Arectangular wave guide with a constant cross section and perfectly conducting walls contains a curved section as sketched below. Also indicated is a local Cartesian coordinate system where the z-axis and y-axis remain tangent and normal to the walls, respectively.

(a) The scalar function Φ satisfies [∇²_⊥ + ω²/c²](y, z) = 0, where 2 = ∂²/∂x² + ∂/∂y². Show that the four vacuum Maxwell equations and conducting wall-boundary conditions are satisfied by time harmonic transverse electric (TE) modes of the form

(b) Suppose that the curvature κ(z) of the side wall at any point on the guide satisfies κa

Separate variables in the Helmholtz equation and show that propagating modes exist in the straight portion of the guide (at least) whenω > πc/a.
(c) Show that at least one mode exists in the curved part of the guide for ω

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