Waveguide questions

I'm a beginner in μW. And, after a lot of reading I've some questions unsolved:

1 - For a rectangular waveguide a≥b, the cutoff frecuency equation is the same for TE & TM modes, right? Why is allways stated that the dominant mode is TE10? Why not TM10, as fc is the same?

2 - How is the history in circular waveguides? I've found different equations for TE & TM modes. And allways based on a constant, like: fcTE11 = 0,586(c/D), but the constant is never explained. I've done some maths, and looks like some of this constants lighty match Π/4 as the ratio for perimeters between a circle an a square. Is this a coincidence?


Thanks a lot!

For the 1st question:
Ez = B sin(mΠx/a) sin(nΠy/b).

If n = 0 or m = 0, then Ez = 0 ==> which means that all fields will be zeros.
That's why there are no TM10.

i.e: n ≠ 0 and m ≠ 0.

Hope this helps

Refer to POZAR P.126