The throat area, mouth area and length all play an important role too. A horn has to be long enough to provide adequate acoustic load at low frequencies and it has to have enough mouth area to release the wavefront without excessive reflection. This mouth termination is really important, as is the length, especially in a conical horn or waveguide, since loading is compromised at low frequencies.
Just out of curiosity, is there any benefit gained from decreasing waveguide depth?