You won't be able to find a single relevant equation, for two reasons:
1 The frequencies depend on how the pipe is restrained. That is the probably single most important thing which determines the frequences.
2 There are many different modes of vibration of the pipe, each with its own (infinite) set of vibration frequencies and each governed by different equations. For example if the pipe was not restrained at all, it could vibrate by bending (like a beam), axially (changing length), torsionally, and radially (not just panting in and out, but also the cross section changing from a circle to an ellipse, or a "wavy" shape with any number of waves round the pipe).
I should further qualify this. Pipe would be unrestrained, and length would hopefully be large compared to diameter and wall thickness. I would be primarily interested in the longitudinal pressure wave resonance, but it would be nice to calculate the predominant radial frequencies for the two strongest modes. Thanks
Ah... I thought you were talking about resonances of the pipe, not the air inside it.
If you assume the air and pipe resonances are not coupled (that's a very reasonable assumption for a "stiff" thick walled pipe) the longtitudinal resonances are just the standard "open and closed organ pipe" formulas, and are independent of the diameter of the pipe.
Re the radial frequencies, the solutions to the wave equation are Bessel functions. If the wavenumber k = omega/c and the radius is r, the lowest frequencies are when kr = 3.832, 7.015, 10.174, ... See http://www.du.edu/~jcalvert/math/cylcoord.htm
Sorry....I'm finding that I am ot communicating clearly at all. I am interested in the resonance of the pipe itself, and possibly an algorithm by which I could arrange the radial frequencies to be an harmonic of the longitudinal reaonance so that the tone would not be unpleasant. Think of giant wind chimes! many thanks!
Finding any formulas for thick shells will be hard. This might give you some leads for thin shells (usually defined as radius/thickness > 10, so your largest radius and smallest thickness are in that range).
"... this thesis presents exact solutions for vibration of closed and open cylindrical shells..." http://library.uws.edu.au/adt-NUWS/public/adt-NUWS20061016.103821/index.html [Broken]
I haven't read all of it (!!!) - and apologies if it tells you a lot more about the subject than you really want to know.