helmholtz resonance.pdf

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A Helmholtz resonator or Helmholtz oscillator is a container of gas (usually air) with an open hole (orneck or port). A volume of air in and near the open hole vibrates because of the 'springiness' of the airinside. 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 soundthat results.)

Some small whistles are Helmholtz oscillators. The air in the body of a guitar acts almost like aHelmholtz resonator*. An ocarina is a slightly more complicated example, because for the highernotes it has several holes. Loudspeaker enclosures often use the Helmholtz resonance of the enclosureto boost the low frequency response. Here we analyse this oscillation, informally at first. Later, wederive 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 ittends to expand back to its original volume. Consider a 'lump' of air at the neck of the bottle (shadedin the middle diagrams and in the animation below). The air jet can force this lump of air a little waydown 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. Thisrarifies the air inside the body, which then sucks the 'lump' of air back in. It can thus vibrate like amass on a spring (diagram at right). The jet of air from your lips is capable of deflecting alternatelyinto the bottle and outside, and that provides the power to keep the oscillation going.

Helmholtz Resonance http://www.phys.unsw.edu.au/jw/Helmholtz.html

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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 theresonator. For the bottles in the animation at the top of this page, the wavelengths are 180 and 74 cm respectively, sothis approximation is pretty good, but it is worth checking whenever you start to describe something as a Helmholtzoscillator. The consequence of this approximation is that we can neglect pressure variations inside the volume of thecontainer: 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 ofair ρ. (Some complications about the effective length are discussed at the end of this page.) If this 'plug' of airdescends a small distance x into the bottle, it compresses the air in the container so that the air that previouslyoccupied volume V now has volume V − Sx. Consequently, the pressure of that air rises from atmospheric pressurePA 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 thecase 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 constantγ, the ratio of specific heats, which is about 1.4 for air. (This is explained in an appendix.) As a result, the pressurechange 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 ithas a frequency which is 1/2π times the square root of the constant of proportionality, so


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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 is130 Hz, which is about the C below middle C. (See notes.) So the wavelength is 2.6 metres, which is much biggerthan 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 endof the neck of the bottle. This is oversimplified. In practice, an extra volume both inside and outside moves with theair in the neck – as suggested in the animation above. The extra length that should be added to the geometrical lengthof the neck is typically (and very approximately) of 0.6 times the radius at the outside end, and one radius at theinside end).

An example. Ra Inta made thisexample. He took a sphericalHelmholtz resonator with a volume

of 0.00292 m3 and a cylindricalneck with length 0.080 m and cross-

sectional area 0.00083 m2. Toexcite it, he struck it with the palmof his hand and then released it. Amicrophone inside the resonatorrecords the sound, which is shownin the oscillogram at left. You cansee that the hand seals the resonatorfor rather less than 0.1 s, and thatduring this time the oscillations areweaker and of relatively highfrequency.

Once the hand is released, anoscillation is established, whichgradually dies away as it losesenergy through viscous andturbulent drag, and also by soundradiation. Close examination showsthat the frequency rises slightly asthe hand moves away from the openend, because this the hand restrictsthe solid angle available forradiation and thus increases the endeffect (or end correction).

The length of the neck is increasedby one baffled and one unbaffledend effect, giving it an effectivelength of 0.105 m. With a speed ofsound of 343 m/s, the expressionabove gives a resonant frequency of90 Hz.

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


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Sound of theresonator being slapped.

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 complicatedbecause the body can swell a little when the air pressure rises inside – and also because the air 'in' the sound hole ofthe guitar has a geometry that is less easily visualised than that in the neck of a bottle. Indeed, in the case of the guitarbody, the length of the plug of the air is approximately equal to the two 'end effects' at the end of a 'pipe' which isonly a couple of mm thick. The end effects, however, are related to and of similar size to the radius of the hole, so themass of air is substantial. The length of the end effect of a cylindrical pipe that opens onto an infinite, plane baffle is0.85 times the radius of the pipe. Although the soundboard of a guitar is not infinite, one would expect a similar endeffect, and so the effective length of the 'plug' of air would be about 1.7 times the radius of the hole. (Some makersincrease 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 theequation 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 bykeeping the body volume constant. When measuring this, a common practice is to bury the guitar in sand, to impedethe swelling or 'breathing' of the body. However, guitars are not usually played in this situation. So the Helmholtzcalculation 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 intothe 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 volumemust 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 withthe 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 thefrequency of the second lowest string, around D on a violin or G-A on a guitar. You can reduce or shift the Helmholtzfrequency substantially by covering all or part of the hole with a suitably shaped pieced of stiff cardboard. If you thenplay a note near the resonance and then slide the card so it alternately covers and reveals the hole, you'll clearly hearthe effect of the resonance.

Is the 0.85r effect reasonable? Ra Inta, who did a PhD on guitar acoustics in our lab, suggests an interestingdemonstration:Damp the strings on your guitar so they don't vibrate (e.g. a handkerchief between strings and fingerboard). Hold thepalm of one hand above the soundhole, and close to it. With a finger of your other hand, strike the soundboard asharp blow near the soundhole and close to the 1st string. You will feel a pulse of air on the palm of your hand. Theblow of your finger pushes the soundboard in and squeezes some air out of the body. Now move your hand graduallyfurther 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


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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.

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helmholtz resonance.pdf

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