Waveguide operation exactly at cutoff In a rectangular waveguide, what happens to the flow of electromagnetic energy at the cutoff frequency?

Introduction / Context:
Propagation in waveguides occurs only above the cutoff frequency for a given mode. At and below cutoff, fields become evanescent and do not transport real power along the guide. Understanding this boundary clarifies why systems specify operating bands well above cutoff to ensure low loss and stable group velocity.

Given Data / Assumptions:

• Rectangular metallic waveguide supporting a specific TE/TM mode.
• Operating at frequency f = f_c (mode cutoff).
• Lossless idealization for basic reasoning.

Concept / Approach:
At cutoff, the longitudinal propagation constant β equals 0, so the phase advance per unit length is zero and group velocity collapses to 0. Poynting vector averaged over a cross-section is proportional to β (through power–velocity relations), so average power flow goes to zero. For f < f_c, β becomes imaginary; fields decay exponentially and still do not carry real power.

Step-by-Step Solution:

Verification / Alternative check:
Measured transmission near cutoff shows rapidly increasing attenuation as frequency approaches f_c from above, and below f_c only near-field (non-propagating) coupling occurs.

Why Other Options Are Wrong:

• Infinite/50%/10%: none aligns with the physics of β = 0 and zero group velocity at cutoff.

Common Pitfalls:
Assuming nonzero power because fields exist—standing fields do not guarantee net forward energy transport; confusing stored reactive energy with transmitted power.

Final Answer:
the flow of electromagnetic energy is zero