Visualize TE/TM mode field distributions in rectangular waveguides in real time. Auto-calculate cutoff frequency, guide wavelength, phase velocity, group velocity, and wave impedance. Dispersion diagram included.
Formulas and key points
Quick answer A rectangular waveguide's cutoff frequency is f_c = (c/2)·√((m/a)² + (n/b)²) (a, b = broad and narrow wall dimensions, m and n = mode indices), and the dominant mode is TE10 for a>b, TE01 for a<b, and both are degenerate for a=b. Above cutoff the guide wavelength is λ_g = λ/√(1 − (f_c/f)²) and the group velocity v_g = c·√(1 − (f_c/f)²); below f_c the field decays evanescently.
What are Waveguide Modes?
Explanation in dialogue
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What exactly is a "mode" in a waveguide? I see TE10 and TM11 in the simulator, but what do those letters and numbers mean?
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Basically, a mode is a specific pattern of electromagnetic waves that can travel through the hollow metal pipe. TE stands for "Transverse Electric," meaning the electric field is entirely perpendicular to the direction of wave travel. The numbers, like 10, are the mode indices m and n. They describe how many half-wavelengths of the field pattern fit across the waveguide's width (a) and height (b). Try selecting TE10 in the simulator and watch how the electric field arrows form a single half-sine wave across the width.
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Wait, really? So for TE10, m=1 and n=0. Does n=0 mean there's no variation in the field along the height? And what's this "cutoff frequency" that keeps coming up?
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Exactly right! For TE10, the field is uniform along the height (b). The cutoff frequency is the most critical concept here. It's the minimum frequency for that specific mode to propagate. Below it, the wave just decays exponentially—it can't travel. The formula depends directly on m, n, and the waveguide's dimensions. In practice, this is why waveguides act as high-pass filters. Try it yourself: set the operating frequency f just below the calculated fc for TE10. You'll see the guide wavelength and velocities go to imaginary values, indicating no propagation.
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That makes sense. So if I design a system, I want to operate in the TE10 mode and avoid other modes. How do I choose the right waveguide size? And what are group and phase velocity for?
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Great questions. You choose dimensions so that only your desired mode propagates within your frequency band. For instance, for a standard WR-90 waveguide (a=22.86 mm, b=10.16 mm), TE10 cuts off at 6.56 GHz. You'd operate well above that but below the cutoff of the next mode (TE20 at 13.1 GHz) to stay single-mode. The velocities tell you about signal and energy travel. Phase velocity ($v_p$) is the speed of wave crests and is always \gt c (light speed)! Group velocity ($v_g$) is the speed of energy/information and is always \lt c. In the simulator, slide the frequency up and see $v_p$ decrease and $v_g$ increase, both approaching c as $f \gt \gt f_c$.
Physical Model & Key Equations
The fundamental governing equation is the cutoff frequency. It determines which modes can exist and is derived from solving Maxwell's equations with the boundary conditions of perfectly conducting walls.
$f_c$: Cutoff frequency (Hz) $c$: Speed of light in the medium (usually air/vacuum, ~3e8 m/s) $m, n$: Mode indices (non-negative integers, both not zero for TM modes) $a, b$: Width and height of the waveguide (in meters for the formula)
Once a mode is above cutoff ($f \gt f_c$), its propagation is characterized by the guide wavelength and velocities. These depend on the ratio of the operating frequency to the cutoff frequency.
$\lambda_g$: Guide wavelength (wavelength inside the waveguide, > free-space wavelength) $v_p$: Phase velocity (speed of constant phase, > c) $v_g$: Group velocity (speed of energy/pulses, < c) $\lambda_0$: Free-space wavelength ($c/f$)
Real-World Applications
Radar and Satellite Communications: Rectangular waveguides are the backbone for connecting high-power radar transmitters to antennas and in satellite ground stations. The TE10 mode is almost exclusively used because of its simple field pattern, low loss, and ability to handle high power. For instance, the feed network for a large satellite dish uses precisely machined waveguide runs.
Particle Accelerators: In facilities like the Large Hadron Collider, waveguides are used to transfer microwave power (at frequencies like 400 MHz or several GHz) to accelerate particle beams. The precise calculation of modes and cutoff frequencies is critical to ensure efficient power transfer without exciting unwanted modes that could cause power loss or heating.
Microwave Ovens: The magnetron tube, which generates the 2.45 GHz microwaves, feeds its power into the oven cavity through a waveguide. The waveguide ensures the RF energy is efficiently transferred from the magnetron to the cooking chamber, which itself is a resonant cavity operating in a mix of modes to heat food evenly.
Medical and Industrial Heating: Systems for hyperthermia cancer treatment or for drying/curing materials in manufacturing often use waveguide-based applicators to focus microwave energy onto a target. Engineers must design the waveguide and applicator to support the correct mode for efficient and controlled energy delivery.
Common Misconceptions and Points to Note
First, do you think "a waveguide is just a metal tube"? In reality, the inner surface's conductivity and roughness directly impact loss. While simulators assume ideal conditions, actual waveguides use copper or silver plating, and at high frequencies especially, the surface is finished like a mirror. For example, a 10% drop in conductivity at 10GHz could increase transmission loss by several percent.
Next, selecting the operating frequency. It's not simply "anything above the cutoff frequency is OK." You must also consider the frequency at which higher-order modes beyond the fundamental mode (TE10) begin to propagate. For instance, in WR-90 (a=22.86mm, b=10.16mm), the cutoff frequency fc for the TE10 mode is about 6.56GHz, but the fc for the next TE20 mode is about 13.1GHz. In practice, to ensure stable single-mode transmission, an operating band around 1.25×fc to 1.9×fc is often used as a guideline. Practice using the simulator: switch to m=2, n=0 to check its fc and find the "safe zone" free from mode interference.
Finally, the difference between "nominal" and "effective" dimensions. The dimensions a and b discussed are the "inner dimensions." However, especially with corner radii or discontinuities at flange connections, the effective electrical dimensions change slightly. For high-precision designs like filters or resonators, you'll need to use simulation to compensate for this shift.
What is Rectangular Waveguide Mode Calculator?
Rectangular Waveguide Mode Calculator is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Enter the internal width a and height b in mm, and operating frequency f in GHz. The usual broad/narrow-wall naming assumes a≥b.
Choose TE or TM, then one of its mode buttons. Results update automatically; there is no Calculate button.
Compare cutoff, guide wavelength, phase/group velocity and wave impedance for the selected mode.
Worked Example
Choose “Fundamental TE10 Mode” for WR-90 (a=22.86 mm, b=10.16 mm, f=10 GHz). The results are f_c=6.562 GHz, λ_g=39.76 mm, v_p/c=1.325, v_g/c=0.755 and Z_TE=499.6 Ω, using c≈3.00×10⁸ m/s and free-space impedance≈377 Ω. The next cutoff is TE20 at 13.123 GHz (TE01 is 14.764 GHz), so only TE10 can propagate at 10 GHz.
Model Limits and Notes
This is an ideal hollow air/vacuum guide with perfectly conducting walls. Metal loss, roughness, connectors and dielectric filling are not calculated.
Determine the single-mode range from dimensions and the next mode cutoff. A fixed multiple of the dominant cutoff is not a general guarantee.
Phase velocity above c does not imply faster-than-light information. In this model v_p×v_g=c².
The field, side and 3D views illustrate the principle. Animation speed and schematic depth are not measured physical time or length.