High frequency propagation in a waveguide with an expanding section

A waveguide whose walls have a constant local impedance is readily analyzed if its cross section is uniform with a standard shape. This is not true if the cross section expands or contracts. The (one-dimensional) Webster horn equation typically is used for low frequency propagation in such configurations, but doing so restricts consideration to situations where the size of the cross section changes slowly over the scale of a wavelength. Somewhat paradoxically, a waveguide consisting of a sequence of different sized uniform segments has been the subject of numerous analyses that are not limited to low frequencies. The present work extends one such analysis, which uses a collocation technique at discontinuous junctions, to a two-dimensional configuration in which a transition section expands linearly. The field in each section is described as a modal series, whose terms are the product of a transverse standing mode and two phasors for axial propagation. Cartesian coordinates are used to describe the field within the uniform sections, whereas a polar coordinate description of the field in the expansion section facilitates satisfying rigid wall conditions. Various schemes for arranging the collocation points are discussed, and convergence of the series is examined.