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Electromagnetic scattering problems in cylindrical wave guides, including free space, involve the calculation of the fields produced in the presence of geometrical discontinuities by arbitrary currents. Such discontinuities may be replaced by equivalent electric and magnetic currents. The over-all calculation then leads to two distinct problems: first, the calculation of the fields produced by the prescribed and induced currents in a discontinuity—free guide; second, the self-consistent determination of the induced currents by the condition that the fields so produced satisfy the boundary conditions at the discontinuity surfaces. The first problem is treated herein by representation of the fields in terms of a complete set of vector modes characteristic of the possible transverse field distributions in the guide cross section. This representation transforms the over-all field problem into one-dimensional modal problems of conventional transmission line form. The eigenvalue problem of finding the characteristic modes is discussed in detail for the case of a uniform guide with perfectly conducting walls. The transformation procedure and the solution of the resulting transmission line problem are treated from an impedance point of view. A typical modal analysis and synthesis is presented for the explicit determination of the fields produced by arbitrary electric and magnetic currents in an infinite and semi-infinite wave guide of arbitrary cross section. The connection with a corresponding dyadic Green's function representation (to be treated in Part II) is pointed out.

Orthogonality relations are given for the ``four-vector'' guided modes of anisotropic uniform wave guides and for the ``six-vector'' guided modes of uniform wave guides containing unidirectional electron beams with axially independent dc velocities. In general, these orthogonality relations involve both the modes of the given wave guide and those of an appropriate ``adjoint'' wave guide. A description is given for the adjoint wave guides associated with wave guides which contain media whose characteristics are restricted only by the requirement that they be axially independent and which, if bounded, are subject to a variety of boundary conditions arising from idealizations of actual boundaries.

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