You try to send a microwave signal through a rectangular metal waveguide. The waveguide has a width of 3 cm. You try two frequencies: 4 GHz and 6 GHz.
What happens?
Only 6 GHz propagates! The waveguide’s cutoff frequency is f_c = c/(2a) = 3×10⁸/(2×0.03) = 5 GHz. The 6 GHz signal is above cutoff and propagates. The 4 GHz signal is below cutoff — it can’t fit half a wavelength across the 3 cm width, so it decays exponentially inside the guide. A waveguide acts as a high-pass filter!
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Sweep the input frequency and watch waves inside the waveguide. Below cutoff, the wave amplitude decays exponentially from the entrance. Above cutoff, standing wave patterns propagate through. Find the exact cutoff frequency where the transition occurs.
f_c = c/(2a) (TE₁₀ mode)
The waveguide acts as a high-pass filter. Below cutoff, waves decay exponentially. Above cutoff, they propagate but dispersively — the phase velocity exceeds c (no violation of relativity since group velocity < c).
Radar transmitters use rectangular waveguides to carry high-power microwave signals from the magnetron to the antenna dish. The waveguide dimensions are matched to the radar’s operating frequency.
Satellite ground stations use waveguides to connect receivers to the antenna feed horn. Different frequency bands (C, Ku, Ka) require different waveguide sizes.
Linear accelerators use waveguides to deliver precisely timed microwave pulses that accelerate charged particles. The waveguide mode’s phase velocity is tuned to match the particle speed.
The microwave oven cavity is essentially a large waveguide/resonator. The metal walls confine 2.45 GHz microwaves. The mesh on the door has holes smaller than the microwave wavelength, acting as a waveguide below cutoff — blocking the microwaves!
“A waveguide tells electromagnetic waves: ‘You must be at least this tall to ride.’ Below the cutoff frequency, entry is denied.”
Waveguide Lab
Explore EM wave propagation in rectangular waveguides. Adjust guide width and input frequency. Visualize field patterns for different modes. See evanescent decay below cutoff.
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Start with a wide waveguide and low frequency. The wave decays exponentially inside — measure how quickly. Now increase the frequency past cutoff and watch the wave suddenly propagate! Try narrowing the guide to raise the cutoff frequency. Explore higher-order modes (TE₂₀, TE₁₁) by increasing frequency further — they have more complex field patterns across the cross-section.
Waveguides enforce boundary conditions that create a cutoff: waves must fit at least half a wavelength across the guide. This transforms free-space waves into guided modes with dispersion, phase velocities above c, and a frequency threshold for propagation.