Waveguide modes

Explore TE modes of an ideal vacuum-filled rectangular guide with perfectly conducting walls. Enter a=1–5 cm, b=0.5–3 cm, frequency 1–20 GHz and integer indices m,n=0–4, excluding TE₀₀. The initial 2.286×1.016 cm dimensions are the WR-90 example and are retained exactly. Select a standard mode or enter both indices. Calculate draws a static field; Play animates it. After the first calculation, valid parameter or mode edits update immediately, paused at phase zero; invalid entries clear old output. Reset restores WR-90, TE10, 10 GHz and the default display. Only exact inputs and colour/arrow choices are retained after reloading; fields and playback are regenerated. Field colour, arrows and phase inspect the same calculated mode.

With c=299792458 m/s and dimensions in metres, f꜀=(c/2)√((m/a)²+(n/b)²). Above cutoff, β=(2πf/c)√(1−(f꜀/f)²), λg=2π/β, vg/c=√(1−(f꜀/f)²) and vp/c=1/(vg/c). The phase velocity above c is not a signal velocity. Below cutoff α=(2π/c)√(f꜀²−f²), with amplitude e⁻ᵅᶻ and 1/e decay distance 1/α; no real guided wavelength or group velocity is assigned. At cutoff β=α=0, λg→∞ and vg/c→0 are limiting values, not a propagating power-carrying mode. A relative neighbourhood of eight binary64 epsilon values is treated as this numerical boundary.

The transverse field follows H_z∝cos(mπx/a)cos(nπy/b): Eₓ∝(n/b)cos(mπx/a)sin(nπy/b), Eᵧ∝−(m/a)sin(mπx/a)cos(nπy/b), E_z=0. Both components share a normalization by max(m/a,n/b), giving peak spatial magnitude 1. Tangential electric field vanishes at the conductor walls. TE₀₁ therefore has horizontal Eₓ, TE₁₀ vertical Eᵧ. Colours show the selected normalized component or magnitude; arrows always show the full transverse vector. Red is negative and blue positive for signed components. The physical a/b aspect ratio is preserved.

The lower graph is a representative normalized component E/E₀ along z: cos(φ−βz) over two guided wavelengths above cutoff; e⁻ᵅᶻcosφ over four decay lengths below it; uniform cosφ at the cutoff limit. The cross-section is at z=0. Each display cycle lasts two seconds in all modes; it does not reproduce GHz time or measured power. The phase control (degrees) pauses playback. Hidden or inactive views pause without catching up. No conductor/dielectric losses, material filling, source coupling, multimode excitation or TM modes are simulated.

Ellingson, Electromagnetics II: TE modes · Propagation characteristics