Circular waveguides offer implementation advantages over rectangular waveguide in that installation is much simpler when forming runs for turns and offsets  particularly when large radii are involved  and the wind loading is less on a round crosssection, meaning towers do not need to be as robust. Manufacturing is generally simpler, too, since only one dimension  the radius  needs to be maintained. Applications where differential rotation is required, like a rotary joint for a radar antenna, absolutely require a circular crosssection, so even if rectangular waveguide is used for the primary routing, a transition to circular  and then possibly back to rectangular  is needed.
Calculations for circular waveguide requires the application of Bessel functions, so working equations with a cheap calculator is not going to happen. However, even spreadsheets have Bessel function (J_{n}) 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 circular 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 firsttime 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 circular waveguide can be viewed fullsize by clicking on the thumbnail to the right. Brian also provided a table for rectangular waveguide.
Quantity  TE Modes  TM Modes 
H_{z}  0  
E_{z}  0  
H_{r}  
H_{ϕ}  
E_{r}  
E_{ϕ}  
β_{nm}  
Z_{h,nm}  
Z_{e,nm}  
k_{c,nm}  
λ_{c,nm}  
Power^{††}  
α^{†}  
† ^{ }  
The expression for α is not valid for degenerate modes.  
Equations derived from "Foundations for Microwave Engineering, R.E. Collin, McGrawHill  
†† Thanks to Patrick L. for finding error where "4" in denominator should be "2." 
n  p_{n1}  p_{n2}  p_{n3} 
0  2.405  5.520  8.654 
1  3.832  7.016  10.174 
2  5.135  8.417  11.620 
n  p'_{n1}  p'_{n2}  p'_{n3} 
0  3.832  7.016  10.174 
1  1.841  5.331  8.536 
2  3.054  6.706  9.970 
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