Waveguide dispersion: Choose the correct relation between guide wavelength (λg), free-space wavelength (λ0), and cutoff wavelength (λc) for a propagating mode in an air-filled waveguide.

Introduction:
Inside a hollow waveguide, the phase constant differs from that in free space due to boundary conditions. This leads to a guide wavelength λg that depends on the operating wavelength λ0 and the mode's cutoff wavelength λc. Correctly relating these three quantities is fundamental for component design and measurement.

Given Data / Assumptions:

• Air-filled, perfectly conducting waveguide.
• Single propagating mode with λ0 < λc (i.e., f > fc).
• Standard dispersion relation applies.

Concept / Approach:

The axial phase constant is β = (2π/λ0) * sqrt(1 − (λ0/λc)^2). Since λg = 2π/β, direct substitution yields λg = λ0 / sqrt(1 − (λ0/λc)^2). This expression shows two useful limits: as λ0 → λc (approach cutoff), λg → ∞; as λ0 ≪ λc (well above cutoff), λg → λ0.

Step-by-Step Solution:

Verification / Alternative check:

Many handbooks also present an equivalent reciprocal-squared form: 1/λg^2 = 1/λ0^2 − 1/λc^2. Care must be taken with the signs; a mistaken reversal produces non-physical results. The option provided here (λg = λ0 / sqrt(1 − (λ0/λc)^2)) is the most direct expression for λg.

Why Other Options Are Wrong:

• a: sign order is incorrect (gives imaginary λg or wrong trends).
• b and d: place the square-root factor incorrectly, yielding λg smaller than λ0 near cutoff, which is unphysical.
• e: simple sum has no basis in waveguide theory.

Common Pitfalls:

Forgetting that λg ≥ λ0 for a propagating mode and that λg diverges at cutoff; the correct formula must reflect these behaviors.

Final Answer:

λg = λ0 / sqrt(1 − (λ0/λc)^2)