The framework for solving the problem of a hollow conducting waveguide is set up; the fields are broken down into longitudinal components and transverse components. Expressions are derived which relate these components to each other. It is shown that transverse electromagnetic (TEM) waves cannot propagate in hollow waveguides, but transverse electric (TE) and transverse magnetic (TM) waves can. Also, the boundary conditions that the tangential component of the electric field vanishes and the normal component of the magnetic field vanishes on the surface of the waveguide are expressed in terms of the longitudinal component of the magnetic field for TE waves and the longitudinal component of the electric field for TM waves.
