Wikipedia:Reference desk/Archives/Science/2023 December 18

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December 18

Organ pipe

If the air temperature rises with height due to poor church HVAC then would there be only one main pitch equal to putting the average speed of sound of the standing wave in the pitch formula or will there be two main pitches one for the speed of sound at each end or will the fundamental be smeared into an infinite number of pitches more or less equally loud all lying between a pitch based on the speed of sound of the most heated part and the speed of sound of the least? 166.199.7.54 (talk) 00:51, 18 December 2023 (UTC)

Are you asking whether a temperature gradient across a column of air causes its resonant standing wave to vary in frequency accordingly? Remsense留 02:11, 18 December 2023 (UTC)

Correct, if a pipe is x Hertz when everything's 20C what sound will come out when it's 20 on average but the air has a temperature gradient? 166.199.7.54 (talk) 02:40, 18 December 2023 (UTC)

I've thought about this for about 15 minutes, and I think your "two frequencies at each end" guess is going to be my guess also. Ideally, there's only meaningfully a "pitch" in the direction that changes in air pressure are propagating (i.e. down the length of the tube) Remsense留 04:38, 18 December 2023 (UTC)

As well as the standing wave producing the sound, there is also movement of air due to the action of the bellows. I suspect that quite quickly the air in the pipe will have a uniform temperature closely related to the intake temperature of the blower. Martin of Sheffield (talk) 09:20, 18 December 2023 (UTC)

Even if the temperature varies considerably in the pipe, there is only one pitch. The pressure waves will travel through the pipe with a varying velocity, the speed of sound, which increases with the temperature. But the pitch is determined by the standing waves, whose frequency only depends on the total travel time of the wave through the pipe. Compare this to a pendulum clock. The speed of the pendulum varies over its trajectory, but the clock ticks with only one rhythm.  --Lambiam 10:15, 18 December 2023 (UTC)

Right! I think my mistake was thinking about the standing wave as if it were a traveling wave, not the sum of traveling waves. Remsense留 15:29, 18 December 2023 (UTC)

When the air temperature rises, the speed of sound in air increases proportionally to the square root of the absolute temperature and the resonant length of the metal Organ pipe expands. The sound pitch does not change as much as either effect alone would cause because they partially compensate one another. Philvoids (talk) 18:45, 18 December 2023 (UTC)

This is not the case: the expansion/contraction of the aerophone body caused by change in temperature is negligible proportional to the corresponding expansion/contraction of the column of air inside. Remsense留 18:51, 18 December 2023 (UTC)

The length of the column of air inside the pipe is essentially the length of the pipe containing the air.  --Lambiam 19:47, 18 December 2023 (UTC)

Right, but the point I was making is that change in the length of the pipe due to the expansion or contraction of the aerophone itself is negligible. I suppose I misspoke when I said the air was expanding/contracting, I should've said it was becoming more or less dense. Remsense留 19:49, 18 December 2023 (UTC)

That is correct. A pipe made of an alloy of tin and lead, as many church organ pipes are, will expand by about 0.05% when the temperature rises by 20 degrees. The speed of sound will increase by about 3.4%.  --Lambiam 20:29, 18 December 2023 (UTC)

I assume that the Fourier analysis in terms of wavelength of the waveform inside the pipe depends only on the shape of the pipe but not on the conditions of the air inside it. The lowest eigenmode (I guess that's what's meant by "standing wave" above) has wavelength twice the length of the pipe (if the later is closed at both ends and essentially one-dimensional). However, the conversion to frequency (the dispersion relation) depends on temperature, as demonstrated in the video linked above. With a temperature gradient, the air molecules should oscillate with varying frequency along the length of the pipe. The next question is how the oscillation inside the pipe is coupled to the air outside. If the coupling is homogeneous along the pipe then we should here a continuous superposition of frequencies over a range determined by the temperature range. If the coupling occurs at a single point (the labium or an open end (are organ pipes open?) come to mind) then the pitch will be determined by the temperature at that point. The actual situation will be somewhere inbetween these extremes, and we won't discuss the effect on timbre. --Wrongfilter (talk) 20:31, 18 December 2023 (UTC)

It feels like we should link to Q factor, a parameter that describes (among other things) the "perfectness" of a resonator - in other words, how ideally the resonating energy is exactly concentrated at one single harmonic frequency. Real physical resonators have a bit of a spread - a "bandwidth" - over which the energy is spread. Things like imperfections or nonuniform conditions along the extent of the resonant cavity (including temperature) will cause the resonator to have a lower "Q" factor - and consequently a wider bandwidth. Audibly, this means a tone that is less "clear" and it will sound different - perhaps even sounding "out of tune" in severe situations. Nimur (talk) 20:43, 18 December 2023 (UTC)

Assuming a constant frequency $f$, the wavelength $\lambda$ is given by $\lambda =v/f,$ in which $v$ is the speed by which the waves propagate – in this case the speed of sound. If that speed varies along the length of the pipe, so does the wavelength, which makes it difficult to assign a meaning to "the Fourier analysis in terms of wavelength of the waveform inside the pipe".  --Lambiam 08:36, 19 December 2023 (UTC)

What do you mean by "constant frequency"? The air molecules obviously do not travel the entire length of the pipe but oscillate by a fairly small amount around a mean position. The frequency of these oscillations can vary along the pipe, and it is these oscillations that eventually couple to the air outside. You can of course always analyse any 1D wave form in terms of Fourier waves (sines and cosines); what's the worst that could happen? Maybe the coefficients become time dependent, but the wavelengths of the modes remain the same. Compare to a guitar string that is plucked in the middle so that predominantly the lowest mode with wavelength twice the scale length of the guitar is excited, this determines the pitch. Now increase the tension of the string; the wavelength remains the same but the pitch increases. Wavelength is determined by geometry, frequency by the physical properties of the oscillating medium. --Wrongfilter (talk) 08:58, 19 December 2023 (UTC)

Sorry, that doesn't sound right. A standing wave is a product of a position-dependent oscillation and a time-dependent oscillation, or a sum of two travelling waves. It can't have a position-dependent frequency, or waves would pile-up somewhere. What happens is that the wavelength varies along the length of the tube. Assuming there's an antinode at both ends of a simple, one-dimensional tube, the node in the middle gets displaced towards the end with the lower sound speed. The period of the fundamental of the produced sound is still twice the sound travel time along the length of the tube.

As for your guitar comparison: consider a guitar where the thickness of the string changes along its length. To excite primarily the fundamental, you have to pluck it near the thick end. PiusImpavidus (talk) 13:15, 19 December 2023 (UTC)

I don't think there's a contradiction here. A standing wave with a displaced node can still be Fourier analysed, it will just have a different overtone spectrum compared to a symmetric wave. And I guess you can rearrange the Fourier sum to arrive at an expression with a wavelength that is formally position-dependent. But that just gives the spatial waveform inside the tube. The main question still remains: what does it sound like? Heating the pipe uniformly does not change the spatial form of the standing wave, but it does change the pitch. Applying a temperature gradient doesn't necessarily change the pitch if applied in the right way, but it will change the spectrum of the wave inside the pipe, as well as that of the sound wave outside the tube. I'm not sure we can get any further than that without a quantitative analysis. --Wrongfilter (talk) 16:26, 19 December 2023 (UTC)

A few thoughts: Organ pipes can be open-ended or stopped, in which case they sound an octave lower. According to a document from Harrison's website, Peterborough Cathedral is heated continously during the winter, and the difference from floor to ceiling is only 2° C. In other cases a difference was noted of 15° C between the player at floor level and the Swell division 15ft above; another tuner noticed a difference of 11° C between Great and Swell divisions. The relative humidity matters almost as much as the temperature. Get both wrong, and this happens: "The organ is in an organ chamber some 25 feet above the only outlet from the forced-air heating. Six months after the heating was installed, every soundboard split, slides stuck, runnings everywhere, and tuning almost impossible. The difference between floor level and the organ was at least 20° F (11° C). Remedial work to the organ cost more than £55,000." Church Heating and the Pipe Organ. MinorProphet (talk) 22:52, 18 December 2023 (UTC)

And pipes can be flue or reed. Flue pipes have an antinode at the window whereas reeds have a node at the reed. And after you've considered flue/reed stopped/open, then the shape of the resonator is critical to the tone: parallel, conical (getting narrower) or conical (getting wider). To get back to the OP's original question though, very shortly after speaking any pipe will be filled by the air from the blower/bellows and will be consistent in temperature and humidity along the length. The pipe, however, is quite a different matter! Martin of Sheffield (talk) 23:05, 18 December 2023 (UTC)

In many English churches, in winter, the temperature of the pipes or the air in the pipes is not an issue. Far more important is the need to warm up the organist to operating temperature. -- Verbarson  ^talk_edits 19:00, 19 December 2023 (UTC)