I want to design a horn antenna to detect the 21 cm hydrogen line. I know how to optimize the antenna dimensions by taking into account some functions that approximate the phase errors and efficiencies. (The method I'm following is proposed by Leandro de Paula Santos Pereira and Marco Antonio Brasil Terada in New Method for Optimum Design of Pyramidal Horn Antennas). They end up with a slightly different horn antenna equation (which is a 4th order polynomial) which has to be numerically solved to get the value of the aperture width B. However, that equation contains the waveguide dimensions a and b, which are assumed to be known.
That leads me to my question: how exactly do I find out what these waveguide dimensions are? Looking around on the internet, I found out about how rectangular waveguides work, but all those formulae are based on the fact that the waveguide width is known. Nowhere is the waveguide height taken into account, and I understand why it isn't, but there's a huge difference between not taking it into account in theory and actually making the waveguide section for the horn antenna.
Can't I just go "Since I want to detect 1.42 GHz, I'll make sure that my cutoff frequency for the waveguide is some x less than 1.42GHz, and I'll make sure that 1.42GHz falls in the middle of the 'operating range' (the average of 125% and 189% - as a side note, why are these the given range of operating frequencies for a rectangular waveguide? It seems a bit arbitrary...). I can find out the cutoff frequency by dividing 1.42GHz by 1.57, and I know that the TE10 is the dominant mode in the waveguide, so I can calculate the waveguide width." How do I calculate the height? If I divide the width by 2, why? Is there any text that explains this? Is there some other way to calculate the optimal height and waveguide thickness?
Also, the folks over at microwavetools.com have the dimensions I'm looking for as a fuction of strictly the required wavelength, and their values are close to the waveguide in horn antennas that others built - but why do their dimensions work? Why do they have seemingly arbitrary multipliers for the wavelength?
Any help would be appreciated, maybe a link to a textbook chapter or a research paper that explains how to calculate waveguide dimensions in detail, as I'm at my wits' end trying to make sense of this. As a side note, since I'm inserting a pin into the waveguide to excite the horn antenna, how long should I make the waveguide? I've heard values from 3\$\lambda\$ (somewhere around 62 centimeters) to 'the length is arbitrary'....