Guitar resonance and soundhole geometry – Part 7: HIGH PERIMETER TO AREA RATIO (P:A) SOUNDSLOTS

PART 7: COMPARISON OF HIGH P:A SOUNDSLOTS WITH ROUND SOUNDHOLES

This is the seventh 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.

So far this series, Parts 1 to 6 have looked into the function of soundholes in general and the performance of the traditional round guitar soundhole. 

Part 5 shows how Helmholtz resonance is detectable in the response of my rigidly-confined airbody, excited by a signal from the loudspeaker sited in the lower bout. 

For this parlour-sized cavity the Helmholtz resonant frequency was at around 150Hz, and varied with soundhole size as theory predicts. The measured values coincide quite closely with predictions made using the equation provided by Gore & Gilet (Contemporary Acoustic Guitar – Design 2-14)

The strongest low frequency resonant response of an airbody is the Helmholtz mode, pictured as a plug of air in the region of the soundhole pushed and pulled back and forth as the airbody inside the soundbox expands and contracts at its fundamental resonant frequency

The most important experimental findings so far have been:

  • Blocking off the soundhole of a guitar reduces the emission of sound by about 30% depending on the instrument (the other 70% is direct transmission of sound to the listener by the soundboard vibration)
  • The larger a soundhole is the better it is at emitting soundwaves, and the more complex the tone it can produce – it has better acoustic conductance
  • Helmholtz resonance is not detectable except very close to the soundhole, so does not directly contribute to the sound of the guitar
  • The oscillating airflow associated with Helmholtz resonance is strongest at the edge of a soundhole and weakest in the centre (the edge effect)

My interpretation of these results is:

  • While the Helmholtz resonant response can only be detected very close to the soundhole, the three spring model indicates that the airbody in a soundbox acts as an elastic connection between the top and back panels. Helmholtz resonance, although not audible, therefore plays an important role in shaping the overall sound qualities of the instrument (loudness, timbre)
  • Much of the sound coming from the soundhole is produced by complex processes inside the soundbox from soundwaves pumped into the box by the vibrating soundboard
  • These internal processes include standing wave resonances, reflection, interference, and diffraction as the sound finds its way out through the soundhole

From Part 7 onwards, the emphasis shifts from round soundholes and detecting Helmholtz resonance to a direct comparison between round soundholes and soundslots.

The question now becomes: will a longer skinnier soundslot (high perimeter : area ratio) be a more efficient radiator than a shorter fatter one, as the Royal Society paper (see Part 1) asserts?

My experiments have shown that with Helmholtz resonance there is indeed higher activity at the edges of a soundhole than in the centre. The Royal Society paper links this to their finding in violins that the overall efficiency of slots (f-holes) is greater than round holes because they have a greater length of edge (perimeter) for their area and thus more scope for the edge effect to come into play.

In order to see if the same effect measured in violins happens in guitars as well, I repeated the original soundhole resonance experiment using a set of rectangular soundslots, all with the same area but with increasing perimeter to area (P:A) ratio – that is, longer and narrower.

These all had the same area as a round 45.4mm (R45.4) radius soundhole for comparison. [1]

SOUNDSLOT GEOMETRY

The soundslots used in the experiment were the following sizes:

SLOT NUMBERDIMENSIONS (mm)AREA (mm2)P:A RATIO
SLOT 3 (RECTANGLE)150 x 436450 0.0598
SLOT 4 (RECTANGLE)170 x 3864600.0644
SLOT 5 (RECTANGLE)190 x 3464600.0693
SLOT 6 (RECTANGLE)210 x 3165100.0740
SLOT 6A (RECTANGLE)250 x 2665000.0849
SLOT 6B (FLAT C SHAPE) 300 x 21.765100.0959
SLOT 7 (FLAT C SHAPE)341 x 1964110.1122
SLOT 8 (FLAT C SHAPE)386 x 1765620.1228
DOUBLE SLOT 1 (SHALLOW S)[380 x 1764600.1263
 MEAN AREA:6480 +/- 1.2% 

The error margin of +/- 1.2% in the slot areas could be improved, but is reasonable for drawing general conclusions.

The flat C shape for slots 6B to 8 was made to fit across the upper bout of the Parlour size cavity. Each of these C shapes added a curved end to a straight slot.

The S shape for double slot 1 fits around the curve of the upper bout edges and is inset by 15mm from the edge of the cavity.


The graph below shows the overall response of the slots. Again, there was little activity above 1,500Hz, so this shows the plot from 0 to 1,000Hz.

Figure 1: Response of soundslots 

The overall form of this graph is similar to that for round soundholes (see Figure 2below), with a strong response between 150 and 300Hz and a scattering of low-order high frequency responses. [2]

Figure 2: Response of round soundholes – note the overall similarity to Figure 1

Both holes and slots each follow the same spectral pattern no matter their size. For both, greater hole area allows stronger radiated power.

From the data represented in Figure 1, the overall strength of the sound emission from each slot can be calculated. 

(Remember that all data is relative to the closed soundbox, so is unfortunately not an absolute measure that can be used to calculate the performance of holes or slots from scratch.)

Figure 3 belowshows how the radiative power of the slots varies with increasing perimeter to area.

Figure 3: Soundslot relative radiated power by P:A ratio

Figure 4presents the same data, showing how the slots compare to the round soundhole (radius 45.4mm, purple data point):

There is clearly a relationship between the radiating power of a soundslot and its P:A ratio. A linear regression fit gives an R2value of 0.953 – a good result.

Figure 4: Comparison of round soundhole to soundslots of the same area by P:A ratio

Figure 5 below gives a clearer view of the total relative radiated power of the slots.

It’s clear that below a P:A ratio of 0.08 the slots have little advantage over the round hole. However as P:A becomes greater than 0.1 there is a large increase in sound conductivity as found in the Royal Society paper.

Figure 5: Comparison of radiated power 50 to 1,000Hz for soundslots[3]

The high P:A ratio slots 7 and 8 show an improvement in conductivity of 60% over the equivalent round hole (R45.4).

But what about the quality of sound produced by the slots? Dividing the radiated power into octave bands shows a consistent increase in power in the Octave 3 band (165 to 330Hz). Most of the increase in power of the high P:A slots is in this octave covering most of the guitar’s range.

Figure 6: Soundslot radiated power by octave. P:A ratio increases from left to right.

Here is the same data expressed as a percentage of total radiation for each slot, again broken down by octave frequency band:

 Figure 7: Soundslot radiated power by octave band as a percentage of total power

This shows a decrease in bass response as the P:A ratio increases, the difference being made up by a similar increase in the higher octave band (covering the notes E5to C6– guitar string 1 is E4).


HIGH P:A SOUNDSLOT CONCLUSIONS

  • This data provides good support for the assertion in the Royal Society paper that a soundhole with a long perimeter for its area will be a more efficient overall radiator that one with a lower P:A ratio, and not just for air-cavity resonance
  • Higher P:A ratios show a decrease in bass response and an increase in treble response
  • To provide a worthwhile advantage over a round soundhole, the P:A ratio should be greater than 0.1

FOOTNOTES

[1]It turns out that fitting soundslots onto the Parlour sized soundboard without compromising strength means a restriction in their total area to this smaller sized equivalent round hole size

[2]Note also the 430Hz peak in Figure 1, which I suggested in Part 6 is a standing wave resonance set up along the 380cm length of the cavity.

[3]Slot6B was slightly under size

Guitar resonance and soundhole geometry – Part 6: THE SOURCE OF SOUND FROM A GUITAR SOUNDHOLE

PART 6: THE SOURCE OF SOUND FROM A GUITAR SOUNDHOLE

This is the sixth 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 focuses on completing the description of the sound projected by a round guitar soundhole. We discovered that the Helmholtz resonance signal could only be detected very close to the soundhole.

In Part 5 we confirmed the presence of Helmholtz resonance in the sound projected from the air cavity of a guitar, but found it is not radiated at all strongly.

HR has a large influence on the sound of a guitar, but not because anybody can hear it directly. This is a subtle point with consequences not intuitively easy to grasp.

We know now that changing the size of a soundhole produces small changes in the frequency of the guitar air-cavity’s HR. The importance of this is that it is the air-cavity that forms the main connection between the soundboard and the back of the guitar (see Part 1for the “three spring” model). 

Without an air-cavity, or with the soundhole blocked, there is no connection and the side/back and soundboard can vibrate in isolation at their own natural frequencies.

Add in the air-cavity by unblocking the soundhole, and now we have not two isolated resonators but three connected ones. [1]

So the importance of knowing about HR and its connection to soundhole size is that it offers a way of tuning the soundbox resonances. In practice a guitar maker can shift soundbox resonances off scale note frequencies if necessary by modifying the soundhole. [2]

Gore and Gilet describe tweaking the response of a guitar (Contemporary Acoustic Guitar 2-15) suffering from this flaw by subtle changes to, among other things, the soundhole size. Changing the air-cavity resonance can alter the coupled top and back resonances enough to move peaks off scale notes.

In this experiment I have not been concerned with questions of impedance [3]or the effect of coupling oscillators together, or top vibrational modes. In fact, I have designed the experiment to eliminate these complications as far as possible. Consequently my results have nothing to say about this aspect of soundhole size.

THE ORIGIN OF THE FAR FIELD SOUND

Figure 1: Partial spectrum of the far-field sound from the rigid guitar air cavity

The broad spectrum sound picked up by the far-field microphone of course originates from the chirp signal fed into the cavity by the loudspeaker.

We now know that we can’t detect the “sloshing in and out” Helmholtz response this far from the soundhole, so what in fact are we looking at in this spectrum?

What gets picked up by the microphone comes via two pathways:

PATHWAY 1:by a direct pathway from the loudspeaker to the microphone without passing through the cavity; and

PATHWAY 2:through the cavity and out of the soundhole to the microphone

Pathway 1is of no interest in the experiment, and is removed from the data by subtracting the closed soundhole signal from every subsequent soundhole spectrum.

Pathway 2is the important one. What happens inside the cavity is very complex, but two broad processes – both mediated by the soundhole – can be distinguished from each other:

  • resonances set up inside the cavity which show as frequency peaks in the spectrum; and
  • complex reflection, interference and diffraction processes (call them RID) that do not owe anything to resonance, which show up as the broadband background in the spectrum for each soundhole.

The resonance processes are very complicated, but for a rigid cavity are in principle not hard to picture – all are variations on a theme of waves reflecting between surfaces and interfering with each other to set up standing waves[4]of a frequency determined by the path-length allowed by the cavity walls (known to acousticians as “room modes”) [5].

Figure 2: Standing wave resonances between reflecting plates (room modes)

The waves you see here are the result of a sound wave of frequency f reflecting backwards and forward between the two walls. The formation of standing waves is too complex to go into here, but there is some good video material available on YouTube that will explain. The main point is that you can see the distance between the walls selects out particular frequencies at which standing waves form.

One requirement for a simple standing-wave resonance to happen is that there be reflecting surfaces parallel to each other and square to the line of travel of the soundwave. Like the resonances in a guitar string, the closed ended air resonances have a node at each reflecting surface.

Due to the curved shape of a guitar body, this limits the number of such resonances that are likely. One strong suspect, though, would be the two ends of the cavity, parallel to each other and 39cm apart.

A quick calculation [6], treating this pathway as a closed-ended pipe, gives a standing wave resonance (n = 1) at about 440Hz.

Figure 3: Actual resonances in the rigid-walled guitar soundbox

Figure 3above does show a strong peak at 430Hz, which supports (but by no means proves) this idea.

Interestingly, this peak is visible in all the experimental runs done with this cavity. Some more experimental work is needed to try and pin down where all the peaks in Figure 3originate.

Keep in mind that these kinds of processes (including those discussed below) only account for about 30% of the sound put out by a real guitar. The rest comes from air being moved by the vibration of primarily the soundboard, but also the sides and back of the soundbox. Those processes aren’t part of these experiments.

RESONANCE / INTERFERENCE / DIFFRACTION PROCESSES

Due to their complexity, the RID processes are near impossible to visualize. In cases where no particular frequency is singled out for resonance in the cavity, the signals fed in bounce off the cavity walls, travel through each other, and eventually find their way out through the soundhole. 

The best I can offer to help visualize this is “evidence” derived from a virtual ripple tank, but this only gives solutions in 2 dimensions rather than 3, and only at one frequency at a time. It does however give some insight into what is happening inside a guitar soundbox when soundwaves travel through it without triggering any resonances.

The virtual ripple tank calculates the progress of soundwaves injected into a simulated guitar soundbox by a source representing the vibration of the bridge in a real guitar. It will only do this for one frequency at a time, so the real picture will be enormously more complex than the images here show.

The first picture shows the simulated soundbox with a soundhole in it:

Figure 4: The virtual ripple tank cavity (a 2d section through a 3d soundbox)

The next picture shows the progress of a soundwave with a wavelength about the size of the box’s depth, just after it has been projected downwards into the cavity by the bridge area of the soundboard vibrating up and down in response to the strings. Red and green represent pressure crests and troughs in the wave.

Figure 5: The soundwave enters the cavity from the bridge area, as it would in a real instrument

In the picture above the first crest (in red) is just beginning to reflect off the back of the box.

The next picture is a little time later after the wave has “explored” the confines of the cavity, bouncing back and forth, and is beginning to find its way out of the soundhole.

Figure 6: The soundwave begins to find its way out of the soundhole after “exploring” the cavity

The picture above shows what happens after the complex set of reflection, interference, and diffraction[7]processes have stabilized into a pattern. The sound coming from the soundhole is about the same wavelength as the original signal, but does not emerge strongly or coherently at this low a frequency.

The next picture shows a higher frequency, shorter wavelength signal being fed in. In this case, two complete wavelengths fit into the vertical box dimension.

Figure 7: A higher frequency wave is injected into the soundbox by the movement of the soundboard and begins its exploration of the cavity

After some time the complex wave interactions again form the stable pattern shown below:

Figure 8: The higher frequency wave sets up a stable pattern after it exits through the soundhole

The final example shows the pattern set up by a high frequency wave.[8]

The radiated sound is much more coherent than for the longer wavelength in the first example.

Figure 9: High frequency wave pattern[9]

In a real guitar, of course, there would be many frequencies entering the cavity simultaneously rather than just one, making for a mind-bendingly complicated and constantly shifting pattern inside the cavity and emerging from the soundhole.

Though very limited, this approach does shed some light on the question of where soundhole radiation comes from[10]. It is the end result of a complex series of interactions within the soundbox in which soundwaves of many different frequencies, generated by vibration of the soundboard, reflect off surfaces, interfere with each other and finally escape out of the soundhole. Some of these frequencies produce resonances inside the air-cavity as well.

SOUNDHOLE SOUND PROJECTION CONCLUSIONS

  • In Part 5we identified the presence of Helmholtz resonance in the response of a guitar soundbox, and found that in itself it contributes very little to the sound you hear coming from a guitar. However, it is important because the resonance acts as the “spring” connecting the vibrating soundboard to the rest of the soundbox
  • We then asked where the sound coming from a soundhole that we do hear originates. While not giving an entirely satisfactory answer, it seems that internal processes of reflection, interference, and diffraction of the soundwaves pumped intothe box by the vibrating soundboard produce most of the sound we actually can hear.
  • A point worth noting is that, as you can see clearly in Figure9, diffraction produces complex interference fringes in the sound coming from a guitar soundhole, implying that your position in front of a guitar will determine to some extent how you hear it. In the “dark” fringes the volume will be reduced at that frequency, for example.

FOOTNOTES

[1]Coupling two resonators like this with a third produces a rather strange effect. In isolation the first resonator might have a natural frequency of 100Hz, and the second 120Hz. Couple them together and these frequencies “repel” each other: resonator 1 might now be at something like 98Hz, and the second at 122Hz. The theory behind this is complicated so I won’t address it here.

[2]Gore and Gilet (Contemporary Acoustic Guitar,2-15)) have a sophisticated analysis of this question. Their research shows that one requirement for a guitar to respond consistently across the spectrum is that no major body resonance should occur at the same frequency as any note of the musical scale. The reason is that if a body resonance occurs on the note A2110Hz, for example, the energy of the string plucked to play A2will flow rapidly out of the string because of the low impedance of the guitar body at this frequency. The result will be a loud, short, thunk of a response rather than a sustained note.

[3]Impedance is the measure of how easily energy can make its way through the interface between two different media.

[4]The Helmholtz resonance as we have seen is a special case where the whole air-cavity “sloshes” in and out of the cavity through the soundhole.

[5]This comes from the website http://www.sengpielaudio.com/StandingWaves.htm

[6]Using the equation  where c is the speed of sound and L is the cavity length

[7]Note how the sound emerging from the soundhole spreads out like a fan – this wave process is called “diffraction” and partially accounts for the way a guitar can be heard from other angles than straight in front of the soundhole. Going back to the Royal Society paper, it also helps explain why the low frequency sound from a violin is described as “monopolar”, as it is similar to how a wave would travel outwards from a point source.

[8]Unfortunately the Virtual Rippletanksoftware doesn’t allow for realistic frequency calculations for real soundwaves. For example, the first wave in this series had a wavelength equal to the box depth, which in a real guitar would be about 0.09m. 

A real soundwave travelling at 340m/s would have a frequency of 3,800Hz – not a realistic figure as we have seen from the experimental evidence that shows the action takes place at between 100 and 600Hz. The software unfortunately doesn’t allow one to choose a longer wavelength than this.

[9]Notice in the column directly underneath the bridge there is a standing wave set up, shown by the regular pattern of the crests and troughs. In a real guitar cavity the resonant frequency would be about 1900Hz.

[10]Remember that soundhole radiation represents only about 30% of the sound coming from a guitar. The rest is made up of soundwaves generated by the complex vibrations in the soundboard.

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. 

Guitar resonance and soundhole geometry – Part 4: THE EFFECT OF SOUNDHOLE SIZE

PART 4: HOW THE SIZE OF A SOUNDHOLE DETERMINES THE WAY A GUITAR AIRBODY EMITS LOW FREQUENCY SOUND FROM 90 TO 1000Hz (E1to E6)

This is the fourth 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 tests the properties of the traditional round guitar soundhole shape. 

The central question here is how the radius of a guitar soundhole affects the loudness and tone of the sound that comes from the soundhole of a guitar.

As I show later in Part 6, airbody resonance is only one source of the sound that emerges from a guitar soundhole.

Part 3 discusses the experimental method in detail.

Figure 1 below shows the resonant cavity used in the experiment. It is the size and shape of one of my Parlour size instruments.

A stiff heavy top was added, including a box for a small speaker in the lower bout and a hole for the drop-in soundholes in the upper.

Figure 1: A view of the resonant cavity used in the experiment

Figure 2: The cavity with the top added – the box contains the speaker used to excite the cavity, and you can see where the different-sized drop in soundholes go

A digital microphone captured the sound coming from each hole for analysis. An acoustic hood was used to damp out stray resonances from the room (see Appendix 4 for an evaluation of its effectiveness).

Figure 3: the experimental setup showing two out of three sides of the acoustic hood

Lead weights were used to further damp any vibrational response in the cavity top and speaker box.

The method I used was:

  • The stimulating signal was a 60 second sine-wave “chirp” rising from 50 to 1,000Hz over a period of 60sec
  • This signal was fed into the resonant cavity from a loudspeaker embedded in the lower bout of the cavity top plate
  • The microphone was held 80cm above the soundhole
  • The signal coming from the soundhole was recorded using Audacitysoftware, then analyzed with the software’s Analyse / Plot spectrumfacility
  • A series of drop-in soundholes allowed me to change the soundhole size

An important part of the method was to start each experimental run with a measurement taken with the soundhole completely blocked. This provided a baseline response and was subtracted from the response of each soundhole.

The downside of this is that it makes all measurements relative to the closed response. However, that is acceptable given that relative responses allow a clear analysis of the differences between soundholes. It also has the advantage of cancelling out the effects of speaker response, since the data used for each soundhole was its response above or below the closed box baseline.

Keep in mind that these results throw light on how the low frequency – bass – tones change as the radius of the soundhole increases. These are the frequencies are produced by the airbody inside the soundbox resonating. 

Having got that out of the way, let’s look at some results.


RESULTS FOR ROUND SOUNDHOLES

Here is the signal spectrum for a series of soundholes covering the range from 35.7 to 54.9mm in radius. The graph shows the whole signal range from 50Hz to 1,000Hz.

Figure 4: Spectral response of round soundholes to chirp signal [1]

The radius of each hole is shown in the legend (R35.7 is a 35.7mm radius).

This shows that the cavity/soundhole combinations act in a very similar way to each other. No combination is capable of transmitting a complete version of the stimulating signal (50 to 1,000Hz sine wave), showing that each selects some frequencies to project while being unable to project others. This is the effect of the resonant frequencies of each cavity/hole combination.

The peaks in the graph are where each cavity/hole combination resonates in response to the signal. The resonant response is very similar for each combination.

By far the most sound energy projected is in the 100 – 260Hz range (notes G#2to C– fret 4 on string 6 to fret 8 on string 1, most of the range of a guitar). Given the evolution of guitar design, this is likely not a coincidence.

The area under each line is a measure of the power of the sound, so it’s clear that the largest soundhole (the green line R54.9) is the best emitter of the five holes.

The next graph zooms in on the 100 – 300Hz range.

Figure 5: View of main resonant peak for round soundhole response

The main difference between the five holes is that the larger ones transmit the signal more strongly than the smaller ones.

This is clearer in the next graph, which simply compares the largest hole with the smallest:

Figure 6: Response of radius 35.7mm and 54.9mm round soundholes to chirp signal

One difference is the larger hole’s activity between 500 and 550Hz, although this peak is not at all strong.

At first sight these results might be puzzling. Many acoustic guitar theory discussions maintain that smaller soundholes give a guitar better bass response, and larger ones better treble. This is not at all obvious from a glance at the data.

However, I will show that even though the spectral contour for all soundholes is the same, the proportionof energy radiated by smaller or larger holes does fit with the theory.

The next graph compares the total (relative)[2]power radiated by each cavity/hole combination:

Figure 7: Relative power radiated from a range of round soundholes over range 50 to 1,000Hz

So it’s clear that the larger a soundhole is, the more effectively it radiates sound from the air cavity – the relationship of intensity to area is linear with a very good fit (R2= 0.9965).

There are some other differences as well that become clear if we look at how well the soundholes radiate at different frequencies.

Given that an acoustic guitar’s fretboard goes from E2(82.4Hz) to E5(659.3Hz) it makes sense to divide the 50 to 1,000Hz spectrum up this way:

OCTAVEFREQUENCY BAND (Hz)GUITAR FRETBOARD RANGE
282 – 165STRING 6 FRET 0 TO12
3165 – 330STRING 4 FRET 2 TO 14
4330 – 660STRING1 FRET 0 TO 12
5660 – 1319STRING 1 ABOVE FRET 12

Here again is the set of responses from different radius round soundholes:

Figure 8: Response of round soundholes of different sizes

While the round soundhole responses show the same contour as each other, they actually radiate different proportionsof their total sound power in different octaves of the spectrum.

The next graph shows this clearly for the three spectral bands I chose (they are the bands based on the note E2at 82.4Hz – the frequency of the bass E string on the guitar – and two higher octaves covering the acoustic guitar fretboard):

Figure 9: Round soundhole radiation by octave for different radius soundholes

Notice how poorly all the soundholes radiate in the 660–1,000Hz band (the pale blue bar is hardly visible).

The next graph shows the same data but by percentage of the total for each frequency band:

Figure 10: Soundhole radiation by octave (% of total for each hole)

You can see that the bass response (in purple) of the 35.7mm hole is stronger than the others as a percentage of the total output– but of course its total output is lower than the rest.

If this behavior is the same in an actual guitar (which, remember, is not a rigid box like the one used to collect this data), soundholes become better at radiating in the 163 – 330Hz octave (Octave 3) the larger they get. Their low frequency response does in fact seem to tail off as a percentage of the total, as conventional wisdom suggests, even as their total radiating power improves.


WHY GUITARS HAVE 50mm RADIUS ROUND SOUNDHOLES

So the bigger the soundhole, the more strongly it radiates, but there will be an upper size limit at which the structural integrity of the soundboard will be compromised and the vibrating soundboard surface area cut into. Making bigger soundholes would quickly begin to have an effect on the top’s vibrational modes as well, particularly the long dipole (see Appendix 8). 50mm is a good compromise.

More prosaically, a 50mm radius hole also makes it easier to get a hand in to make repairs than a smaller one. [3]

So generations of guitar makers have found a good size for round soundholes – but as is always the case there’s more to be said.

PART 4 SUMMARY

To sum up so far:

  • Round guitar soundholes radiate most strongly in the band from 100 to 600Hz, and hardly at all in the higher frequencies
  • All soundhole/cavity combinations select particular frequency bands that they are able to radiate strongly – the are unable to respond at any but there resonant frequencies
  • The greater the area of a round soundhole, the more effective it is at radiating sound – the relationship between radiative power and area is linear (double the area gives double the projection)
  • The greater the radius, the better the sound radiation becomes in Octave 3, central to the scale range of the guitar
  • At this point we don’t know if these conclusions hold for a real guitar soundbox, which is not rigid – this needs to be checked (see Part 10)

[1]CHIRP 12 in the graph title refers to the experimental run number using the chirp signal

[2]Remember that all readings are relative to the blocked hole response

[3]I used to employ my granddaughter with her small hands to help install pickups in my guitars because I was using 45mm radius soundholes at the time, thinking they would give better bass response.

Guitar resonance and soundhole geometry – Part 3: MY SOUNDHOLE EXPERIMENTS

PART 3: THE SOUNDHOLE EXPERIMENTS

This is the third 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 describes the technical aspects of my experiments, and can be skipped by readers not interested in this.

The questions I ask in these experiments on guitar soundhole performance are:

o   Are there differences in the way a resonant cavity responds to the same stimulating signal when the soundhole size is changed?

o   What part does Helmholtz resonance actually play in the sound radiating from a guitar soundhole? Can soundhole area be used to control Helmholtz frequencies?

o   What is the source of the sound that radiates from the soundhole of a guitar?

o   Do soundslots (holes with a high perimeter to area ratio) perform better than round soundholes of the same area?

o   If slots are preferable to round holes, as the Royal Society paprer claims (see Part 1), designing a single long and narrow soundslot with area equivalent to the traditional 50mm radius soundhole is difficult without compromising soundboard strength. Is there any disadvantage in dividing the slot into two segments?

o   How well do experimental results derived from a rigid cavity translate to a real, elastic guitar soundbox?


THE EXPERIMENTAL RESONANT CAVITY

The experimental method uses a cavity made from sheet polycarbonate bent around the mould used for my Parlour guitar design. This cavity sits inside an outer box with the gap filled with sand to help damp out any vibration, since I am interested only in the response of the air-cavity as it is shaped by the soundhole. The sides and bottom of the containment box are 16mm MDF.


Figure 1: The rigid-walled resonant cavity

In operation the lower bout is topped with 12mm thick plywood, which has a speaker mounted in a box at roughly the position of the bridge. The upper bout is removable, and the square hole allows different size “soundholes” to be fitted and changed. This upper bout is held in place with thumbscrews to help damp out vibration, and heavy lead masses further restrict any vibrational response.

Figure 2: Resonant cavity with top and speaker box fitted

The next picture shows the operating setup. The lead weights (1.3kg each) are there to damp out any resonant response from the speaker box and cavity containment, allowing only the airbody to respond to the input signal. You can see one of the test soundholes in place – in practice I tried to overlap the corners of each soundhole with the lead weights.

Figure 3: Lead weights used to damp vibration in the top and the speakerbox – note the drop-in soundhole.

THE ACOUSTIC HOOD

This setup is placed inside a hood lined with acoustic foam to minimize room resonances that might otherwise affect the results. [1]

Figure 4: Ready to record (the near side of the hood has been removed for clarity)

You can see the microphone at the top of the sound hood

ESTABLISHING A BASELINE FOR SOUNDHOLE RESPONSE

The first reading for each measurement run begins with the hole blocked completely to establish a baseline, then “opening up” holes and measuring the change in response of the system relative to the closed box. The closed box signal is subtracted from the signal recorded for each soundhole.

Using the closed box signal as the measurement baseline also subtracts out the effect of any direct transmission of sound from the speaker box to the microphone.

CHIRP SIGNAL USED TO EXCITE THE CAVITY

The signal used to excite the airbody is a “chirp” generated by the software package Audacity. This is like a rising siren note starting on 50Hz and ending on 1000Hz. It is a pure sine wave signal with a 60 second duration, fed into the box from an iPod Classic through a small power amplifier. (Speaker and amplifier details can be found in Appendix 3)[2]

Audio file 1: The chirp signal used to excite the airbody

As you play this audio file, you may notice that your speaker or headphones respond better at some frequencies than others, so the sound seems to vary in level as it goes up the slide.

This is one of the problems in doing my experiment, because of course the speaker I used was just as incapable of reproducing the signal accurately at all frequencies as any other. No speaker I know of has an absolutely flat frequency response.

This is the reason why my experiment design relies on the closed soundbox signal as its baseline. The microphone at the top of the sound hood will pick up the signal from the soundbox, but also the signal that goes directly to the microphone from the speaker box. This “closed box” signal is subtracted from every open sound hole’s response, so that the final data set shows how much above or below the baseline signal the soundhole response is. This goes a long way towards erasing the speaker response as a component in the experiment.

The downside to this approach is that each sound hole’s response is relative, and can only be used comparatively – but that’s okay, because it’s comparisons I’m interested in.

DETECTING AND RECORDING THE SIGNAL

The soundbox response is picked up 80cm above the soundhole by a digital microphone (see Appendix 3) linked to a MacBook computer. Audacity is used to record the response signal at a sampling rate of 96kHz.[3]

Each run is repeated three times and the average used in analysis. 

Each run starts by recording the background noise for 5 seconds, followed by 60 seconds of soundhole signal.

Air temperature and relative humidity are noted because it is important to calculate the speed of sound for predicting the Helmholtz resonance response.

ANALYSING THE SIGNAL

Next, Audacity does a fast Fourier transform (Analyse / Plot spectrum) to produce a spectrum of the radiated sound. This is exported at a 1.5Hz resolution text file for each run, which can then be imported into Excel for analysis and graphing. 

Figure 5: typical soundhole response (Audacity Analyse/Plot spectrum function)

Once the data is in Excel, it is analysed this way:

  • Data sets (initially 0 to 30kHz) are trimmed to cover the range 0 to 5kHz. This culling of data makes it easier for Excel to operate, and there is little if any radiated signal above 2kHz anyway.
  • Because it is digitally recorded, the sound levels are in deciBels (dB), and are negative values. This is because digital recorders are often used for producing music, and aim to avoid digital clipping. They set 0dB as their upper limit.
  • The three sets of response data for each run are averaged.
  • The minimum level each averaged data set is subtracted at each measured frequency – this converts the readings to equivalently spaced positive dB values. But of course also disqualifies the data from any claim to being absolute. Relative readings are okay in this experiment because I am only interested in comparing soundhole responses to see if slots perform better than round holes.
  • The positive levels (β) are then converted from dB levels into radiated power readings in W/m2 so they can be added and subtracted validly   (II010(β/10))
  • Because the closed box response is the baseline (it also contains speaker response information and any direct transmission of sound from the speaker box to the microphone), it is subtracted from each data set to show the response of each hole or slot relative to each other and the closed box.
  • An analogue for total radiated power (again relative to the closed box) is found by adding up the radiated power calculated for each soundhole for each frequency band (resolution is 1.46Hz), essentially an integration across the spectrum.

After this analysis it is possible to plot graphs showing:

  • The spectral response of each soundhole
  • The total radiated power (relative to the closed box) across the spectrum, or any section of it, for each soundhole

This allowed me to understand the overall effectiveness of each soundhole as a radiator, and the tonal qualities of the radiation.

The disadvantage of the method is that it is not possible to say anything about the absolute values for radiated sound power.


SUMMARY OF PART 3

  • I am unaware of any experimental data describing the performance of guitar soundholes, so I set out to make my own measurements
  • The method used a rigid-sided guitar shaped air cavity with a “chirp” signal pumped into it to measure the resonant response of the airbody/soundhole combination across a range of frequencies
  • I changed the soundhole size and shape using a set of “drop-in” holes
  • The first experiment looked at the performance of round soundholes to establish some basic information (see Part 4) about how the sound projected varied with the size of hole
  • The motive is to discover if the traditional round guitar soundhole is the best design, or whether other shapes might offer better performance

WHAT’S NEXT?

In Part 4 we move on to analyse the response of round soundholes of different sizes, to establish a baseline on the way to finding out if soundholes of other shapes can outperform them. We’ll look at both the relative loudness of different sized holes, and also at the quality of the reproduction of the input signal≥


[1]See Appendix 4for an evaluation of the effectiveness of the hood – it certainly removed some room resonances.

[2]The speaker response is rated at 70Hz to 7kHz. In practice the signal was easily audible at below 50 to 70Hz, so the low activity of the resonator in this range is most likely a real measurement. (That statement hints at the limitations of the low-tech equipment used in these experiments…)

[3] If needed, Audacitycan subtract the background noise from the signal (Effects / Noise removal), so a new version of each data file can be saved after doing this, keeping the raw data as well. Trials showed that this made little difference to the results, so was not used.

Guitar resonance and soundhole geometry – Part 2: THE AIR-CAVITY RESPONSE

THE AIR-CAVITY RESPONSE IN MORE DETAIL

This is the second in a series of ten 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.

The first post of this series put forward a simple model to help understand what happens when an acoustic instrument is played.

The model represents the soundbox as an interaction between three soundbox elements, the soundboard, the air contained inside the soundbox, and the sides/back of the box:

Figure 1: the three mass/spring model for a soundbox

Any up and down movement in the soundboard transfers into the whole connected system via the springs, and causes all three elements to vibrate – each has its own natural frequency, so they “fight it out” and find a way of responding that works for all three.

Now, Part 2 here we come…

In this post we’re building a theoretical knowledge of guitar acoustics, focusing on Helmholtz resonance.

PART 2 is all about a fundamental type of resonance that powers the acoustic response of a guitar. You may quite reasonably choose to skip this post and move on the the next few, but you will probably need to come back to it later to fully grasp how a guitar works.

HELMHOLTZ RESONANCE

One part of a real airbody’s response is to “slosh” in and out of the soundhole, pumped back and forth by the vibrating soundboard. This “sloshing” or “breathing” response is called Helmholtz resonance.

A more familiar example of Helmholtz resonance happens in a car when one of the back windows is lowered while driving. The air in the car vibrates in and out through the window at a frequency determined by the size of the car interior and the amount the window is open (see Appendix 7). The signal that excites the airbody in the car is the turbulent airflow rushing past the window opening.

An important thing to keep in mind is that Helmholtz resonance is an air pressure variation inside a rigid container – in a car, you can feel the rapid pressure changes in your ears because they’re quite low in frequency.

Figure 2: the car as a Helmholtz resonator

This spectrum shows a strong response peak at 15Hz. Human audition can’t experience frequencies below 20Hz as continuous tones, so the car’s Helmholtz resonance is an unpleasant pressure cyclic pressure variation.

Helmholtz resonance in a car is somewhat like blowing over the top of a bottle to make it sound a note. The air in the bottle has its own particular natural frequency at which it will vibrate in and out through the neck. If this frequency is present in the airflow over the neck, the air in the bottle will feel it and respond to it by resonating. (See Appendix 9 for more detail about air resonances in bottles and pipes.)

You can perhaps also begin to see the similarity between what happens in the car and what happens in a guitar soundbox. In the case of the guitar, the driving impulse is provided by the strings vibrating and moving the soundboard up and down.

So it’s no surprise that discussions of the air-cavity/soundhole response revolve around Helmholtz resonance (which I’ll refer to as HR for the rest of this paper). 

Helmholtz Resonance takes place when air in a container with a small opening in it is excited by an external signal of some kind. A guitar soundbox excited by the strings is an example.

 

THE HELMHOLTZ EQUATION

Professor von Helmholtz himself derived an equation to predict the resonant frequency for a rigid spherical flask with a short neck (see diagram below). 

(c = speed of sound; S = area of the neck aperture; V = volume of resonator body; L = length of neck)

His experiments led to this equation that predicts the frequency the flask selects out for resonance

The Helmholtz equation predicts that the wider the flask’s neck (giving a larger area S), the higher the resonant frequency will be. This will be important when we see what part Helmholtz resonance plays in a guitar soundbox.

The larger the volume of the sphere V, the lower the frequency – an inverse relationship. This is the same for the neck length L as well.[1]

You will have noted right away that a though a guitar soundbox isn’t spherical, and the soundhole doesn’t have a hollow neck projecting from it [2], it isa resonant cavity with an aperture.

Figure 3: the Helmholtz resonator ( http://newt.phys.unsw.edu.au/jw/Helmholtz.html)

Helmholtz thought of the air contained in the bottle’s neck as separate from the air in the cavity: he used a mass/spring model where the neck air is the mass, and the airbody inside the bottle is the spring.

The model predicts the fundamental Helmholtz frequency very well.

There is one complication to consider. The effect of the resonance actually extends a small distance outside the neck, so a factor called the “end effect” needs to be added in to the basic equation before it will predict with full accuracy. It adjusts the size and mass of the “neck plug” to make the model more realistic.

The UNSW website has this to say:

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.

In my own spherical Helmholtz resonator experiment I used the UNSW figure to apply the effective lengths of the different necks I used, and it resulted in a very good match between calculation and measurement.

I made a 128mm internal diameter Helmholtz resonator (I used a small globe of the world) with an adjustable-length neck (I sawed bits off as I went…) to confirm the equation.

I blew across the neck to excite the air-cavity, digitally captured the response and analysed it with Audacityas described in the Experimental method section below. This is the response of the spherical airbody with a 13mm long neck:

Figure 4: measured response of 128mm diameter Helmholtz resonator with 13mm neck

This is the spectral response of the resonator with a 13mm long neck. There is a clear series of resonance peaks between 0 and 1000Hz – my first surprise. Why? Well, the Helmholtz model predicts only one resonance, not a series of them. The spring/mass model used in deriving the simple equation can have only one solution, and it has no place for a harmonic series [3].

Here is the same data, showing a closeup the response from 0 to 600Hz:

Figure 5: harmonic series produced by 128mm diameter Helmholtz resonator with 13mm neck

The resonant peaks occur at 117.2Hz, 237.3Hz, 354.5Hz, 471.7Hz, and 585.9Hz. These are very close to whole multiples of the 117.2Hz fundamental, so clearly represent a harmonic series. [4]

But how well does the frequency of the first harmonic compare with the model? The next graph shows the data for the fundamental frequency for the whole set of different resonator neck lengths, next to the model’s prediction.

Calculations from the Helmholtz equation used the UNSW recommendation for end effect correction:

Effective neck length = 0.6R + R + L

HOW WELL DOES THE EQUATION WORK?

I do recognise that unless you’re as much of a Physics tragic as me, you don’t really care. Anyway, these are the fundamental frequencies I found for the spherical resonator with a range of different neck lengths (H) from 13mm to 58mm:

Figure 6: Helmholtz resonance fundamentals for 128mm spherical cavity

The predictions match the calculated values very well.

Figure 7: Response of 128mm diameter Helmholtz resonator compared to theory

The variation between the measured results and the calculated model predictions ranges from 1 to 6% – quite good agreement.

Gore and Gilet (Contemporary Acoustic Guitar, Vol 1 1-29) are correct when they say that:

…The Helmholtz effect is important because it largely determines the low frequency response of a guitar.

But while developing analytical models for guitar response, Gore and Gilet warn that the resonant frequency for a real guitar cavity is not straightforward to calculate.

With the guitar, there are several complications: a guitar isn’t spherical; it doesn’t have a neck stuck on the soundhole; it isn’t rigid; and the airbody is only one out of three resonators that make up a soundbox.

So Helmholtz’s simple model may not seem to have much chance of working very well, but two of the four issues are reasonably easy to deal with.

Firstly, my experimental guitar-shaped resonant cavity is designed to be very rigid.

There is indeed no bottle neck attached to the soundhole [5], but the physics recognizes the “end effect” that takes into account the fact that air beyond each end of the pipe participates in the resonance, making a kind of invisible “neck”. 

In the case of a guitar, “neck length” is the thickness of the soundboard at the soundhole, plus a fudge factor (oh, did I say that?) for end effects..

SUMMARY OF PART 2

Part 2 is about Helmholtz resonance, a vibrational mode of the airbody contained in a soundbox. This is introductory theory underpinning the experiment on how soundhole geometry can affect the resonant response of a guitar.

You may need to come back to this later to fully understand what the importance is – it took me several months before I could truly grasp why it is important.

WHAT’S NEXT?

In the next post we look into the process of measuring the effect of changing soundhole geometry on the response of a guitar soundbox. As far as I know, this is new research for guitar acoustics.

Here’s a picture of the rigid guitar-shaped cavity I used for my experiments:

Figure 8: the experimental rigid guitar-shaped cavity

This is made to be heavy and rigid. You can see that the space between the cavity walls and the box is packed with sand, and in operation the top is heavily damped by lead weights. 

The effect in terms of the mass/spring model is to clamp the soundboard and sides/back so they can’t move – that way, the only response I measure will be from the airbody.

Figure 9: The experimental equipment effectively “clamps” the soundbox so only the airbody can resonate

FOOTNOTES


[1]This equation is often applied to non-spherical resonators with a caution that it will be an approximation only (see http://newt.phys.unsw.edu.au/jw/Helmholtz.html).[1]

[2]…but what if it did? Perhaps this could be an easy way of tuning the soundbox resonances.

[3]Many oscillators – like guitar strings – can resonate at a series of different frequencies, characterized by multiplying the fundamental frequency by 2, 3, 4,…,n –  this is called a harmonic series

[4]While of interest, I didn’t find any evidence that there is a similar harmonic series in the airbody response of a guitar soundbox, so I’ll leave it at that.

[5]But what if there were…? Just a thought.

Guitar resonance and soundhole geometry – Part 1: A MODEL FOR HOW ACOUSTIC GUITARS WORK

SOME BASICS ABOUT HOW ACOUSTIC GUITARS WORK

This is the first 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.

The questions I ask and try to answer in this series of posts on guitar soundhole performance are suggested by my own experience as a guitar maker, and my physics-based research into how acoustic instruments actually produce sound.

The questions are:

o  What effect does soundhole geometry (size, shape, and placement) have on how a guitar sounds? (see Part 4)

o  What is the source of the sound that comes from the soundhole of a guitar?

o  If soundslots do perform better than round holes, is there any disadvantage in dividing the slot into two segments? (Designing a single long and narrow soundslot with area equivalent to the traditional 50mm radius soundhole is difficult without compromising soundboard strength.) 

o  Do soundslots (holes with a high perimeter to area ratio) perform better than round soundholes of the same area? [1]

Given my plan to use a rigid box for my experiment, how well do experimental results derived from a rigid cavity translate to a real, flexible guitar soundbox?

 

AN ASSUMPTION…

The assumption underlying this paper is this: 

Guitar making need not be a matter of reproducing traditional geometries and methods.

Guitar makers can be designers and innovators if they understand better how the instrument works. New materials are worth investigating (such as bamboo and carbon-balsa-carbon laminates for bracing). 

So are different soundbox geometries.

…AND ITS DANGERS

I can best express the dangers in my assumption by saying that I have come to see an acoustic guitar as a box with strings that makes sounds our brains can recognise as “guitar”. “Guitar” equates to a particular sound, not to a particular box.

There’s nothing wrong with an instrument sounding unusual, but there comes a point when you have to come up with a new name for the sound because it’s no longer “guitar”. Electric guitars were unacceptable for a while before people were willing to expand their definition of “guitar”.[2]

INTRODUCTION

The traditional shape for a guitar soundhole is circular, with a radius of about 50mm.

As with all traditional design solutions, it is worth asking whether round soundholes of this size have been proved by trial and error to be the best design, or whether contemporary designers simply accept tradition and no longer wonder whether better designs are possible.

An article published by Royal Society Publishing Proceedings Ain 2015 called

The evolution of air resonance power efficiency in the violin and its ancestors(Hadi T Nia, Ankita D Jain, Yuming Liu , Mohammad-Reza Alam, Roman Barnas, Nicholas C Makris)

The authors trace the evolution of violin-type instrument soundholes from the 10th to the 18th century, from the initial round shape to the familiar classical f-hole. 

The Royal Society paper argues that violin-style f-holes are in fact considerably better sound radiators than round soundholes.

The paper is wide-ranging, so here are the main points from my point of view:

By determining the acoustic conductance of arbitrarily shaped sound holes, it is found that air flow at the perimeter rather than the broader sound-hole area dominates acoustic conductance (my emphasis)

….As a result of the former, it is found that as sound-hole geometry of the violin’s ancestors slowly evolved over centuries from simple circles to complex f-holes, the ratio of inefficient, acoustically inactive to total sound-hole area was decimated, roughly doubling air-resonance power efficiency (my emphasis).

F-hole length then slowly increased by roughly 30% across two centuries in the renowned workshops of Amati, Stradivari and Guarneri, favouring instruments with higher air-resonance power, through a corresponding power increase of roughly 60%…

Here’s a quick summary of what I find interesting as a guitar maker and designer:

  • It turns out that air can move in and out through a soundhole more effectively if the hole is longer and narrower than if it is shorter and rounder. This will particularly influence the instrument’s low frequency response.
  • The air contained in the soundbox of an acoustic instrument contributes to the overall sound by vibrating in a simple way. When the strings make the soundboard vibrate, the air inside responds by expanding and contracting in and out through the soundhole. This is sometimes called the “breathing mode” [3], because the soundboard moving up and down acts just like the diaphragm at the bottom of our chest cavity driving air in and out of our lungs.[4]
  • The reason for this is that most of the air movement happens at the edges of the hole rather than towards the middle. Compared to the air movement at the edges, the centre is acoustically inactive. [5]
  • Making a soundhole longer and narrower reduces the size of the acoustically dead central area of a soundhole and can “roughly double air resonance power efficiency”.

We need to carefully note that the RS article refers specifically to low frequency sound radiation from the violin air cavity only. The frequencies involved are around 100Hz – the low E string on a guitar sounds at 82Hz. A violin radiates much more complex sound from other surfaces than that generated by the air-cavity alone, as does a guitar.[6]

As an added bonus, the results of the resonance experiments I have carried out show that the scope for re-designing guitar soundholes in fact goes beyond just control of the low frequency air-cavity response.

Let’s ask a practical question to start with. What happens if you block off the soundhole of a guitar?

This is what [7]:

Figure 1: Change in response of a Jumbo guitar when its soundhole is blocked

The blue line is the tap response with the soundhole open. It’s clear that the open hole response of this guitar is strongest at 92Hz (very close to F#2). But this response is choked off when the soundhole is blocked, as shown by the red line. 

The measurements show that the radiated power of the sound coming from the guitar over a wide frequency range drops by about 30% when you close off the soundhole (0.027W/m2open drops to 0.019W/m2closed).[8]

Here is the result of an identical experiment with a violin:

Figure 2: Change in response of a violin when the f holes are blocked

The air-cavity response for the much smaller violin body is at 271Hz (about C#3), and is drastically choked off when the f holes are closed.

Although the responses of the two instruments are different, they share the same air-cavity resonance mechanism as part of their bass tone production. There’s reason to believe, then, that guitar designers can profit from the knowledge violin makers have arrived at by evolution over a number of centuries.

So air-cavity responserefers to the ability of the air contained in a stringed instrument’s soundbox to resonate at certain frequencies.

HOW IMPORTANT IS THE AIR CAVITY RESPONSE?

We’ve established that when listening to an acoustic guitar, about 30% of the sound you hear comes from the soundhole.

As I will show later, only a very small portion of this total 30% comes from the air-cavity itself vibrating. This may seem confusing right now, but it’s important to keep it in mind – I will explain. For the moment we’ll leave the question of where the rest of that 30% comes from.

This series of posts attempts to explain why a resonance that contributes such a small part of the overall projected sound is so important, and how the geometry of the soundhole can influence it.

The other roughly 70% of the sound you hear from a guitar comes from the vibrating soundboard pushing on the air next to it. Pressure waves travel out into the environment from the moving board surface. These are more complex and about twice as strong as the soundhole signal [9].

Soundboard signals are multipolarbecause the board has several different modes of vibration available to it (monopole, cross dipole, long dipole etc; see Appendix 8andContemporary Acoustic Guitar1-74).[10]

GUIDES FOR INNOVATION

To avoid trial and error, some scientific background is important to understanding what might be fruitful lines to work along for innovation to be successful.

The best source I have found for understanding the principles is Contemporary Acoustic Guitar Design Volume 1by Trevor Gore and Gerard Gilet. Its mathematical modeling approach can be a little intimidating, but you can skip the maths if you want and concentrate on their conclusions. 

Much of what is in this report is inspired and informed by their work.

RESONANCE AND RESONATORS – SOME BASICS

If you haven’t come across the term resonance in a technical sense before, Appendix 1 summarises most of what you need to know to understand the majority of this report. Here are some basics:

A simple resonator is something that vibrates in a predictable way after it has been displaced from a stable position. A pendulum is an example, as is a guitar string or a drumhead.

Resonance happens when a resonator responds to exactly-timed external pushes that cause the resonator to continue or build up its motion rather than coming back to its rest position.

SIMPLE PENDULUM

In the case of real-world examples, often an audio resonator will respond to impulses of one (or more likely several related) frequency that it selects out of a complex sonic environment. [11]

Think of a child’s swing – when you give it a push it swings away from the push, slows, stops and then moves back toward you. If you leave it alone it keeps swinging but eventually comes to a stop at its original stable position.

The important requirement for something to vibrate around a central position is a restoring force

In the case of a swing, the restoring force is gravity, or the weight of the swing. As the swing moves away from its stable rest position, its weight begins to pull it back. The swing slows down, stops, and then accelerates back toward the centre. When it arrives back at the central position it is moving at its fastest and overshoots, slowing again as it moves out towards the other extreme of the cycle.

RESONANCE

If you want to keep a vibration going for longer, or even build it up, everyday experience shows us that you need an external force to push at just the right moment in each swing – it demands a particular frequency of push before it responds, and it also demands that you push at the right moment in its cycle. [12]

Think of pushing a kid on the swing to build up the amplitude.

For the purposes of this paper, it is the need for a particular frequency of driving impulse to get a response is one of the most important characteristics of a resonator.

In the case of an acoustic guitar, my experiment focuses on the resonance of the air contained inside the soundbox – the airbody.

The airbody is one out of the three most important resonators that make up a guitar.

HOW CAN A FLABBY LUMP OF AIR RESONATE?

This question is crucial to understanding my experiment and its results.

If it’s contained in a soundbox, air can “slosh” in and out through the soundhole because it’s a fluid. It will slosh with a particular frequency determined by the size of the box and the size and shape of the soundhole.

In a real instrument the vibration of the soundboard pushes the air in and out through the soundhole.

When the soundboard moves down it pushes air out, and when moving up draws it back in. Even without going into more detail at this point, it’s clear that there’s a very strong connection between soundboard resonance and airbody resonance.

In my experiment, the airbody is driven to resonate not by a vibrating soundboard but by a loudspeaker pumping a pressure wave into a guitar-shaped cavity. When the pressure in the cavity is high, air is pushed out through the soundhole, then drawn back in when the pressure is low. Details of the experiment are in Part 3

COUPLED RESONATORS

As Gore and Gilet show [13], acoustic guitars can be modeled quite accurately by thinking of the soundbox as three coupled resonators: 

  • the soundboard
  • the air body contained by the soundbox 
  • the sides/back of the box. 

Picture these three components as three masses connected by springs:

Figure 3: Three coupled resonators: an analogy for an acoustic guitar soundbox

If you pull the top mass down and let it go (as in tapping the soundboard of a guitar), you disturb all three and they will all necessarily start to vibrate. The three resonators will “fight it out” between themselves until they come up with a compromise that will allow them all to vibrate in harmony with each other.

They are coupled and they form a single system, although each one keeps its own resonant characteristics.

If you now change the nature of one of the resonators, the others must also change in response because they are coupled. For example, if you were to change the mass or the stiffness of the soundboard, the resonant frequency of all three would change in response.

The fundamental principle of conservation of energyis very important here. The energy you put into the system by moving the top mass and stretching the springs is spread among all three as they begin to vibrate. 

Resonance does not create energy. In a guitar the only energy the soundbox has to work with is the amount given to the strings when you pluck or strum them – the job of the instrument is to turn this rather small amount of mechanical energy into sound energy as efficiently as it can.

With the three mass/spring model of the guitar soundbox, what happens if you block off the soundhole? That would be like clamping the centre mass so it can no longer vibrate:

Figure 4: Three mass model with the soundhole blocked

Clamping the central mass changes the system drastically. Both the soundboard and the side/back elements revert to their own resonant characteristics – it’s no longer a coupled system. [14]

Keep in mind that for a system to vibrate it must have both mass (inertia) and elasticity (more strictly, a force that acts to return the resonator to its stable position). Both force and mass determine its behavior.

Anything that affects one element in a system will affect the others. In the case of the three coupled resonators that make up a soundbox, a change to one will affect the sound of an instrument as a whole.

This series of posts concentrates entirely on the characteristics of:

  • the airbody’s mass – determined mainly by the size of the soundbox 
  • the airbody’s elasticity – determined mainly by the soundhole.

Blocking off the soundhole has the effect of clamping the second of the three resonators –called decoupling.

You can see the effect of the decoupling in the graph below. This is the spectral signature of a Jumbo sized guitar, produced by tapping, and there is a lot of information about the instrument contained in it. [15]

Figure 5: The effect of blocking the soundhole of a guitar

It’s well worth another look at this spectrum now that we know more about what’s happening when you block a guitar’s soundhole.

The blue line shows the response to tap testing with the soundhole open, and the red with it blocked off. Blocking the soundhole stops the air-cavity vibrating, and hence removes the main connection between the top and the back – hence coupled responseanduncoupled response

The air-cavity response (the strong peak at 91Hz on the blue line) is cut off when the soundhole is blocked.

The red/blue peak to its right at just under 176Hz is the main soundboard response, which is unchanged except for a slight frequency shift (from 176Hz open to 174Hz closed). Just to the right of that you can see the main back response at 204Hz, which also disappears when the soundhole is blocked. [16]

The challenge in designing and building a good guitar is to set the mass of the soundbox parts and their stiffness to make the best use of the energy being pumped in by the strings. 

The purpose of this report is to describe the effects of soundhole geometry on guitar resonance, beginning with the air-cavity response.

By blocking off the soundhole of an instrument it is possible to measure that the air-cavity/soundhole contributes about 30% of the projected sound, though this varies one instrument to another. The remaining 70% of the sound is projected by the vibrating surface of the soundboard. 

Clearly a doubling of the air-resonance power efficiency in a guitar would usefully increase its projection – taking the 30% figure as an example, a doubling would produce an extra 15% in total radiated power.[17]

The air-cavity resonance is most marked at low frequencies, so increasing its efficiency will also alter the instrument’s tonal balance toward the bass end. 

This offers a way of shaping the instrument’s sound at the design stage.

The experiments I am describing in the next sections of this paper are designed to investigate the airbody/soundhole response of a guitar body in isolation.

To do this the airbody has to be contained in a box that is too solid to easily vibrate. Looking at the triple mass/spring model in Figure 7, I set up the apparatus to be the equivalent of holding the top and bottom masses firmly in clamps to isolate the airbody response. This uncouples the system and allowed me to study the airbody response by its self as I changed the size and shape of the soundhole.

Figure 6: “Clamping” the soundboard and sides/back of box to isolate the airbody response

You may quite likely be wondering what the springs linking the airbody to the soundboard and the sides/back component represent in a real soundbox. This will be explained more in Part 2, using the concept of Helmholtz resonance.

SUMMARY

  • The experiment I am reporting on in this series is designed to give information to help in the design of the soundhole of a guitar
  • A grasp of the concept of resonance is important in reading this experimental report
  • About 30% of the sound coming from a guitar is radiated from the soundhole; however only a small part of this comes directly from airbody resonance
  • A good way to visualize the workings of an acoustic instrument’s soundbox is to think of it as three coupled oscillators: the soundboard, the airbody, and the sides/back
  • Air cavity resonance is determined by the size of the cavity and the properties of the soundhole
  • The importance of the air cavity resonance is in providing an important part of the coupling between the top and back/sides resonators
  • The experiment described used a rigid cavity to isolate the airbody response, allowing study of changes to the soundhole geometry

WHAT’S NEXT?

Part 2 in the series delves into soundbox resonance in more detail, and introduces the important concept of Helmholtz resonance.

FOOTNOTES


[1]  This question comes from reading a paper published by the Royal Society titled The evolution of air resonance power efficiency in the violin and its ancestors

[2]I personally love the idea of new sounds, but guitarists have legitimate expectations about what they expect from an instrument.So it comes down to the riddle of when a guitar is not a guitar.  In my opinion there is a certain amount of mythology about instrument sounds. For example, blind tests of highly experienced professional violinists show that picking a Stradivarius from supposedly lesser instruments is not as clear cut as we might think. Psychoacoustics plays a large part in it.

[3]See http://www.acs.psu.edu/drussell/guitars/hummingbird.htmlfor an animation of the breathing mode – more on this later in this series.

[4]You can make a guitar blow out a candle by pointing the soundhole at the flame and tapping the soundboard.

[5]The reason for can be found in hydrodynamic theory, often referred to as the Bernoulli effect

[6]“Monopolar” means that an instrument’s soundhole acts as a single point source that radiates uniformly outwards in all directions like the ripples from a pebble dropped in a pond – this is the simplest form of radiator. 

[7]This information comes from the tap method, which uses a small padded hammer to tap the instrument

[9]A hint about the radiation from the soundhole: the soundboard also projects sound intothe soundbox.

[8]These figures, like all thepower numbers in this paper are relative only, as I’ll explain later.

[10]Though it is not the focus of this paper, an excellent set of animations to show these more complex modes can be found at:

https://www.acs.psu.edu/drussell/guitars/hummingbird.html

[11] I will talk a lot about Helmholtz resonance in this paper. Before electronic equipment existed, Helmholtz used banks of audio resonators to analyse the spectral content of complex sounds, each one responding to a narrow, known, frequency band.

[12]The timing of the driving impulses is known as phase – in phase maintains or builds up resonance, where out of phasedamps it out.

[13]Contemporary Acoustic Guitar Vol 1, P 2-3

[14]It isn’t quite that simple in reality, because of course the top and sides and back are glued together, so they do remain mechanically connected together. Strangely, this connection is often less important for low frequency resonance than the airbody connection.

[15] The reason this graph looks different from Figure 1is that I have left the results in deciBels (dB) rather than convert to relative radiated power

[16]This isbecause I tapped the soundboard only. In its decoupled state, with no strong connection to the top, the back doesn’t respond strongly to the tap.

[17]This last point isn’t evident from Figure 6because the strength of the tap was inevitably different between the open and closed measurements. A “standard tap” would help here, and it illustrates one of the difficulties of trying to make direct comparisons between instruments using the tap method.