Guitar resonance and soundhole geometry – Part 5: HELMHOLTZ RESONANCE AND THE GUITAR

PART 5: HELMHOLTZ RESONANCE AND RADIATION FROM A SOUNDHOLE

This is the fifth in a series of nine posts summarising the results of an experiment I carried out during 2017 and 2018 to try to increase my understanding of resonance in acoustic guitars, and in particular how the design of soundholes could be improved.

This section continues to test the properties of the traditional round soundhole shape. The particular focus is on detecting and measuring the Helmholtz resonance response of the guitar shaped cavity.

We’ve seen that Helmholtz resonance (see Part 2) is the way the air contained in a cavity will “slosh” in and out of any hole in that cavity. It does so at a rate determined by the size of the cavity and the characteristics of the hole, such as its area and the effective length of its “neck”.

The Helmholtz equation tells us that there is a well-defined relationship between these quantities, suggesting that were we to detect HR happening in a guitar soundbox, it would show as a unique peak in the sound spectrum for each size of hole. We can even use a form of the Helmholtz equation to predict where HR peaks should show themselves in different positions along the frequency axis.

None of these peaks fits the HR prediction, which is that for each hole there should be a peak at the frequency unique to that hole. There is no evidence here that HR has been picked up by the microphone 80cm away.

But rather than rely on just looking at some spectra and finding nothing, luckily a more rigorous process is possible [1].

PREDICTING THE HELMHOLTZ FREQUENCY OF A GUITAR CAVITY

Before tackling the question of whether we should expect HR to be heard in the signal from a guitar cavity, we need to see what the prediction of the HR resonant peaks would be.

Gore and Gilet present a useful discussion of applying the Helmholtz equation to the guitar (Contemporary Acoustic Guitar – Design2-14). They derive what they disarmingly call a “fiddle factor”,α, an effective length figure based on soundhole radius rather than soundhole thickness that applies to a Dreadnought body specifically, and suggests that the Helmholtz equation for a guitar should be:

where:

  • c is the speed of sound in m/s
  • R is the soundhole radius in m
  • S is the soundhole area in m2
  • V is the cavity volume in m3
  • α is a factor derived from experimental data for a particular guitar shape. For a Dreadnought they measure this as 1.63[2]

Applying this version of the Helmholtz equation to the experimental cavity for four of the round holes gives the following prediction (T = 15.6 °C; RH = 53%; c = 340.4m/s):

SOUNDHOLEfH (Hz)
R35.7237
R45.4274
R50.1290
R54.9306

Figure 1: Prediction of Helmholtz peaks using the Gore equation

Here is the experimental result showing the spectral response of the series of round soundholes of different diameters relative to the closed hole response. To collect this data the microphone is 80cm above the soundhole (far field):

Figure 2: Overall spectral response of different round soundholes

By far the strongest radiation is in the range from 100Hz to 600Hz, with very little activity above that.

The next plot gets us closer to the range of frequencies (80 to 200Hz) where we would expect some HR activity to be visible, and now shows only four of the holes for more clarity. This is the far field response, with the microphone 80cm above the soundhole. The vertical dotted lines show the predicted HR frequencies for each hole size:

Figure 3: Main response for round soundholes with distant microphone showing expected Helmholtz frequencies (vertical lines)

As well as noting again the relentless similarity in the contours of the spectrum for each size of hole, it’s clear there is no sign of Helmholtz resonance in this data, otherwise the dotted vertical line in each colour would line up with a peak of the same colour.


DETECTING HELMHOLTZ RESONANCE– THE NEAR FIELD RESPONSE

If Helmholtz resonance is air “sloshing” in and out of the soundhole, maybe pressure differences produced by it can only be picked up by a microphone suspended closely over the hole.

The next graph shows the near field response for a number of soundholes, collected by a microphone 10mm directly above the soundhole:

Figure 4: Near field response of round soundholes (microphone very close to hole)

Figure 4shows the expected range if we are to get a glimpse of HR. The higher side of the double peak is between 160 and 180Hz, and none of the holes show any variation to the same pattern. 

The smaller of each twin, to the left of the main peak in each data set, is a different matter. You can see that each hole’s lesser peak is at a different frequency, and that the frequency drops as the hole gets smaller, just as HR theory predicts.

It looks promising, but let’s see how the data looks from a more analytical point of view. The second column shows the predicted Helmholtz frequency using the Gore equation. 

The third column shows the value measured from the spectrum shown in Figure 4.

SOUNDHOLEGORE PREDICTION fH (Hz)NEAREST PEAK fH (Hz)VARIATION FROM PREDICT (%)
R19.499NONEN/A
R25.9115NONEN/A
R35.7134NONEN/A
R41.1144150.8-4.5%
R43.1148150.8-2.1%
R45.4152152.3-0.4%
R50.1159152.34.4%
R54.9167155.36.9%

Figure 5: Analysis of near field peaks for round soundholes

It certainly isn’t perfect, but there is a level of agreement between prediction and measurement – the predicted values are within a range of +/- 7%, a reasonable error range as these things go.

The three smallest holes did not produce a measureable Helmholtz peak.

The agreement is certainly not perfect but still strongly suggestive that we have tracked down the HR, occurring as it does very close to the soundhole in the “neck” of the air cavity.


THE EDGE EFFECT

The Royal Society paper (see Part 1) makes the point that the oscillating airflow involved in Helmholtz resonance is stronger at the edge of a soundhole than in the middle for fluid dynamics reasons, which is why a larger perimeter : area ratio makes for a more efficient low frequency resonator.

The plot below supports this point. The red line shows the response of the 50.1mm round soundhole at its edge, and the blue line at its centre:

Figure 6: Demonstration of edge effect in a round soundhole

The next graph shows a comparison of the activity at the centre and the edge of the soundhole in arbitrary units, derived from the graph above, measured very close to the hole:

Figure 7: Comparison of activity at the centre and edge of a soundhole (arbitrary units)

Lack of a suitable small microphone unfortunately meant I could not look into this in more detail across the diameter of the hole, but it does support the idea that the edge of a soundhole shows more activity than the centre.


HR IS DETECTABLE AT THE SOUNDHOLE BUT NOT FROM A DISTANCE

So it turns out that for a guitar HR is only detectable if the sensor microphone is placed directly over the top of the soundhole as close to it as possible without blocking the hole. HR, in other words, does not contribute significantly to the volume of sound projected from a guitar soundhole. A listener at even a small distance from the instrument will not hear it at all.

Then why all the fuss about HR as a formant of a guitar’s overall sound?

The importance of HR is that it is the resonant frequency of the “spring” between the top and back plates, and its natural frequency will help to shape the response of the soundboard and back, through coupling (see Part 1 for the mass/spring model).

It certainly seems that a guitar designer can enlist soundhole size into the range of variables that can be drawn on to craft the overall tone of an instrument. The other main variables are the volume of the air cavity and the elasticity and mass of the top and back plates.

Every guitar top has the same vibrational modes available to it (see Appendix 8), the main ones being the top monopole, the cross dipole, the long dipole, and the cross tripole, each having a higher frequency than the last. It is the existence of the same set of modes in every guitar that makes up what we immediately recognise as the guitar sound.[3]

The violin family has its own set of modes, and hence have their own distinctive sound when plucked, and certainly when bowed.

This is an easy principle to state, but of course quite complex in practice when considered alongside the need for strength and robustness to resist long-term string loading and daily wear and tear.

SUMMARY OF HELMHOLTZ RESONANCE IN THE GUITAR:

  • HR is detectable in soundhole activity, but only very close to the soundhole
  • HR directly contributes very little to the audible sound radiated from a guitar – that’s not why it is important
  • The importance of the air-cavity is its role in coupling the soundboard and the back of an instrument; a better understanding of just how it can be varied offers a way of tuning the soundbox response for better performance
  • The Helmholtz equation, suitably modified, makes reasonable predictions for the resonant frequency of an air-cavity with a soundhole of particular size
  • The air movement into and out of the guitar soundbox as a result of HR is more marked at the edge of the soundhole than in its centre, supporting the idea that a soundhole with a large perimeter to its area will encourage HR
  • While it won’t be heard directly itself, more efficient HR will form a better “connecting spring” between soundbox elements

WHAT’S NEXT?

Having detected the presence of Helmholtz resonance with a good agreement to the theory, we also discovered that it contributes directly very little to the sound level projected by a guitar soundhole.

We know that about 30% of the sound coming from a guitar comes from the soundhole. So if not Helmholtz resonance, what is the source of this sound?


FOOTNOTES

[1]You might choose to miss the next section if you’re already convinced, and want to move on to why HR has such importance in this experiment.

[2]How they arrive at this figure is a little complex for this discussion, but I have used it as a starting approximation for the Parlour-size cavity (against their advice).

[3]Appendix 8gives a very simple overview. Better still, there is an excellent discussion of these modes in Gore and Gilet (Contemporary Acoustic GuitarDesign, 1-75, 1-82). The important point is that different makes and models of guitar have their distinctive sounds as a result of a different balance between the same available set of vibrational modes. 

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s