Questions: Propagation in Rectangular Waveguides

Question 1

A rectangular waveguide has width a = 2 cm and height b = 1 cm. What is the cutoff frequency of the dominant TE₁₀ mode, and what frequency range supports single-mode propagation?

• A. f_c = c/a = 15 GHz; single-mode operation above 15 GHz
• B. f_c = c/(2a) = 7.5 GHz; single-mode operation between 7.5 GHz and 15 GHz
• C. f_c = c/(2b) = 15 GHz; single-mode operation between 15 GHz and 30 GHz
• D. f_c = c/(a + b) ≈ 10 GHz; single-mode operation above 10 GHz

Answer: (B)

The cutoff frequency of the TE_mn mode is f_c = (c/2)√[(m/a)² + (n/b)²]. For TE₁₀ (m=1, n=0): f_c = c/(2a) = (3×10¹⁰ cm/s)/(2×2 cm) = 7.5 GHz. The next lowest cutoff is TE₂₀ at c/a = 15 GHz (or TE₀₁ at c/(2b) = 15 GHz). Single-mode operation — only TE₁₀ propagating — requires operating above 7.5 GHz but below 15 GHz. Below 7.5 GHz, no mode propagates; above 15 GHz, multiple modes coexist.

Question 2

Measurements show that the phase velocity of the TE₁₀ mode in a waveguide slightly exceeds the speed of light c. What does this imply?

• A. The measurement is in error — special relativity forbids any signal from exceeding c
• B. The phase velocity can exceed c near cutoff, but the group velocity (energy velocity) remains less than c, so no information or energy travels faster than light
• C. Both phase and group velocity exceed c in waveguides, since the mode travels in a guided medium
• D. The waveguide is malfunctioning — in normal operation, phase velocity never exceeds c

Answer: (B)

Phase velocity v_p = c/√[1 − (f_c/f)²] exceeds c for all frequencies above cutoff, approaching infinity as f → f_c. This is not a violation of special relativity because phase velocity is not the velocity of energy or information. The group velocity v_g = c√[1 − (f_c/f)²] is always less than c and represents the actual speed of energy transport. The product v_p × v_g = c² — a fundamental waveguide relation. Near cutoff, phase velocity is very large while group velocity is near zero; at very high frequencies, both approach c.

Question 3

A mode whose operating frequency is below its cutoff frequency does not propagate through the waveguide — its fields decay exponentially along the propagation direction.

• T. True
• F. False

Answer: True

Below cutoff, the propagation constant k_z = (2π/λ)√[1 − (f_c/f)²] becomes imaginary, since (f_c/f)² > 1. An imaginary propagation constant means the fields decay as e^(−|k_z|z) — exponential attenuation along the guide rather than oscillatory propagation. These are called evanescent modes. They still exist as near-field patterns near a source or discontinuity, but they carry no net power down the guide. This is why choosing an operating frequency below the dominant mode's cutoff completely prevents propagation.

Question 4

The TM₁₀ mode exists in a rectangular waveguide and has the same cutoff frequency as the TE₁₀ dominant mode.

• T. True
• F. False

Answer: False

The TM₁₀ mode does not exist. TM modes require E_z = sin(mπx/a)·sin(nπy/b). With n = 0, this becomes sin(mπx/a)·sin(0) = 0 identically — no field exists. The same applies to TM_m0 and TM_0n modes: at least one of m and n must be nonzero for TM modes to exist. The lowest-order TM mode is TM₁₁. This is why TE₁₀ is the unique dominant mode — there is no TM counterpart at the same cutoff frequency.

Question 5

Why is single-mode operation desirable in a rectangular waveguide, and how do the dimensions a and b determine the frequency range over which it is achievable?

Model answer: Single-mode operation means only one field pattern (TE₁₀) propagates down the guide. This is desirable because multiple coexisting modes travel at different phase velocities, causing modal dispersion — signal distortion as different modes arrive at different times — and modal interference, which creates unpredictable field patterns. The frequency range for single-mode operation is determined by the gap between the TE₁₀ cutoff (c/2a) and the next mode's cutoff. For a guide with b ≤ a/2, the next mode is TE₂₀ at c/a, giving a usable bandwidth of c/(2a) to c/a (a factor of 2 in frequency). Making a larger lowers the single-mode band to lower frequencies but keeps the factor-of-2 bandwidth. Making b smaller raises TE₀₁ above TE₂₀, maximizing the single-mode range.

The choice of waveguide dimensions is an engineering tradeoff: larger a supports lower frequencies but also allows more modes at higher frequencies. Standard waveguide families (WR-90, WR-62, etc.) are specified with a ≈ 2b to place the TE₀₁ cutoff at the same frequency as TE₂₀, maximizing the single-mode bandwidth to a factor of 2. Real systems operate well within the single-mode band (not close to either edge) to avoid the high attenuation near cutoff and the risk of exciting the next mode due to manufacturing tolerances.