Why no TEM mode; first four modes

Why a hollow waveguide cannot support TEM

A TEM wave has E_z = H_z = 0, so both E and H lie in the transverse plane and the field is like a plane wave. Inside a hollow, single-conductor guide:

• The transverse H lines must form closed loops in a transverse plane (∇·B = 0). Maxwell's equations then require a longitudinal conduction or displacement current inside the guide for them to circulate around.
• There is no centre conductor to carry axial conduction current, and a TEM wave would have no E_z for displacement current either.
• Equivalently, a TEM field would need a transverse electrostatic potential between two conductors, but a single closed conductor is at one potential, so the transverse E would be zero.

So only TE or TM modes exist (H_z or E_z ≠ 0), and each has a cut-off frequency.

Cut-off frequencies (45 × 90 mm)

Take a = 90 mm = 0.09 m (the larger side), b = 45 mm = 0.045 m.

f_c = 1.5×10⁸ √[(m/0.09)² + (n/0.045)²]

TE₁₀:  1.5×10⁸ × 11.11 = 1.667 GHz
TE₂₀:  1.5×10⁸ × 22.22 = 3.333 GHz
TE₀₁:  1.5×10⁸ × 22.22 = 3.333 GHz
TE₁₁/TM₁₁: 1.5×10⁸ × √(123.5 + 493.8) = 1.5×10⁸ × 24.85 = 3.727 GHz

Answer: 1.667 GHz, 3.333 GHz (TE₂₀ and TE₀₁), and 3.727 GHz (TE₁₁/TM₁₁).