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. Click here for a transmission lines & waveguide presentation. A Cavity Resonance Calculator is included in RF Cafe Calculator Workbook for FREE. Rectangular Waveguide Cutoff FrequencyThe 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):  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) 
 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  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 
 The lower cutoff frequency (or wavelength) for a particular TE mode in circular waveguide is determined by the following equation: , 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 
 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 

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