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Impedance concepts in wave guides have been discussed rigorously and have been applied to the analysis of a plane discontinuity in the form of a junction between two guides of arbitrary cross sections separated by an infinitely thin diaphragm having arbitrary openings. The problem is described by a four-terminal network whose elements are given by the solution of an infinite number of simultaneous equations, any finite number of which may be solved to give a uniformly converging approximation to the true solution. In any specific application, it is merely necessary to substitute the characteristic eigenfunctions and eigenvalues in these equations without repeating their formulation. In many cases the analysis gives a lower bound to the true impedance, and in the case of a simple obstacle (no change of cross section) an alternative analysis is developed to yield an upper bound. The general equivalent circuit is represented as a T network, but it is shown that there is one important category of problems where this T section reduces to an ideal transformer plus a shunt element, which in turn may be reduced to a pure shunt element. The formulation of a pi network is discussed. The theory is applied to a transverse wire, capacitative and inductive windows, and capacitative and inductive changes of cross section in a rectangular guide, and approximate expressions for their impedances are deduced.