Supplements for Chapter 13
Musical instruments based on Helmholtz resonators:
The Udu drum is a remarkable and simple instrument, seemingly easily discovered by any group making ceramic pots, i.e. hundreds of different societies in the past thousands of years. However, it seems only in Nigeria was the design discovered or exploited.
An Udu drum has a Helmholtz resonator shape, usually with a long neck, and an extra hole somewhere on body, both holes perhaps half to a quarter as big as a palm. The Helmholtz modes are exited by slapping the holes with the palm in various ways, and the Helmholtz frequency is controlled by modulating the effective hole diameters with proximity of the palms to the holes. So, it is a drum with a variable pitch. The pitch rises as the hands are removed from the vicinity of the holes . An excellent demonstration is seen here. Most Udu players also use the high pitched vibrations of the body itself, excited by a slap, including perhaps metal rings on fingers as a percussive element. Their lower body vibration modes couple weakly but often not inaudibly to the Helmholtz mode, and perhaps (?) to other internal modes of the neck and cavity (an interesting project).
Pre-Columbian whistling jars, and double jars
It seems to be difficult to get good information on these mysterious (in their purpose) Pre-Columbian musical instruments based on Helmholtz resonators. Many of the jars were held for years without realizing that they possessed a hidden, internal fipple that made them whistle at the Helmholtz mode frequency (and perhaps overblown to higher modes of the chamber?). The double jars (this fine pre-Columbian example courtesy of the Indiana University Art Museum) sometimes are designed to work only when water is introduced inside them, partially filling the chambers. As the water is poured back and forth, the displaced air is forced to move across a fipple, causing the jar to whistle. Various animal sounds were deliberately arranged, including bird chirps, that could be created if one of the air streams entered under water, causing bubbles. As water enters one chamber from another, the pitch will glide up or down, because the free air volume of the resonator chamber is changing - ingenious.
A good scholarly article on Peruvian whistling bottles may be found at J. Acoust. Soc. Am. Volume 62, Issue 2, pp. 449-453 (1977); (5 pages). You will need access to the journal through a library or online through an institution.
An excellent article by Brian Ransom here.
The Ocarina is at least 12,000 years old. Mayans, Aztecs, Chinese, Indian, but oddly not European until the Ocarina was brought back by Aztec performers sent by Cortez to Emperor Charles V to perform at the royal court. The Italian Guiseppe Donati made the sweet-potato shaped ocarina variant with more holes than before and with more accurate tuning in the 19th century. This is called the "modern Ocarina" but many earlier varieties are still held in high favor.
The ocarina is a multi-holed Helmholtz resonator with a fipple as a source of driving frequency and power (described in chapters 13 and 14). The fipple is usually part of the interior or exterior of the main body.
Cooperative resonance of the fipple and resonator:
The fipple aims a stream of air to strike an edge or blade. This air stream is unstable due to vortex formation. Its tendency to oscillate above and below the blade is easily influenced by local pressure and velocity changes. The fipple air stream typically passes over a hole but the air stream from it enters the main body of the resonator. The instability of the stream against the blade is a huge factor in the ubiquity of the fipple, since it gladly sets up its oscillations to coincide with those of a resonant cavity; the Helmholtz cavity in this case, but the resonant open tube on the face of a recorder or pennywhistle (that are not traditional Helmholz resonators).
Just as we discovered in chapter 13 , the Helmholtz frequency is independent of the shape of the cavity, allowing a lot of freedom of design of an ocarina, something not lost on those who make them. The requirements are so flexible that the modern artist/musician sound sculptor extraordinare Brian Ransom makes absolutely gorgeous sculptures that are also ocarinas, such as his Peace Deity series. The one shown here has a fipple incorporated into an arm. (Ransom also makes other musical instruments that are not ocarinas, but perhaps qualify as "experimental".)
Brian Ransom recognized the importance of pre-Columbian whistle jars. Often they were mistaken as silent vessels having a connection with marriage ceremonies. His article on the jars is given here.
Mountain Ocarinas has a nice web-based explanation of the workings of an ocarina.
Brian Ransom demonstrates a marvelous paired ocarina (two side by side) of his own making, each with a large hole (Udu-drum like) that can be tuned continuously in pitch by proximity of the palm, as in an Udu drum. The beating of two notes is then completely under the performers control, with wonderful effect. This is about 16 minutes into the video.
Project: Ad Hominem Helmholtz resonators
Why You Hear What You Hear mentions that we all carry around resonators with us. The easiest to hear perhaps is the oral cavity, closed off at the back of the tongue (so you can't breathe at that moment). The mode may be excited by popping the pursed lips with a flicked finger or flicking a cheek with a finger. The pursed lips trick can sound quite a bit lower in frequency than the cheek version; this is because the covering finger is not removed fast enough. To prove this to yourself, use the flicked cheek and put a finger or part of the hand close to the mouth at various distances - and hear the pitch go down as the hand closes off the lip opening.
Helmholtz Resonators and Architectural Acoustics
We mentioned the passive Helmholtz resonators still used in air ducts to block the transmission of certain frequencies down the duct - an example of Helmholtz resonators in architectural acoustics. We also discussed their possible use in ancient Greek theaters.
Helmholtz resonators have been incorporated in modern walls for sound damping purposes (figure 13.yyy). This has been going on in churches for hundreds of years, seemingly in some cases with the express purpose of suppressing reverberations. Although it has not been mentionedbefore, to my knowledge, one can imagine a tuned resonant cavity (some have been found with ash in them, presumably to dampen sound and increase the range of frequencies over which it is effective, by broadening the resonance) set to dampen an annoying standing wave resonance near a pulpit, for example. See
Resonant Cavities in the History of Architectural Acoustics
Robert G. Arns and Bret E. Crawford
Technology and Culture
Vol. 36, No. 1 (Jan., 1995), pp. 104-135
Published by: Society for the History of Technology
Helmholtz resonators and short circuiting, proximity resonance, pendulum analogs,...
This figure from chapter 13 shows three situations involving one or a pair of identical Helmholtz resonators, run as a simulation in Ripple, that you can easily reproduce or vary in any way you like. At the left is seen a pair of identical resonators initially loaded in phase with each other (see the movies below); as the sound escapes, each oscillates with the same period. The amplitudes add from each of the resonators, and since they are in phase they reinforce each other. The amplitude far away (seen as the brighter colors representing amplitude the figure) increases by a factor of 2 compared to one resonator alone. The power goes as amplitude squared, so a four-times faster decay of stored energy takes plase compared to one resonator alone. A factor of two faster decay would be expected since two resonators are decaying, with twice the initial stored energy of a single resonator. The other factor of two is due to constructive interference of the two resonators. Q is not determined by the total power radiated, but rather by how fast that power decays. (Suppose I have one damped mass and spring with a Q of 12. Now we collect 10 such systems, not interacting with each other. The Q of this collective system is still 12, but it dissipates 10 times the power of one mass and spring.)
In chapter 10 we learned that Q can be calculated as
From this we see again that Q is independent of the initial stored energy , since power dissipated is proportioal to the initial stored energy.
Suppose we artificially reduce the initial total energy of the two resonators, making it the same as the single resonator was initially. Then a factor of two increase in power radiated, due to constructive interference, still remains for the two resonators, and will cause them to decay with a rate twice as fast as the single resonator. This faster decay by a factor of 2 translates to a smaller Q by a factor of two, compared to the single resonator. If the single resonator Q was 18, as was measured in the simulation by the techniques/formulas of chapter the double is predicted to be 9; this is exactly what is seen in the simulation.
A single Helmholtz resonator is a monopole source. A dipole source is created by placing another such resonator right next to it, but breathing out of phase with the first one. The air being exhaled by one source is at the same moment inhaled by the other, diminishing the air pressure fluctuations that escape beyond the tubes. Air flows from high pressure to low, and in so doing lowers the highs and raises the lows; an acoustical short-circuit. Consequently there is a much slower radiation of amplitude and therefore power. This causes the energy to remain longer in the resonators and give a higher Q. The Q depends on the distance between the mouths of the two resonators; the smaller the distance the more short circuiting and the larger the Q. Perhaps you can verify that the power radiated decreases by a factor of the square of d, where d is the distance between the resonators. This means the Q is proportional to the square of d. It also can be shown that Q is proportional to the square of the wavelength of the sound emitted, for the out of phase resonators.
Connection with coupled pendula
Note too that the frequency of the out-of-phase motion (500) is higher than the in-phase (482), in complete agreement with the double coupled pendulum oscillator model with both pendula the same frequency before they were coupled, i.e. figure 10.9 (yyy), middle panel. Here, each resonator alone is analogous to a lone oscillator. The coupling analogous to the connecting spring in figure 10.9 is the air in the vicinity of the mouths. Finally, the single pendula on resonance (i.e. here the Helmholtz resonators alone) have a frequency lying between the symmetric and unsymmetric motion of the paired pendula, here about f=490.
Click on the figures below to load the corresponding movies.
Next, we consider driving a Helmholtz resonator on and off resonance from the outside, showing some of the manipulations on screen as well. Halfway through, the drive is turned off and you can see the resonator subsequently dump (radiate) its stored energy; if you were nearby you would hear it ring out its resonant frequency well after the far away source has ceased.
A deceptive point
Helmholtz resonators of the type made by Koenig and intended for frequency detection are almost always photographed and shown like this:
This leads to the impression that they are a chamber with a single small, narrow open neck at the top. In fact they are resting on a much wider and shorter neck, almost hidden, that is the actual port to the resonant chamber. The short nipple is inserted into the ear and is nearly completely closed off thereby. The eardrum however senses pressure changes nearly equal to those inside the resonant chamber and therefore very loud! This is what makes the Helmholtz-Koenig resonators such exquisite detectors, ones that few people have actually heard. The following drawing makes these points clear:
Project: The Q of a glass tea bottle (or other cavity)
Listen to the bottle here, after it was "popped" with a flat object at the mouth:
Resonator on raised seating: Greek theatre?
The Greeks supposedly specified their resonant urns be placed in the seating, upside down with the neck seated in a slot in the seats. Although the video of a Ripple simulation of this is hardly definitive, it is suggestive:
Is this effect on the URL below a Helmholtz resonance, or something else?