ABSTRACT
Guided propagation, which means sending almost all the energy conveyed by an electromagnetic wave along a chosen path, inevitably requires an inhomogeneous medium. Propagation in waveguides without sources is described by homogeneous equations. In any mathematical problem of the type, it is important to discuss linear independence of solutions. This chapter discusses some relationships which assess linear independence of different solutions in its strongest possible form, namely orthogonality relationships. Many numerical methods have been developed on the premises, and can in fact study waveguides of any shape within reasonable processing times and memory occupation, but it remains extremely hard to assess the completeness of a set of solutions obtained in this way. Waveguides of rectangular cross-section provide an excellent example of application for the previous results, being at the same time technically significant and mathematically simple.
