Hi all,

I am a pipe organ restorer, with 20 years of experience, so I think I
can give some suggestions on the subject.
First of all, the explanations given by John Payson are generally right
with some inaccuracies. The frequencies generated by an open flue pipe
are given by F=(v*n)/(2L+k), where v is the speed of sound, which in
turn is equal to 20.055*sqr(T) m/s; T is the temperature in Kelvin
degrees (sqr is square root, of course); n is an integer giving the
order of the overtone, and k is a constant which takes into account the
diameter of the pipe; for open pipes it is equal to 2*sqr(S), where S is
the section of the pipe. For closed pipes the frequencies are
F=(v*(2*n+1))/(4L+k), with the same constants. If we set n=1 we obtain
the fundamental; for n>1 (integer) we have the overtone frequencies,
which are exact multiples of the fundamental, as can be verified in
actual fact. For closed pipes, only odd overtones are allowed.

After theory, let's go to the practical stuff

>I want to measure a maximum level difference of 2m with a
>maximum frequency deviation of 1 octave. What should the
>diameter be?

Before you think to the diameter you must determine the length of the
pipe. If you want to measure 2 m. of difference within an octave, you
need a pipe 4 m. long; in fact, the resonant length at maximum level is
half the length at minimum, so the frequency is double (one octave
range). A pipe partially immersed in a liquid can be regarded as a
closed pipe, if we assume that the liquid is dense enough with respect
to the air contained in the pipe. When designing organ pipes, a ratio of
1/3 or 1/4 the circumference with respect to the length is considered a
good  proportion for closed pipes, then for a pipe 4 m. long we should
use a circumference of about 1 m., hence a diameter of more or less 30
cm.. This is the right proportion for a pipe that should produce a
musically pleasant tone; for the purpose of measurement a smaller
diameter can be used, but take into account that the narrower the
diameter, the stronger the overtones, then the higher the risk that the
oscillator hooks with the overtones, giving false measurements.
Maybe 4 m. are too much for your application. I can suggest to bend the
tube one or more times, so it can be arranged as you want. Bending the
tube does not affect the resonant frequency, provided that the diameter
is constant in the curves.
Consider also that with this pipe dimension you work with very low
frequencies, as you can easily calculate.
What about enlarging the frequency range to more than an octave?

Best regards

Leonardo Perretti
leo.perretti@projectpp.it