The cross-section of a waveguide rarely remains uniform from the source to the load. Many kinds of discontinuities can occur, such as transition from a rectangular to a circular guide, transition between a waveguide and a coaxial line, differences in height, attenuators, diaphragms used to provide a filter effect, and surface roughness left from the finishing process. As we have already seen, the electromagnetic field of the principal mode can no longer satisfy the boundary conditions at such points, and higher modes are produced locally but disappear again at some distance from the discontinuity. The mathematical treatment of such problems is very complicated, and an exact solution is not always possible, except in certain cases where the geometry is extremely simple. Approximate methods, which are often quite complicated, must normally be used to deduce the field configuration near such a discontinuity. Fortunately, we are not usually interested in the short-range effect of discontinuities, but rather in the effect at some distance. The overall effect can be described with the aid of complex reflection and transmission coefficients (see Section 14.3). If the discontinuity has an appreciable length in the axial direction of the waveguide, we must of course specify a reference plane, i.e. the plane at right angles to the axis at which the coefficients describing the effect of the discontinuity have been determined.
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