Rectangular waveguides, as opposed to circular and elliptical waveguides, are by far the dominant configuration for the installed base of waveguides for compact systems like radar and inside equipment shelters. That is probably due to the generally greater rigidity of rectangular structures because the wall thickness can be easily made thicker than with circular. It is also easier to route and mount in close quarters, and attaching penetrating objects like probes and switches is much simpler.

Most rectangular waveguide calculations can be performed on any calculator that has trig function keys. Calculations for circular waveguide, on the other hand, requires the application of Bessel functions, so working equations with a cheap calculator is not going to happen. However, even spreadsheets have Bessel function capability nowadays, so determining cutoff frequencies, field strengths, and any of the other standard values associated with circular waveguide can be done relatively easily. The formulas below represent those quantities most commonly needed for rectangular waveguides. Please see the figure at the right for variable references.

Note: I received the following note from Brian Sequeira, of the Johns Hopkins University Applied Physics Laboratory. "I reviewed tables on rectangular and circular waveguides, and based on my experience of what confuses first-time readers and what does not, I made adjustments to notation & symbols, corrected a couple of sign errors, and put expressions in a form that make their units more apparent." The table for rectangular waveguide can be viewed full-size by clicking on the thumbnail to the right. Brian also provided a table for circular waveguide.
 

Quantity	TE Modes	TM Modes
Hz		0
Ez	0
Ex
Ey
Hx
Hy
Zh,nm
Ze,nm
kc,nm
βnm
λc,nm
α†
†

The expression for α is not valid for degenerate modes.
Equations derived from "Foundations for Microwave Engineering, R.E. Collin, McGraw-Hill