A TEM (Transverse Electromagnetic) wave is characterized by electric and magnetic fields that are both perpendicular to the direction of wave propagation. In a rectangular waveguide, which has a rectangular cross-section, it is not possible to support a pure TEM wave.
The reason for this limitation lies in the waveguide's geometry. In order to sustain a TEM wave, the electric and magnetic fields must be entirely transverse (perpendicular) to the direction of propagation and have no longitudinal (parallel) components. However, in a rectangular waveguide, the electric field can have a longitudinal component due to the presence of metallic walls or conducting surfaces.
When an electromagnetic wave propagates in a rectangular waveguide, the boundary conditions imposed by the conducting walls cause the electric field to have both transverse and longitudinal components. The presence of the longitudinal electric field component means that a pure TEM wave cannot exist in a rectangular waveguide.
Instead, a rectangular waveguide typically supports a combination of TE (Transverse Electric) and TM (Transverse Magnetic) modes. These modes have either the electric or magnetic field purely transverse to the direction of propagation, while the other field has both transverse and longitudinal components.
It's worth noting that there are waveguides, such as coaxial cables, that can support TEM waves due to their specific geometry and the absence of conducting walls parallel to the wave propagation. However, rectangular waveguides, with their metallic walls, do not support pure TEM wave propagation.