It's pretty complicated, and to some degree might include as much "art" as "science."
For a flared tube the effective tube length varies with frequency. Thus we are not dealing with the usual physics of a cylindrical tube closed at one end and open at the other. If we were dealing with the simple tube, then the harmonic series would be based upon an odd number of quarter wavelengths in the tube. But flared brass instruments, by letting the flare automatically change the effective length of the tube, create a new effective harmonic series that enables the musician to make the instrument play frequencies characteristic of both odd and even harmonics (except for the first harmonic).
This means the flare is not any old flare. The flare has to be "tuned" so that the effective length changes in the correct way to allow the musician to play an in-tune harmonic series.
This means we can't calculate the playable frequencies simply by basing a harmonic series on a tube of a certain length. The rate of flare is just as important as the physical length of the tube.
Not having built any brass instruments myself, I am just guessing that early instrument manufactures had to experimentally determine the correct flare to make an instrument that could be played in tune. At instrument manufacturing/design companies today, I do not know if the proper flare is theoretically calculated, or if it is experimentally determined. Does anyone on this list know?
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu