Why does TE₁₀ depend only on the broad dimension a?

For **TE_mn**, **k_c² = (mπ/a)² + (nπ/b)²**. The **lowest** cutoff is **m = 1, n = 0**, giving **k_c = π/a**—there is no **y** variation for **n = 0**, so **E_y** is uniform across the **b** dimension in this ideal mode.

What do ⊙ and ⊗ mean in the cross-section?

The figure looks along **+z** (out of the screen). **H_z** points along **z**, so positive **H_z** is drawn as **⊙** (toward you) and negative as **⊗** (into the page), matching a common textbook arrow convention.

Why does the bottom panel change above vs below cutoff?

It colours **|E_y|** on an **x–z** cut at **y = b/2**. Above cutoff the pattern travels as **sin(ωt − βz)** (horizontal dashed lines mark one **λ_g** in **z**). Below cutoff the same time harmonic multiplies **e^{−αz}**, simulating decay away from a source.

Are WR presets exact measured dimensions?

They are **nominal** interior sizes for teaching. Real waveguides have tolerances, losses, and excitation details not modelled here.

Rectangular Waveguide TE₁₀

Dominant mode fields (E_y, H_x, H_z), cutoff f_c = c/(2a√ε_r), β and guide wavelength λ_g.

About this model

A lossless rectangular waveguide with inner broad wall a, height b, filled with a homogeneous isotropic medium ε_r (relative permittivity), supports TE and TM modes. The dominant mode is TE₁₀, with cutoff wavenumber k_c = π/a and cutoff frequency f_c = c/(2a√ε_r) in the usual convention where the wide dimension is a. For operating angular frequency ω and wavenumber k = nω/c with n = √ε_r, the mode propagates when k > k_c, giving β = √(k² − k_c²) and guide wavelength λ_g = 2π/β. When k < k_c, the longitudinal dependence is evanescent with α = √(k_c² − k²). The simulator sketches the standard field dependencies (time-harmonic, +z propagation): E_y ∝ sin(πx/a) sin(ωt − βz), H_x with the same transverse factor for real power flow along z, and H_z ∝ cos(πx/a) cos(ωt − βz), omitting wall losses, coupling, higher modes, and frequency dispersion of ε_r. Presets approximate common WR band dimensions in millimetres for quick exploration; the physics uses SI formulas with c in vacuum.

Who it's for: Undergraduates in electromagnetics, microwave engineering, or applied physics who are learning hollow-waveguide modes beyond TEM on two-conductor lines.

Key terms

Rectangular waveguide
TE₁₀ mode
Cutoff frequency
Phase constant β
Guide wavelength λ_g
Evanescent decay α
Dominant mode

How it works

Dominant TE₁₀ mode in a hollow rectangular waveguide: transverse E and H patterns, cutoff frequency, and guide wavelength when propagating.

Frequently asked questions

Why does TE₁₀ depend only on the broad dimension a?: For TE_mn, k_c² = (mπ/a)² + (nπ/b)². The lowest cutoff is m = 1, n = 0, giving k_c = π/a—there is no y variation for n = 0, so E_y is uniform across the b dimension in this ideal mode.
What do ⊙ and ⊗ mean in the cross-section?: The figure looks along +z (out of the screen). H_z points along z, so positive H_z is drawn as ⊙ (toward you) and negative as ⊗ (into the page), matching a common textbook arrow convention.
Why does the bottom panel change above vs below cutoff?: It colours |E_y| on an x–z cut at y = b/2. Above cutoff the pattern travels as sin(ωt − βz) (horizontal dashed lines mark one λ_g in z). Below cutoff the same time harmonic multiplies e^{−αz}, simulating decay away from a source.
Are WR presets exact measured dimensions?: They are nominal interior sizes for teaching. Real waveguides have tolerances, losses, and excitation details not modelled here.

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About this model

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Frequently asked questions