---
title: "Modes in rectangular waveguide"
subject: "Electromagnetism"
chapter: 6
key: "semester_3/electromagnetism:modes-supported-rectangular-waveguide"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/modes-supported-rectangular-waveguide
site: BCT Engineering
---

# Modes in rectangular waveguide

Electromagnetism · Chapter 6: Wave guides

- Asked in: 2073 Shrawan, 2 marks; 2069 Chaitra, 2 marks; 2068 Chaitra, 4 marks; 2070 Chaitra, 3 marks (asked 4 times)
- Video: [Modes in Rectangular Waveguide (TE Mode, TM Mode, Dominant Mode, Evan Scant Mode, Degenerative Mode)](https://www.youtube.com/watch?v=yW8QUzt78tI) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/modes-supported-rectangular-waveguide
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-6

## Question

Explain the modes supported by a rectangular waveguide.

## Short answer

A rectangular waveguide supports TE_mn (E_z = 0, m,n not both zero) and TM_mn (H_z = 0, m,n ≥ 1) modes. TEM cannot propagate. Each mode has cut-off frequency f_c = (c/2)√((m/a)²+(n/b)²). The dominant mode TE_10 has the lowest cut-off c/(2a).

## Answer

A hollow rectangular waveguide of cross-section a × b (a > b) supports a set of discrete field patterns called modes. Each is labelled by integers m and n, the number of half-wave variations of the field along x (width a) and y (height b). Waves travel along z.

#### TE modes (transverse electric)

- E_z = 0; the electric field has no component along the propagation direction.
- H_z = H₀ cos(mπx/a) cos(nπy/b) e^{−jβz}
- m = 0, 1, 2, ...; n = 0, 1, 2, ...; but m and n are not both zero.
- Lowest: TE₁₀. Wave impedance Z_TE = η/√(1 − (f_c/f)²).

#### TM modes (transverse magnetic)

- H_z = 0.
- E_z = E₀ sin(mπx/a) sin(nπy/b) e^{−jβz}
- m = 1, 2, ...; n = 1, 2, ... (both must be non-zero, otherwise all fields vanish).
- Lowest: TM₁₁. Wave impedance Z_TM = η√(1 − (f_c/f)²).

#### Cut-off frequency

A mode propagates only when f > f_c:

```text
f_c = (c/2) √[(m/a)² + (n/b)²]
```

Below f_c the wave is evanescent (it attenuates rapidly).

#### TEM mode

A hollow guide has a single conductor, so it cannot support a TEM wave (E_z = H_z = 0): a TEM wave would need two conductors.

#### Dominant mode

TE₁₀ has the lowest cut-off frequency f_c = c/2a and is the dominant mode. A guide is normally used in the band f_c(TE₁₀) < f < the next cut-off (TE₂₀ or TE₀₁), where only one mode exists.

#### Degenerate modes

TE_mn and TM_mn (m, n ≥ 1) have the same cut-off frequency and are called degenerate modes.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why must m and n both be at least 1 for TM modes?
   A. E_z = E0 sin(m pi x/a) sin(n pi y/b), so if either is zero all the fields vanish
   B. Because H_z must be non-zero
   C. Because TM modes need two conductors
   D. Because cut-off is zero otherwise
   Answer: A. A zero index makes E_z, and hence all components, vanish. (Key point: TM mode indices)

2. Why can a hollow rectangular waveguide not support TEM?
   A. Because its walls are too thin
   B. Because TEM has cut-off
   C. Because a > b
   D. It has a single conductor, and TEM needs two
   Answer: D. TEM waves need two conductors. (Key point: TEM)

3. Which is the dominant mode?
   A. TM11
   B. TE01 with f_c = c/2b
   C. TE10, with the lowest cut-off f_c = c/2a
   D. TE20
   Answer: C. TE10 has the lowest cut-off and TM11 is the lowest TM mode. (Key point: Dominant mode)

4. What happens when f < f_c?
   A. It propagates with higher velocity
   B. The wave is evanescent and attenuates rapidly
   C. It converts to TEM
   D. It reflects with Gamma = 0
   Answer: B. A mode propagates only above its cut-off. (Key point: Below cut-off)

## Related in Electromagnetism

- [Why no TEM mode; first four modes](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-no-tem-four-modes) — Electromagnetism, chapter 6
- [First four modes, 2.5×1.2 cm guide](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Afirst-four-modes-air-copper-waveguide) — Electromagnetism, chapter 6
- [Short note: TE and TM modes](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Ate-tm-modes-rectangular-waveguide) — Electromagnetism, chapter 6, asked 2×

## Related in other subjects

- [Rectangular waveguide modes and critical parameters](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-modes-parameters) — RF and Microwave Engineering, chapter 4
- [Circular waveguide TE equations and degenerate modes](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Acircular-te-rectangular-modes) — RF and Microwave Engineering, chapter 4
- [Rectangular waveguide TM mode field equations](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-tm-equations) — RF and Microwave Engineering, chapter 4, asked 11×
- [Rectangular waveguide TE mode field equations](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-te-equations) — RF and Microwave Engineering, chapter 4, asked 11×