Phase Velocity in a Rectangular Waveguide (General Medium) Let μr be the relative permeability and εr the relative permittivity of the filling medium. Let fc be the cutoff frequency of the chosen mode. The phase velocity vp (speed of a constant-phase point) in a rectangular waveguide at operating frequency f is given by which expression?

Introduction / Context:
Unlike TEM transmission lines, waveguides support TE/TM modes with dispersive propagation. The phase velocity exceeds the medium wave speed as frequency approaches cutoff, a key concept in microwave engineering.

Given Data / Assumptions:

• Uniform rectangular waveguide supporting a specific mode with cutoff fc.
• Filling medium characterized by μr and εr (lossless for this analysis).
• Operating frequency f > fc.

Concept / Approach:

The phase velocity in a waveguide is vp = ω / β. For TE/TM modes, β = k0 * sqrt(μr * εr) * sqrt(1 - (fc / f)^2), where k0 = ω / c. Therefore vp equals the medium wave speed divided by sqrt(1 - (fc/f)^2). Since the square root in the denominator is less than 1 above cutoff, vp exceeds the medium speed.

Step-by-Step Solution:

Verification / Alternative check:

Near cutoff (f → fc+), the denominator → 0+, so vp → ∞, consistent with dispersion theory. At high f ≫ fc, vp approaches the medium speed.

Why Other Options Are Wrong:

Option B inverts the dispersion; C and D misuse μrεr factors; E is dimensionally incorrect and ignores μr, εr.

Common Pitfalls:

Confusing phase velocity with group velocity; assuming vp cannot exceed c (it can; energy is carried at group velocity, not phase velocity).

Final Answer:

vp = (c / sqrt(μr * εr)) / sqrt(1 - (fc / f)^2)