Acoustic Resonance

I recently read an article about acoustic resonance frequencies within the different chambers of the Great Pyramid. I found that the speed of sound in air at a temperature of 20°C is about 343 metres per second. Now \(343 = 7^3\) and this got me thinking about the acoustic resonant frequencies of cylindrical tubes.

There is one particular formula for a tube that is closed at one end:$$f=\frac {nv}{4(L+0.4d)}$$where \(v\) is the speed of sound, \(L\) is the length of the resonant tube, \(d\) is the diameter of the tube, \(f\) is the resonant sound frequency. Here \(n\) is an odd number (1, 3, 5...) because this type of tube produces only odd harmonics and has its fundamental frequency an octave lower than that of an open cylinder (that is, half the frequency).

After a little experimentation, I found that using \(v=343\) and tube dimensions, in metres, of \(L=11.75\) and \(d=1.25\) the resultant acoustic frequencies (in hertz) were multiples of 7. Figure 1 shows a screenshot of the SageMath code that I used to confirm this:

Figure 1: calculations for a cylinder closed at one end

The formula for the acoustic resonance frequencies for a cylinder that is open at both ends is given by:$$f=\frac {nv}{2(L+0.8d)}$$and, if the length L is doubled so that \(L=23.5\) and the diameter remains the same, then the same multiples of 7 frequencies emerge as shown in Figure 2. Note that all the frequency multiples of the fundamental can occur (2, 3, 4, ... ) not just the odd multiples (3, 5, 7, ... ).

Figure 2: calculations for a cylinder open at both ends

So the takeaway from all this is that a cylinder, closed at one end and with a length of 11.25 metres and a diameter of 1.25 metres, will have a fundamental acoustic resonance frequency of 7 Hz. Similarly, a cylinder that is open and both ends but twice as long will have this same fundamental frequency. Now this has opened up a whole can of worms because this frequency has some interesting properties and takes one into the world of infrasound. Here are some interesting comments about 7 Hz and other infrasonic frequencies taken from this source.

Infrasound is low frequency audio beneath the human range of hearing. Infrasound constantly surrounds us, generated naturally; wind, waves, earthquakes and by man; building activity, traffic, air conditioners and so-on. Low frequency sound is used by marine mammals to communicate over vast distances and by birds to determine migration patterns. 

At higher volumes infrasound of around 7-20 Hz can directly affect the human central nervous system causing disorientation, anxiety, panic, bowel spasms, nausea, vomiting and eventually unconsciousness (supposedly 7-8 Hz is the most effective being the same frequency as the average brain alpha wave). The effect is unintentionally (or not?) generated by the extreme low frequencies in church pipe organ music, instilling religious feelings and causing sensations of “extreme sense sorrow, coldness, anxiety, and even shivers down the spine” in the unsuspecting congregation. Low frequency sound generated naturally or by building work and traffic is said to be the cause of reported apparitions and hauntings – blamed on the ghostly 19 Hz frequency which matches the resonating frequency of the human eyeball.

A cylinder, open at both ends with a diameter of 2.5 metres and a length of 7 metres, will produce a fundamental frequency of 19.06 Hz, the so-called "ghost frequency". I won't pursue any of this further in a mathematically-focussed blog such as this, but I'll investigate it further in one of my other blogs.