Carl Kleinsteuber on modifying an Eb tuba to F:

http://home.planet.nl/~tubaness/How2.htm
"The speed of sound is 343.5 meters per second, assuming relatively normal barometric pressure and a temperature of 20 degrees centigrade. (Higher temperatures require longer wavelengths, and at lower temperatures wavelengths will be somewhat shorter. As can be imagined, when figuring wavelengths for brass instrument construction, one must take into account the warmth of the player's breath. For a temperature of 28 degrees centigrade as compared to 20 degrees C, one must add an additional 1% to the length of the instrument, corresponding to a revised speed of sound of 347 meters per second.) The equal-tempered frequency of FF (F1, or "pedal F") is 43.654 cycles per second, so the length of the wave that produces pedal F is 7.8687 meters long. However, brass instruments act acoustically as "closed pipes", so only half this length is required to produce this note, thus 3.93435 meters. This is further complicated by the fact that a conical pipe contains more air than a comparable cylindrical pipe; it has greater volume. A physicist would say "The pitch produced by a vibrating medium (air!) depends on its weight per unit length." That's why the tubing of a euphonium is shorter than the tubing of a trombone. Yet another factor to consider when planning the correct length of a brass instrument: the diameter of the bell rim. During playing, the acoustical standing waves produced inside a brass instrument actually extend beyond the end of the bell. This effect varies with frequency and the width of the bell. For almost all common purposes, this effect can be expressed thusly:

effective tube length = actual tube length + (0.6 * bell diameter)

In the case of my F tuba, whose bell diameter is 39 centimeters, this means 23.4 centimeters of tubing must be left out of the body of the horn in order for it to come out in the desired pitch. Not an insignificant figure!

By informed guesswork, I determined I needed 92 centimeters (for the 2nd and 3rd branches), flaring from an interior bore diameter of 20.5 mm to 37 mm. Elementary school geometry told me that the following rule applies:

circumference = diameter * 3.1416 "

Boy, I gotta get out more!