Hmmm - 4 replies to my question. At last I have time to reply - I have been so busy lately.
Greg Locock mentions four effects, all of which I think I considered when I did this experiment almost 20 years ago. I think I gave all the relevant data, except for the bulk modulus of the oil - I have always used the rule of thumb of .5%/1000 psi, or 5*10^-6 (psi)^-1 for rough calculations. Assuming this value, for effect #1, the frequency change due to the oil compressibility alone would amount to a reduction of less than one part per thousand. Effect #3 produces an increase in frequency of only 3 parts per hundred thousand. I did not bother to calculate the result for effect #2, but it also is extremely small. All these effects are insignificant. We now come to effect #4. Blevins (formulas for natural frequency & mode shape) gives the following formula for the frequency ratio increase under a tension P, for this particular case : sqrt(1 + P*L^2/(E*I*pi^2)). This effect is NOT negligible. If this were the only significant effect, it would produce a frequency increase of about 32%. I remember checking the axial extension of the tube with a dial indicator when I did this experiment to confirm that I really had the correct axial load, which in this case is 7854 pounds.
And now to the result of the experiment: The pressurization produced no measurable change in frequency whatsoever !
Greg Locock is therefore correct when he says "my argument is that 4 won't matter", although apparently he does not say exactly what his argument is. Prex's conclusion is also correct, although his explanation is somewhat misleading in my view.
I believe the correct explanation is as follows :
If one analyzes the problem from first principles, along the lines of the classic derivation of Blevins' formula, one procedes by considering the integration of a number of infinitesimal segments of the tube, each of which, with the tube in its deformed state, would look like a very short banana shape, or toroidal segment, having a curvature coincident with the local beam curvature. The two end faces of the segment will have forces applied to them, normal to the faces, each equal to 7854 pounds (ignoring any minute variation in tension along the length of the beam, which is valid to a first order of approximation). If the element is of finite size, the two forces are at a slight angle to each other, and it is this slight inclination that produces a radial force component which tends to pull the element back towards the undeflected position. The situation as considered so far does not differ at all from the Blevins case with externally applied tension. However, there are two other forces due to the internal oil pressure which act opposite to each other in the radial direction. These forces are not equal to each other if the beam is deflected, since because of the curvature, the projected areas on the two halves of the inner pressurized wall are not the same. It turns out that this effect produces a radial force (transverse to the beam) that exactly cancels out the axial tension effect. Actually, this is immediately obvious if one assumes the oil volume is in equilibrium, but it helps to understand the whole picture.
Now I ask this : Assuming it were practicable, if the inside wall of the tube were isolated from the radial oil pressure, by inserting an additional thin wall tube inside and pressurizing that, what would then be the resulting change in natural frequency ? (I have not done this experiment).
Although none of this is directly relevant to the original question, it does underscore the need to think about each situation in detail - rather than just making general statements to the effect that "stress increases natural frequency".
