Waveguide Mode Bench

TE / TM rectangular-waveguide explorer · student lab build v1.0

TE10
propagating

Cross-section · |Et| envelope + instantaneous E arrows

Read: heatmap = |Et| envelope (cividis, colorblind-safe) · white arrows = instantaneous E (they breathe when Motion is on) · thin lines = |E| level sets · dashed amber = E-field nulls (nodes), dashed circles = isolated null points · hatched border = PEC wall · dimensions to scale.

Side view · propagation along z

Mode fingerprint · Hz (TE) / Ez (TM)

Controls

Mode indices
1
0

Guide dimensions (mm)
22.9
10.2
Operating frequency f₀ (GHz)
10.00

Invalid (m,n) are shown on purpose — the bench explains why they cannot exist.

Readouts

Mode chart at f₀ m,n ≤ 5 · click to load

Guided challenge · predict → observe → explain

1 · Baseline TE₁₀ — predict the pattern

Set TE, m=1, n=0, WR-90, f₀ = 10 GHz. Before studying the canvas: which way do the arrows point? Where is |E| biggest, and where must it be exactly zero?

2 · m → 2 (TE₂₀) — count the nodes

Predict first: how many interior node lines appear? What do the arrows do at x = a/2? What happens to f_c?

3 · n → 1 (TE₂₁) — add height variation

Does the field still vary only across the width? Which new arrow component appears? Watch the fingerprint panel too.

4 · TE₀₀ — break it on purpose

Set m = 0 and n = 0 in TE. Why is there literally no field? Why can't it quietly become a TEM wave?

5 · Switch to TM — who is allowed?

Try TM₁₀, TM₀₁, TM₀₀ (watch the inspector), then TM₁₁. What do m ≥ 1 and n ≥ 1 guarantee about E_z at the walls?

6 · TE₁₁ vs TM₁₁ — degenerate twins

Load both from the chart (same f_c!). Compare arrow topology: swirl vs radial. Same cutoff, similar null locations, completely different directions — why?

7 · Find the dominant mode

With a > b, which chart row has the lowest f_c? Now set a = 10 mm, b = 22.9 mm — who becomes dominant? And with a = b?

Name that mode 0/0 · streak 0

Press “New pattern”, count the half-waves, guess.

Physics notes & formulas

fc = (c/2)·√((m/a)² + (n/b)²)  ·  λc = 2/√((m/a)² + (n/b)²)  ·  kc = 2π/λc
f₀ > fc (propagating): β = k₀√(1−(fc/f₀)²), λg = 2π/β, ZTE = η₀/√(1−(fc/f₀)²), ZTM = η₀·√(1−(fc/f₀)²)
f₀ < fc (evanescent): α = kc√(1−(f₀/fc)²), fields decay as e−αz — no power is delivered.

TE modes (Ez = 0), m,n ≥ 0, not both 0 (phasors ejωt−jβz, signs follow one common convention — only relative structure matters for the pattern):

HzH₀ cos(mπx/a) cos(nπy/b)
Ex(jωμ/kc²)(nπ/b)H₀ cos(mπx/a) sin(nπy/b)
Ey−(jωμ/kc²)(mπ/a)H₀ sin(mπx/a) cos(nπy/b)
Hx(jβ/kc²)(mπ/a)H₀ sin(mπx/a) cos(nπy/b)
Hy(jβ/kc²)(nπ/b)H₀ cos(mπx/a) sin(nπy/b)

The transverse E is proportional to ẑ × ∇tHz: it curls around the Hz extrema (swirl pattern) and vanishes wherever Hz is locally flat.

TM modes (Hz = 0), only m ≥ 1 and n ≥ 1:

EzE₀ sin(mπx/a) sin(nπy/b)
Ex−(jβ/kc²)(mπ/a)E₀ cos(mπx/a) sin(nπy/b)
Ey−(jβ/kc²)(nπ/b)E₀ sin(mπx/a) cos(nπy/b)
Hx(jωε/kc²)(nπ/b)E₀ sin(mπx/a) cos(nπy/b)
Hy−(jωε/kc²)(mπ/a)E₀ cos(mπx/a) sin(nπy/b)

Et ∝ ∇tEz: the arrows look radial, meeting the Ez cells head-on. Ez must be zero on all four walls while still varying inside — that is exactly why m,n ≥ 1.

Usage · README · self-check rubric

Run: open this single HTML file in a browser. Controls: family toggle → m, n sliders (0–5) → a, b sliders or a dimension preset → f₀ slider. Mode presets and the chart rows load modes instantly. Keyboard: focus any slider and use ←→; Tab reaches every control; chart rows are buttons.