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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Rectangular Waveguide Cutoff Frequency 
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):
(Hz)
(m)

Rectangular Waveguide TE_{m,n} Mode
Cutoff Frequency Calculator
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).

where 
a = b = m = n = ε
= µ = 
Inside width (m), longest dimension Inside height (m), shortest dimension Number of ½wavelength
variations of fields in the "a" direction Number of ½wavelength variations of fields in the "b" direction
Permittivity (8.854187817E12 for free space) Permeability
(4πE7 for free
space) 

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.
End View (TE_{10})
Side View (TE_{10})
Top View (TE_{10})
____ Electric field lines p
_ _ _ Magnetic 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.
End View (TM_{11})
Side View (TM_{11})
____ Electric field lines
_ _ _ Magnetic 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:
(m), where p'_{mn} is
0 
3.832 
7.016 
10.174 
1 
1.841 
5.331 
8.536 
2 
3.054 
6.706 
9.970 

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
0 
2.405 
5.520 
8.654 
1 
3.832 
7.016 
10.174 
2 
5.135 
8.417 
11.620 

