Pipes and HarmonicsWhy do closed conical bores have the same set of resonances as open cylindrical bores of the same length, whereas closed cylindrical bores of the same length have only odd harmonics starting one octave lower?The bores of three woodwind instruments are sketched below. The diameters are exaggerated. The flute (top) and clarinet (middle) are nearly cylinders. The oboe (right) is nearly conical (as are the saxophone and bassoon). The clarinet is about the same length as the flute, but plays nearly an octave lower. The oboe is closed like the clarinet, but its range is close to that of the flute.

For a background to this discussion, it is worth looking at the difference between closed and open pipes, which is explained inOpen vs closed pipes (Flutes vs clarinets), which compares them using wave diagrams, air motion animations and frequency analysis.To compare cylindrical, conical, closed and open pipes, let's look first at diagrams of the standing waves in the tube.

Three simple but idealised air columns: open cylinder, closed cylinder and cone. The red line represents sound pressure and the blue line represents the amplitude of the motion of the air. The pressure has a node at an open end, and an antinode at a closed end. The amplitude has a node at a closed end and an antinode at an open end. These three pipes all play the same lowest note: the longest wavelength is twice the length of the open cyclinder (eg flute), twice the length of the cone (eg oboe), but four times the open length of the closed cylinder (eg clarinet). Thus a flutist (diagram at left) or oboist (diagram at right) plays C4 using (almost) the whole length of the instrument, whereas a clarinetist (middle) can play approximately C4 (written D4) using only half the instrument. If you have a flute or oboe and a clarinet, this experiment is easy to do. Play the lowest note on the flute or oboe, and then compare this with the lowest note on half a clarinet (ie removing the lower joint and bell).Important: in all three diagrams, the frequency and wavelength are the same for the figures in each row. When you look at the diagrams for the cone, this may seem surprising, because the shapes look rather different. This distortion of the simple sinusoidal shape is due to the 1/r term, which is discussed below.

An import proviso: no instrument is a complete cone. If a conical bore came to a point, there would be no cross-section through which air could enter the instrument. So oboes, bassoons and saxophones are approximately truncated cones, with a volume in the reed or mouthpiece approximately equal to that of the truncation of the cone.For musicians who are not mathematicians, the following simplified argument is probably helpful. However, be warned that you will have to concentrate. Einstein is credited with the quote "Everything should be made as simple as possible, but not simpler". I think that I have made this argument as simple as possible. But not simpler. (Mathematicians, physicists, engineers etc, may omit the following and go here.)

When a sound wave travels down the bore of an oboe or saxophone, the wavefront is spread out over an area (the bore cross-section) that increases with distance along the bore. (See Travelling waves and radiation if you have trouble with this paragraph.) The same happens when sound radiates in the open air in spherical symmetry. We can picture the cone as a section of a sphere so, in both cases, the intensity (which is power dived by area) goes as 1/r2, where r is the distance from the apex of the cone or the centre of the sphere. That's why sounds get less loud as you get further away. Double the distance away, the sound power is spread over four times as much area, so the power coming into your ear is four times less. Now the intensity of a sound wave is proportional to the square of the amplitude (pressure or velocity) and so the pressure and velocity are proportional to 1/r in the case we've just discussed. On the other hand, a wave which travels down a cylinder (constant cross- section) is a plane wave. If we neglect the small losses of energy, its amplitude is constant all the way along the cylinder.

So the sound wave in a flute or a clarinet must be made up of sin or cos terms (which don't change in amplitude as you move along), but for saxophones, oboes and bassoons, there must be a 1/r factor to account for the cross sectional area which goes as the square of the distance from the reed. (Think of how loud the sound is at the bell, then imagine going up the bore and concentrating that sound over successively smaller areas: the sound gets louder and louder as you approach the reed.)Now we add what mathematicians call the boundary conditions (i.e. the physical constraints at the ends.) For the flute, we open the tube to the atmosphere at both ends so the pressure here is atmospheric, and the sound pressure (difference from atmospheric) is close to zero. So for the flute we want a zero in pressure at both ends, and that is met by sine waves with wavelength 2L/n where L is the length of the instrument and n is an integer (see the diagram above for the open cylinder, where this condition is met). (In practice, the end of the instrument is not quite a node, and so the effective length is longer than L by a small amount, usually about 0.6 times the radius.)For the clarinet, we want a zero at the open (bell) end, and a maximum at the reed. This is met by cosine waves with wavelength 4L/m where L is the length of the instrument and m is an ODD integer. (see the diagram above for the closed cylinder)

For the conical tubes (oboe et al) we also want a zero at the bell and a maximum at the reed, but we have to fit spherical waves, which have terms involving (1/r) and (1/r2) times the sine and cos functions. For example, the standing wave in pressure has an envelope which is (1/r) times a sine wave with a wavelength which is 2L/n, where L is the length of the instrument and n is an integer. The sine goes to zero at r = L, and(1/r) sin r has a maximum at the reed, as required. Note that it has the same harmonics and the same bottom note as an open cylinder of the same length. (Some serious simplification has been made here: the oboe is only approximately a cone and the cross sectional area does not fall to zero in the reed.)

Thus

the flute and the oboe (approximately the same length) have similar bottom notes (actually one or one half tones apart) while the clarinet (also approximately the same length) plays nearly one octave lower (it is less than an octave because of the non-cylindrical bell. If you replace the bell with a roll of paper to make it a cylinder, the agreement is quite good). It should be mentioned that there are also complications due to end corrections.The behaviour can be quantified, both theoretically and experimentally, using the acoustic impedance spectrum, defined as the ratio of the acoustic (varying) pressure required at the input to the acoustic flow it produces. The figure at right (from a paper by Chen, Smith and Wolfe, 2009) shows the measured impedance spectra for a cylinder (bottom), a flute, a clarinet, a soprano saxophone and a cone (whose apex is replaced by a cylindridal section of the same volume, so as to allow a measurement). In all cases, the impedance becomes small at very low frequency: little pressure difference is required to pump air through a short pipe at low frequency. The first peak corresponds to the frequency of the animation shown on theweb page about cylindrical pipes. In each case, we have chosen the pipes to have the same effective length, which is very roughly the distance from the input to the first open tone hole (for the instruments) or to the other end of the pipe (for the simple geometries).These curves show that, for typical notes, the flute and clarinet only resemble a cylinder at low frequencies, and the saxophone only resembles a cone at low frequencies. To look at the impedance curves for any note on these instruments, see our sites for the flute, the clarinet and the saxophone.For a detailed discussion of the acoustics of open cylindrical, closed cylindrical and closed conical instruments, see:

Introduction to flute acoustics , Introduction to clarinet acoustics , and Introduction to saxophone acoustics .

See also How harmonic are harmonics? and Flutes vs clarinets. And, on our FAQ page, see questions about the transition fromtruncated cone to cylinder and varying the cone angle.

Explanation for those with a mathematical background. The cylinder has a plane wave solution to the wave equation and is written in terms of cos and sin terms. (This is an oversimplification: they have other Bessel functions too, and these are important musically, but not in the first order explanation). For the standing waves in a conical pipe, we need only consider axial motion and variation in pressure. So we may consider the cone as a section of a sphere. So for the conical bore, solutions to the wave equation are expressed in terms of spherical harmonics. Note however the proviso above about truncation of the cone. At low frequencies, the volume in the reed or mouthpiece may be considered as a compliance: the pressure in that region is approximately uniform. This compliance thus contributes part of the boundary condition on the truncatioin.

What is acoustic impedance and why is it important?

Acoustic impedance, which has the symbol Z, is the ratio of acoustic pressure p to acoustic volume flow U. So we define Z = p/U. Z usually varies strongly when you change the frequency. The acoustic impedance at a particular frequency indicates how much sound pressure is generated by a given air vibration at that frequency.This introduction explains its use in understanding the operation of musical instruments. A more general introduction, in physical terms, is given in Sound: impedance, power and intensity. For a review of techniques for measuring acoustic impedance, seeDickens et al (2007). For the techniques developed in our lab, see Acoustic impedance measurements.The specific acoustic impedance z is a ratio of acoustic pressure to specific flow, which is the same as flow per unit area, or flow velocity. In all cases, 'acoustic' refers to the oscillating component. With this proviso, we can say that acoustic impedance Z = pressure/flow and specific acoustic impedance z = pressure/velocity.We discuss acoustic impedance on this music acoustics site because, for musical wind instruments, acoustic impedance has the advantage of being a physical property of the instrument alone – it can be measured (or calculated) for the instrument without a player. It is a spectrum, because it has different values for different frequencies – one can think of it as the acoustic response of the instrument for all possible frequencies. For instance, we measure it at the embouchure of an instrument because it tells us a lot about the way the player's lips, reed or the air jet from the mouth will interact with the instrument itself. So it tells us about the acoustic performance of the instrument, in an objective way that is independent of who might play it, and it allows us to compare subtle differences between instruments. So what is it?An analogy. Many people find helpful the analogy with electrical impedance. Spatial variations in electrical potential (differences in voltage V) give rise to moving charge (electrical current i) and the electrical impedance z = V/i. Here, spatial variations in acoustic pressure (p) give rise to air flow (U) and z = p/U. Here is an introduction to electrical impedance. Resistance is a particular (and rather boring) example of impedance, which is the general term for a ratio of voltage to current. DC (direct current) means constant or slowly varying current. AC (alternating current) means any current in which the movement is alternately backwards and forwards (oscillating) with no overall motion. AC is more interesting because the impedance can vary with

the frequency of oscillation of the current. (These analogies are limited: air is compressible so the flow does not obey Kirchoff's law about conservation of current.)

Consider the opening to a duct. The acoustic impedance Z at that opening is the ratio of the acoustic (or AC) pressure p at the entry to the duct to the volume flow of fluid U into it. Like electrical impedance, acoustic impedance may be a strong function of frequency. The graph shows the measured input impedance of a simple cylindrical pipe (325 mm long, 15 mm in diameter) in Pa.s/m3, as a function of frequency. The magnitude of Z varies by more than a factor of a thousand over this range of frequency. More about this below.The acoustic impedance is complicated by the fact that the current and pressure are not necessarily in phase – the maximum pressure may be ahead of the maximum flow, or vice versa. As in electricity, we use complex numbers to handle this, where the real part represents the in-phase component and the imaginary part the out-of-phase component.Units. The unit of pressure is the pascal – one newton per square metre. A pascal is a big unit for sound: an oscillation of one Pa is usually a very loud sound indeed. (In DC the Pa seems a small unit: atmospheric pressure is 100,000 Pa or 100 kPa.) Flow is measured in cubic metres per second. (A very gentle breeze coming in your window could be 1 m3/s. But for 1 m3/s to flow down a pipe, either the pipe must be big – think ventilation ducts – or the speed must be high.) The units for impedance are therefore Pa.s/m3, which we call the acoustic ohm Ω. For musical instruments, it is a rather small unit, so we use megohms: MPa.s/m3. Finally, sound pressures have a large range. For this and other, psychophysical reasons we use logarithmic scales for sound level and impedance. (For a linear quantity like sound pressure or impedance, you can convert to dB by taking the log of any ratio, then multiplying by 20.)

For an infinitely long pipe, with cross sectional area S and filled with a medium of density ρ and speed of sound c, the acoustic impedance is ρc/S. (See Sound: impedance, power and intensity) So, for an infinitely long pipe with diameter 10 mm filled with air, the acoustic impedance is 5 MΩ. In our lab, we routinely use pipes whose effective length is infinite as references, as well as other calibration. This allows us to make measurements of impedance spectra over 9 octaves (10 to 4000 Hz) using a single impedance head. See Dickens et al (2007) .Specific acoustic impedance*. In contrast to Z, the quantity z = ρc doesn't depend on the size of the pipe: it is a property of the medium alone, called the specific acoustic impedance, and its units are Pa.s/m. The specific acoustic impedance is the ratio of p to the flow per unit area U/A. U/A equals the acoustic velocity: the velocity of particles in the medium due to their motion in the sound wave. It is worth warning that, in some fields such as ultrasound imaging, practitioners sometimes informally say 'acoustic impedance' when they mean specific acoustic impedance. The specific acoustic impedance for air is about 420 Pa.s/m (the exact value depends on temperature and pressure) and about 1.5 MPa.s/m in water, i.e. about 3600 times higher for water. Virtually all condensed phases have high values of z, which accounts for the very high reflection coefficients at gas-condensed phase or condensed phase-gas interfaces.* Specific acoustic impedance is sometimes also called the 'characteristic acoustic impedance of a medium'. Be careful, because 'characteristic acoustic impedance' has another meaning: for a pipe of cross section S and undetermined length, the characteristic acoustic impedance of the pipe is Z0 = ρc/S. Sometimes, in informal use, specific acoustic impedance is even called the acoustic impedance of a medium. This last use is potentially very confusing!

Note

that we have referred to an infinite pipe in the example above. In such a pipe, a varying pressure causes a wave which is a varying flow of air, and which never comes back.In a finite pipe, however, the wave reflects at the far end (whether it be closed or open) and comes back, reflects again, and gives rise to standing waves or resonances (see pipes and harmonics). This causes the impedance to be much higher or lower than the value calculated above, depending on whether the pressure of the returning wave is in phase or out of phase with the driving pressure.The figure at right (from a paper by Chen et al, 2009) shows the measured impedance spectra for a cylinder (bottom), a flute, a clarinet, a soprano saxophone and a cone (whose apex is replaced by a cylindridal section of the same volume, so as to allow a measurement). In all cases, the impedance becomes small at very low frequency: little pressure difference is required to pump air through a short pipe at low frequency. The first peak corresponds to the frequency of the animation shown on the web page about cylindrical pipes.The acoustic impedance of musical wind instruments varies spectacularly with frequency because these instruments are designed to produce one or several frequencies only in a particular configuration. For example, the flute is played with the embouchure hole (at least partly) open to the atmosphere, so the pressure at the embouchure hole is very near to atmospheric pressure. Thus the acoustic pressure (the varying part) is nearly zero. The flow is provided by a jet of air from between the player's lips. Oscillations of air flow in the flute can cause this jet to deflect upwards (outside the flute) or downwards (inside) so that the acoustic flow (the AC component) can be large. Thus the flute operates atminima of Z: a small pressure and a large flow. Most other wind instruments have a reed which is sealed by the player's mouth and they operate at maxima of Z: the varying part of the pressure is large, but the oscillating part of the air flow is small at the reed. SeeFlutes vs Clarinets and Pipes and resonances for more details to examine simple cases. Many examples of impedance spectra are given on our sites for the flute, the clarinet, thesaxophone and brass instruments.In each of the impedance curves for the flute, there are at least a few rather deep, sharp minima, and the flute will usually play a note with a frequency near each of those deep minima. The ease of playing and the stability of the note depend on the depth and narrowness of the minima.Conversely, in each of the impedance curves for the clarinet, saxophone and brass instruments, there are at least a few rather high, sharp maxima, and those instruments will often play a note with a frequency near each of those high maxima. The ease of playing and the stability of the note depend on the

height and narrowness of the maxima. For instance the very highest notes are hard to play, and you can see on the spectra that at high frequency the maxima and minima are weaker—they "help the player less". There is more to it than this, however. For the instrument to play properly a note with frequency f, it sometimes needs an extremum at f, and also extrema at 2f, 3f, 4f etc. The reason for this is that the vibration of the air jet or reed and the sound made by the flute are not simple sine waves. Their waves are periodic waves (that is they repeat in time) and they contain a fundamental and a harmonic series. It is important to the performance of wind instruments that the various minima that help produce a particular note are in the harmonic series. See "How harmonic are harmonics?" and "How do woodwind instruments work?"On our sites for the flute, clarinet and saxophone, it is helpful to look at the Z curve for the lowest note for an explanation of some of the general features of that instrument's impedance spectrum.

Why is Acoustic Impedance Important?

The acoustic impedance of an instrument for any particular fingering is one of the major factors which determines the acoustic response of the instrument in that fingering. It determines which notes can be played with that fingering, how stable they are and it also helps determine whether they are in tune.The acoustic impedance also has a large influence on the sound produced. To see some examples of this, have a look at the two different fingerings for the same note. On the flute, one could look at A#4 (the "long" fingering using the RH index finger, and the "short" fingering using only the LH thumb). Look in particular at the relative depths of the 3rd and 5th minima in the impedance spectrum, and at the strengths of the 3rd and 5th harmonics of the sound produced. Look also at the two different fingerings given for A4, the relative depths of the harmonic minima, and the effect that they have on the timbre produced. Another difference that is worth looking at is the difference made by a "split E" mechanism. Look at the different Z spectra for E6 with and without the mechanism, and compare them with that for the note A5. Then try the experiment of slurring between A5 and E6 on flutes with and without the mechanism. Our databases for clarinet and saxophone also provide many examples.Another thing which has a big influence on the sound is the player, but that is another story and it is rather more complicated. Indeed a big advantage in measuring Z is that it gives us an objective measurement of the instrument

alone. In that way it is in some ways more useful to scientists and to makers than the sound of the instrument. If you get a poor sound from an instrument, it might be because the player is poor, or it might be because the instrument is poor. With our industrial collaborators, Terry McGee and The Woodwind Group, we are working to obtain objective comparisons of different instruments, to analyse the differences in their acoustic properties and to explain their different musical performance.

More about acoustic impedance

An introduction to acoustic impedance in the context of physics is given in Sound: impedance, power and intensity. For the measurement techniques developed in our lab, see Acoustic impedance measurements. For a review of techniques for measuring acoustic impedance, see Dickens et al (2007). Acoustic impedance is somewhat analogous to electrical impedance.

Helmholtz ResonanceA Helmholtz resonator or Helmholtz oscillator is a container of gas (usually air) with an open hole (or neck or port). A volume of air in and near the open hole vibrates because of the 'springiness' of the air inside. A common example is an empty bottle: the air inside vibrates when you blow across the top, as shown in the diagram at left. (It's a fun experiment, because of the surprisingly low and loud sound that results.)

Some small whistles are Helmholtz oscillators. The air in the body of a guitar acts almost like a Helmholtz resonator*. An ocarinais a slightly more complicated example, because for the higher notes it has several holes. Loudspeaker enclosures often use the Helmholtz resonance of the enclosure to boost the low frequency response. Here we analyse this oscillation, informally at first. Later, we derive the equation for the frequency of the Helmholtz resonance.

The vibration here is due to the 'springiness' of air: when you compress it, its pressure increases and it tends to expand back to its original volume. Consider a 'lump' of air at the neck of the bottle (shaded in the middle diagrams and in the animation below). The air jet can force this lump of air a little way down the neck, thereby compressing the air inside. That pressure now drives the 'lump' of air out but, when it gets to its original position, its momentum takes it on outside the body a small distance. This rarifies the air inside the body, which then sucks the 'lump' of air back in. It can thus vibrate like a mass on a spring (diagram at right). The jet of air from your lips is capable of deflecting alternately into the bottle and outside, and that provides the power to keep the oscillation

going.

Now let's get quantitative:

First of all, we'll assume that the wavelength of the sound produced is much longer than the dimensions of the resonator. For the bottles in the animation at the top of this page, the wavelengths are 180 and 74 cm respectively, so this approximation is pretty good, but it is worth checking whenever you start to describe something as a Helmholtz oscillator. The consequence of this approximation is that we can neglect pressure variations inside the volume of the container: the pressure oscillation will have the same phase everywhere inside the container.Let the air in the neck have an effective length L and cross sectional area S. Its mass is then SL times the density of air ρ. (Some complications about the effective length are discussed at the end of this page.) If this 'plug' of air descends a small distance x into the bottle, it compresses the air in the container so that the air that previously occupied volume V now has volume V − Sx. Consequently, the pressure of that air rises from atmospheric pressure PA to a higher value PA + p.

Now you might think that the pressure increase would just be proportional to the volume decrease. That would be the case if the compression happened so slowly that the temperature did not change. In vibrations that give rise to sound, however, the changes are fast and so the temperature rises on compression, giving a larger change in pressure. Technically they are adiabatic, meaning that heat has no time to move, and the resulting equation involves a constant γ, the ratio of specific heats, which is about 1.4 for air. (This is explained in an appendix.) As a result, the pressure change p produced by a small volume change ΔV is just

Now the mass m is moved by the difference in pressure between the top and bottom of the neck, i.e. a nett force pS, so we write Newton's law for the acceleration a:

substituting for F and m gives: So the restoring force is proportional to the displacement. This is the condition for Simple Harmonic Motion, and it has a frequency which is 1/2π times the square root of the constant of proportionality, so

Now the speed c of sound in air is determined by the density, the pressure and ratio of specific heats, so we can write:

Let's put in some numbers: for a 1 litre bottle, with S = 3 square centimetres and L = 5 centimetres, the frequency is 130 Hz, which is about the C below middle C. (See notes.) So the wavelength is 2.6 metres, which is much bigger than the bottle. This justifies, post hoc, the assumption made at the beginning of the derivation.

Complications involving the effective length

The first diagram on this page draws the 'plug' of air as though it were a cylinder that terminates neatly at either end of the neck of the bottle. This is oversimplified. In practice, an extra volume both inside and outside moves with the air in the neck – as suggested in the animation above. The extra length that should be added to the geometrical length of the neck is typically (and very approximately) of 0.6 times the radius at the outside end, and one radius at the inside end).

An example. Ra Inta made this example. He took a spherical Helmholtz resonator with a volume of 0.00292 m3 and a cylindrical neck with length 0.080 m and cross-sectional area 0.00083 m2. To excite it, he struck it with the palm of his hand and then released it. A microphone inside the resonator records the sound, which is shown in the oscillogram at left. You can see that the hand seals the resonator for rather less than 0.1 s, and that during this time the oscillations are weaker and of relatively high frequency.Once the hand is released, an oscillation is established, which gradually dies away as it loses energy through viscous and turbulent drag, and also by sound radiation. Close examination shows that the frequency rises slightly as the hand moves away from the open end, because this the hand restricts the solid angle available for radiation and thus increases the end effect (or end correction).The length of the neck is increased by one baffled and one unbaffled end effect, giving it an effective length of 0.105 m. With a speed of sound of 343 m/s, the expression above gives a resonant frequency of 90 Hz.

Sound of the resonator being slapped.

Waveform (top) and sound spectrum (the latter on a log-log scale) of the impact response of a

Helmholtz resonator.

Helmholtz resonances and guitars

* I said above that the air in the body of a guitar acts almost like a Helmholtz oscillator. This case is complicated because the body can swell a little when the air pressure rises inside – and also because the air 'in' the sound hole of the guitar has a geometry that is less easily visualised than that in the neck of a bottle. Indeed, in the case of the guitar body, the length of the plug of the air is approximately equal to the two 'end effects' at the end of a 'pipe' which is only a couple of mm thick. The end effects, however, are related to and of similar size to the radius of the hole, so the mass of air is substantial. The length of the end effect of a cylindrical pipe that opens onto an infinite, plane baffle is 0.85 times the radius of the pipe. Although the soundboard of a guitar is not infinite, one would expect a similar end effect, and so the effective length of the 'plug' of air would be about 1.7 times the radius of the hole. (Some makers increase this by fixing a short tube below the soundhole, with equal radius.)A couple of people have written asking how big the sound hole should be for a given instrument. Well, we can use the equation above to start to answer that question. However, the swelling of the body is important. This makes the 'spring' of the air rather softer, and so lowers the frequency. The purely Helmholtz resonance can be investigated by keeping the body volume constant. When measuring this, a common practice is to bury the guitar in sand, to impede the swelling or 'breathing' of the body. However, guitars are not usually played in this situation. So the Helmholtz calculation will give an overestimate of the frequency of resonance for a real, flexible body.Let's assume a circular sound hole with radius r, so S = πr2, and L = 1.7r as explained above. When we substitute into the equation for the Helmholtz frequency, using c = 340 m/s, we get:

Notice that we are using standard SI units: we have used the speed of sound in metres and seconds, so the volume must be in cubic metres and the frequency in Hertz, to give an answer in metres.It is more complicated when the tone holes are not circular, because the end effect is not equal to that of a circle with the same area. PhD student and luthier John McLennan is writing up a report of some measurements about this, which we'll post here soon.On guitar and violin family instruments, the Helmholtz (plus body) resonance is often near or a little below the frequency of the second lowest string, around D on a violin or G-A on a guitar. You can reduce or shift the Helmholtz frequency substantially by covering all or part of the hole with a

suitably shaped pieced of stiff cardboard. If you then play a note near the resonance and then slide the card so it alternately covers and reveals the hole, you'll clearly hear the effect of the resonance.Is the 0.85r effect reasonable? Ra Inta, who did a PhD on guitar acoustics in our lab, suggests an interesting demonstration:Damp the strings on your guitar so they don't vibrate (e.g. a handkerchief between strings and fingerboard). Hold the palm of one hand above the soundhole, and close to it. With a finger of your other hand, strike the soundboard a sharp blow near the soundhole and close to the 1st string. You will feel a pulse of air on the palm of your hand. The blow of your finger pushes the soundboard in and squeezes some air out of the body. Now move your hand gradually further away from the hole, and continue tapping with the finger. When do you cease to feel the movement of the air? This will give you a rough estimate of the length of the 'end effect' in the case of the sound hole.

Tuning the Helmholtz resonance

Among the publications of John McLennan, a PhD student in this lab, is an article in which he varies the Helmholtz resonance by varying the speed of sound.

McLennan, J.E. (2003) "A0 and A1 studies on the violin using CO2, He, and air/helium mixtures." Acustica, 89, 176-180.

Some pictures of historical Helmholtz resonators provided by Thomas B. Greenslade, Kenyon College, Ohio.