Speed of sound in a tube

At high frequencies, it's obviously the speed of sound in free air. Imagine a wave that's an inch long traveling in a 100 foot diameter pipe. The stuff ripping down the center doesn't even know the walls are there. In a long pipe, there will be secondary effects from some of the wave hitting the walls.

Or if the wavelength of the sound is much larger than the pipe diameter, it's just a pressure wave fighting its way through the pipe, with lots of viscous drag against the walls. Slow and lossy.

I suspect it is probably all three of the above depending on just how precisely you can measure the velocity of sound. It would be surprising if the speed of sound didn't change a bit when the wavelength was comparable with the dimensions of the tube for instance.

And the exact resonance frequency of an open tube also depends slightly on its diameter as well as its length.

The boundary conditions must generate a small drag on propagation speed in much the same way as a waveguide does for microwaves.

However, I also think that the extent of this effect is too small to have any practical importance unless you are trying to measure the speed of sound to several places of decimals.

May not be completely relevant, but it's not often you get the chance:

Google "Kundt's Tube"

;-)

Reasoning from the electrical waveguide analogy, one would expect that the phase velocity would go up, and the group velocity would go down.

I don't think that's the case with sound, however, because the boundary condition is different.

In a waveguide, the tangential E field has to be zero at the walls, so the lowest-order mode has to have half a cycle's worth of curvature across the waveguide cross-section, i.e. k_x = pi/D, where D is the width of the guide. Since k**2 = k_x**2 + k_y**2 + k_z**2, setting a minimum k_x means that there's less available for the longitudinal variation, i.e. the actual wave propagation part. That makes the longitudinal component k_z smaller than in free space, so that the phase velocity omega/k_z is higher than in free space. Similarly, since k_z goes from zero at the cutoff frequency asymptotically to k_0 at high frequency, it varies more rapidly with omega than k_0 does. The group velocity the reciprocal of this, i.e. d(omega)/dk_z, and so is less than c. That's all because of the electric-field boundary condition.

Contrariwise, for sound in a tube, the field variable is the sound pressure, and pressure doesn't have to go to zero at the tube walls. Thus sound in a tube doesn't necessarily have any transverse variation at all. Since there's no enforced k_x, k_z can be proportional to omega, and sound in a pipe ought to travel at the same speed as in free space.

The one fly in this ointment is friction at the walls. Normally one expects that the gas velocity goes to zero at the walls, so there's some curvature in the velocity, as there is in the E field case. This doesn't necessarily affect the pressure, but one must keep in mind that acoustics is the small-signal limit of the real thermodynamic behaviour of gases, which is highly nonlinear at large signals, so friction may make a difference besides signal loss.

However, it's a common observation that you can put low acoustic frequencies through a pipe--if that weren't so, none of our gas mains would work, for a start. You can't do that with a waveguide.

So my vote is for the speed of sound in a pipe being nearly the same as in free space, and nearly constant with frequency until you start getting multiple transverse modes in the pipe.

Cheers

Phil Hobbs

Interesting. You have prompted me to work out the frequency with which our local gas holder oscillates. If fills and empties about once a week so the frequency is in the region of 1.65 micro c/s

Thanks. That is helpful as the pipe is considerably smaller than the shortest wavelength.

Adrian, Perhaps google on organ pipe dimensioning? ...Jim Thompson

Does it matter? The "Q" is low enough that one can use the speed of sound in air as a starting point and add the open-end effect correction

- and get excellent results. Just look up the many "build it yourself" articles that give lengths. Or look to organs..

The problem with nearly all the measurements I have found with Google is that they need a compensation for 'end effect' which is probably greater than the effect of any velocity change.

The end-effect on a multi-tube microphone is the same for every tube, so it dosn't matter (unless there is a length-dependent factor which I have never heard of). Where it does matter is when the speed of sound is measured by an open-ended pipe, unless corrections are made, the calculated speed willbe significantly in error.

The problem with the microphone is that the tube lengths range from about 37" to about 1", so, if a 12 Kc/s wave entering the longest tube is slowed by 1/72th of the free air speed it will cancel with the same wavefront which has only travelled through the shortest tube.

My brain may be a bit dusty these days, but I rarely forget anything.

Your post triggered a memory of something I haven't thought about for pushing 50 years - a project I *almost* built from an article in the June '64 issue of Popular Electronics. I'd gathered many of the parts, but my skill at music was increasing exponentially at the time and I was beginning to play in downtown Washington, DC, for petty decent money...so the project went on the back burner, and in time was forgotten.

I found an article on building an improved version of the original Popular Electronics device, which also contains a scan of the June '64 construction project article. The modern article starts on page 66, and the scan of the original starts on page 79.

Enjoy... ;-)

Lord Valve

You trying something like this?

G=B2

The way I see it, and I did full around with some tubes when I saw the first way back.

The device does amplify sound. To get any directivity, you need matching length pipes, at least three, In a wide circle. For sound to be directional, you need resonant pipes of the same frequency, not different frequencies. Phase adding and cancellation cannot occur at different frequencies. The larger the diameter of the whole thing, the more directionality you will get.

Greg

I would think the speed would be the same, but depending what your looking at, like phase, over time, resonance in the pipe could change phase.

Greg

It's something you could check experimentally without a lot of effort, setting up the tube as a resonator, driving it with a sinewave source and checking the resonant frequency with a counter.

The chances are that the velocity could be impacted by the presence of the tube, I'd be surprised if the difference was great enough to impact on a sensible microphone design.

The end effects will cause huge errors.

That's the starting point for my design, but I hope to improve upon it.

in a

It is my understanding that a 'tube' microphone does NOT amplify sound, rather focusses it.

Simply changes the lobe pattern of sensitivity, so that...hey, that does semm like amplification doesn't it?

I am confused. To me it seems the original design more or less arbitrarily picked 37 tubes and having a 1 inch change between the lengths of the tubes. It seems to me that the change in length should be a percentage of the length, not a linear amount. And why pick 37 inches for the longest tube and have two 1 inch tubes. Why not pick

If you assume the original design was flawed, then all the fuss about changes in sound velocity is meaningless.

Dan

Measuring an individual tube, I found that plugging the end with a microphone gave a resonance with the 3dB points spaced apart by about 3% of the centre frequency. This was with the open end of the tube cut at an angle to reduce the 'Q' as much as possible. On that basis, it would require over 100 tubes to achieve a flat frequency response over a reasonable audio range.

I then realised that the 'closed' ends of the tubes weren't actually plugged solid, but communicated with a 'mixing chamber' before reaching the microphone diaphragm. This allows sound energy to flow from one tube to another, so the effect is similar to coupled tuned circuits and the response flattens out. In that case, the effect of a small discrepancy in the number of tubes at one particular fequency would be a lot less than one might initially suppose. The extra tube in the published version is there for ease of construction but I suspect that that version of the microphone was just intended to demonstrate the principle and wasn't supposed to be an optimised design (the extra tube might even help to augment a falling response at the H.F. end of the spectrum).

As for the ratio of tube lengths, I was initially puzzled by this until I realised that it is the frequency distribution which needs to be a geometric progression, not the wavelength distribution. By having equal increments of tube length, an approximation to geometrical frequency distribution is achieved.