy

(c) Repeat part (b) when both modes are TM.
(d) Repeat part (b) when one mode is TE and one mode is TM.

A dr Vp(r)q(r) = 0 (p #2g).