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The following equations and images describe electromagnetic waves inside both rectangular waveguide and circular (round) waveguides. Oval waveguide equations are not included due to the mathematical complexity.
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The lower cutoff frequency (or wavelength) for a particular mode in rectangular waveguide is determined by the following equations (note that the length, x, has no bearing on the cutoff frequency):

Rectangular Waveguide TE_{m,n} Mode This example is for TE_{1,0} (the mode with the lowest cutoff frequency) in WR284 waveguide (commonly used for Sband radar systems). It has a width of 2.840" (7.214 cm) and a height of 1.340"(3.404 cm).




TE (Transverse Electric) Mode The TE_{10} mode is the dominant mode of a rectangular waveguide with a>b, since it has the lowest attenuation of all modes. Either m or n can be zero, but not both.
____ Electric field lines 
TM (Transverse Magnetic) Mode For TM modes, m=0 and n=0 are not possible, thus, TM_{11} is the lowest possible TM mode.
____ Electric field lines 
TE (Transverse Electric) Mode  
The lower cutoff frequency (or wavelength) for a particular TE mode in circular waveguide is determined by the following equation: , where p'_{mn} is


TM (Transverse Magnetic) Mode  
The lower cutoff frequency (or wavelength) for a particular TM mode in circular waveguide is determined by the following equation: (m), where p_{mn} is

Related Pages on RF Cafe
 Properties of Modes in a Rectangular Waveguide
 Properties of Modes in a Circular Waveguide
 Waveguide & Flange Selection Guide

Rectangular & Circular Waveguide: Equations & Fields

Rectangular waveguide TE_{1,0} cutoff frequency calculator.
 Waveguide Component
Vendors
 Waveguide Design Resources

NEETS  Waveguide Theory and Application
 EWHBK, Microwave Waveguide
and Coaxial Cable