Because a guitar soundboard is fixed along its edges and has a characteristic waisted shape with a large lower bout, there are only a certain number of ways in which it is free to vibrate. These are called “top modes”, and they are the same in every flat top guitar. Together they produce the sound we recognize as “guitar”.
NODES AND ANTINODES
First, some definitions. An object vibrating in a complex way, such as a guitar string, will show points of maximum movement, interspersed with stationary points.
In the case of a guitar string, the fixed point at each end is always a node, but an elastic string can support a number of different modes of vibration. Here is a diagram of the first four possible modes for a guitar string:
Points of maximum displacement from the string’s rest position (a straight line, of course) – are called antinodes. In the top diagram you can see four antinodes.
Points of no movement are called nodes. In the top diagram the string mode has five nodes.
The bottom picture shows the simplest, or fundamental mode with an node at each end and one antinode in the centre.
The number of antinodes is called the harmonic number n, so here we see n = 1, 2, 3, and 4.
A string has many higher modes available to it when plucked, but the higher the harmonic number the shorter-lived they are. This is why the attack when you pluck a string is vibrant and complex. As the string rings over time, the higher modes die out one by one, eventually leaving the fundamental by itself.
GUITAR TOP MODES OF VIBRATION
Flat top guitar soundboards also show an unvarying pattern in the ways that they are able to vibrate.
The diagrams in this section give a basic idea of how the soundboard of a guitar responds when it resonates at particular frequencies. The + sign indicates the maximum possible upwards deflection from the normal flat surface, and a – sign is a maximum downward deflection.
The dotted line each diagram is the node, the points that stay fixed during the vibration.
What’s important to realise is that all guitars with a flat, relatively thin soundboard show these modes. The main reason why different guitars sound different from each other (different timbre) is that while they all support these same modes they do it to different degrees.
As you look at the diagrams below, you can see how the soundboard bracing can have a huge effect on the guitar’s response.
The monopole is the simplest vibrational mode, and is associated with the the coupled air-cavity resonance frequency at around 150Hz. The top (and often the back) surface acts like a bellows forcing air in and out of the soundhole. 
The top monopole (one antinode), which is the lowest frequency “breathing” mode
It is possible to get a guitar soundbox to blow out a candle by placing the soundhole next to the flame and tapping the bridge firmly. This produces a puff of air that can blow the flame out.
THE CROSS DIPOLE
The cross dipole is a response to higher driving frequencies than the monopole, typically around 300Hz. There are two antinodes (hence “dipole”), and as one is moving up the other is moving down.
The cross dipole (two antinode) mode: each side of the lower bout vibrates separately and out of phase
THE LONG DIPOLE
The long dipole mode is typically found at about 400Hz. Again, there are two opposite-phase antinodes
The long dipole mode, where the two antinodes vibrate out of phase
THE CROSS TRIPOLE
The cross tripole mode is typically found at around 450Hz.
The cross tripole mode, the centre pole out of phase with the two outer ones
These simple diagrams represent only the briefest overview of the top modes, and there a further resonances at higher frequencies with as many as 8 antinodes.
But as for a string, the more complex the mode the shorter-lived it is. One reason a bowed instrument sounds different to a plucked one is that a bow provides a continuous supply of energy to the soundboard, and that tends to prolong the higher modes. The violin’s timbre is more complex than the guitar’s.
THE TOP BOUT
Notice in the diagrams above, the area of the soundboard near the soundhole is pretty inactive. No guitar is as simple as these diagrams, but the top bout area doesn’t contribute a great deal to the guitar’s sound because it is usually heavily braced to withstand the force on the neck trying to make the guitar fold into two.
One potential advantage of the double sound slot geometry is that it may free up more of the upper bout to contribute to the guitar’s sound.
A NOTE OF CAUTION
If you look into this on line, you may come across diagrams showing different and more complex modes derived from Chladni plates. These are metal plates set vibrating by using a violin bow.
The patterns you see from Chladni plates will be different from the ones I’ve just shown simply because a Chladni plate has free edges, whereas a guitar soundboard is connected to the rest of the soundbox all around its edge.
Different guitars have different timbres largely because each instrument has its own particular balance between the top modes produced, both in strength and frequency. The differences are mainly due to the top bracing system used, the mass of the soundboard, its stiffness – and, as we have now seen, by the soundhole geometry.
In my opinion, the material used for the back and sides has only a subtle effect on the sound produced by a guitar, compared with the properties of the soundboard. Many would disagree, and I am happy to concede that others may have a better ear than mine.
These points of maximum displacement from rest in either direction are called “antinodes”.
 In fact there are three separate frequencies associated with this mode, as explained by Gore and Gilet (Contemporary Acoustic Guitar Design, P1-82). The reason behind this is beyond a simple description of top modes.
Two oscillators moving in step are “in phase”; if moving in the opposite sense they are “out of phase”. These differences are measured by “phase angle” where a complete cycle of vibration is 360 degrees, half a cycle is 180 degrees, and so on.
Here are some results from measurements I made of HR in a friend’s car. First we got the HR going at around 80kph by winding down a rear window. I recorded the resonance, which we could feel as an unpleasant buffeting in our ears – the low frequency pressure variations driven by the passing turbulent airstream outside the window.
I then waited by the side of the road as my friend drove past a couple of times with the resonance going, and again recorded the sound of the car passing.
Analysing the inside signal gave a frequency of 15.5Hz. Analysing the signal from the side of the road showed a signal of the same frequency, even though it wasn’t audible to me at the time.
Helmholtz resonance from inside and outside a car
The Helmholtz signal shows as a spike at 15.5Hz in both lines. From outside the car the spike is about 15 to 20dB (up to 100 times) stronger than the background noise of the car engine and tyres.
I was surprised that the HR was detectable at a considerable distance from outside the car, and for a while I expected to find the same for a guitar soundhole. Of course, if you’ve read my report you’ll know I found the opposite.
[I surmise that for the car, the intensity of the HR driving impulse – the turbulent airflow blowing forcefully over the open window – is proportionally much greater than for a guitar soundhole. Whether this is true or not I don’t know.
APPENDIX 4: THE VALUE OF THE ACOUSTIC HOOD IN THE EXPERIMENT
This graph shows the recorded far field signal from a soundhole with (blue) and without (red) the hood in place:
The effect of the acoustic hood on recorded response from a 45.4mm radius round soundhole
Without the hood in place there are strong resonances at 170Hz, 461Hz, and 494Hz (2.00m, 0.74m, and 0.69m wavelengths). These are present for all four holes tested without the hood, and correspond to dimensions in the room close to the apparatus, indicating that they are most likely room standing wave resonances. The hood is quite effective at suppressing these.
Most people are familiar with the unit decibel (dB), which is commonly used to describe environmental sound levels.
The dB scale ranges from 0 (the softest sound that most human ears can pick up) to 130dB (the sound level so loud the it causes pain rather than the accurate perception of a sound).
Everyday examples of dB levels are: 30-40dB in a library; 70-80dB traffic on a busy road; 110-120dB pneumatic drill. For correct use in acoustics, the application of the scale is complex because it has to take into account the fact that the human ear responds differently to different frequencies, as well of other technical factors. This corrected scale is called the dB(A) scale.
However, the important thing to know about the dB unit is that the scale is not linear, but logarithmic. What this means is that a sound at 100dB pushes energy into your ear at not twice the rate of a 50dB source, but 100,000 times (x105).
You might think that 100dB would be twice as loud as 50dB, but in fact a sound twice as loud as 50dB measures at about 53dB.
This isn’t done to be difficult. It’s because the range of human hearing is so great that a linear scale just can’t easily encompass the numbers when expressed as the rate at which energy hitting your ear.
Without going into any more details, here are some examples to think through:
plus 3dB means approximately a doubling of the sound intensity
60dB is ten times the intensity of 50dB
70dB is one hundred times more intense than 50dB
30dB is one hundredth of the intensity of 50dB
47dB is half the intensity of 50dB
One effect of this is that you can’t add or subtract dB as if they’re familiar numbers (50dB plus 50dB is roughly 53dB). For this and other reasons you’ll see that I express my results in power terms – that is the amount of energy falling onto the eardrum every second. The unit is Watts/square metre (W/m2), and the symbol is I.
I also do this because during my analysis of raw data I subtract closed box response from the signal I get from the microphone, and I need to use a non-logarithmic scale.
To give you and idea of how this works, here are some examples of dB readings converted to Intensity:
0dB is I0or 1 x 10-12 W/m2(0.000000000001 W/m2)
3dB is about 2 x 10-12W/m2
10dB is 1 x 10-11W/m2
50dB is 1 x 10-7W/m2
100dB is 1 x 10-2W/m2
Here is a more complete set of conversions:
I also think that unless you are really familiar with how the dB scale works, looking at measurement results that use them can be quite misleading – what looks like a small dip of 3dB actually represents a halving of the sound intensity, for example.
This discussion isn’t central to understanding my results, but as a reader you might be expecting to see the dB scale being used.
There is one final complication that you need to be aware of. I am very careful to present my data and conclusions in terms of RELATIVEresponse. I have only measured how the various aircavity/soundhole combinations respond relative to the response of no soundhole at all – a closed box.
You can only use my numbers to compare, not to predict the actual response of a real guitar soundhole.
In addition, Audacityrecords sound levels on a minus scale. Most digital recording apps do the same, because in recording quality sound such as music it is important to avoid clipping – distortion introduced by a signal that is too intense for the software to handle. Using 0dB on a recording as the maximum level possible helps to identify and control clipping.
It also complicates dealing with data in my experiments. Because I must work in Intensity (W/m2), you can probably see that in the real world 0dB is the lowest sound a human can detect, and anything in the subzero range is rather meaningless.
Unlike the experimental rigid cavity used for the measurements reported in this series of posts, real guitar bodies have elastic – springy – top and back plates. This raises the question of whether the resonant properties of a real soundbox equate with my experimental results. If not, the lessons from the experiment have far less practical use to the guitar designer and builder.
The two degrees of freedom (2DOF) model developed by Gore and Gilet treats the soundbox as three coupled resonators: the top plate, the air-cavity, and the back plate.(see Part 1)
Each of these three oscillators has its own elasticity and mass, with the air-cavity providing the coupling between the top and the back. The experiment shows us that the air-cavity response can be varied by changing the size and geometry of the soundhole because the Helmholtz resonance will vary with the size of the hole.
Because the air-cavity acts as the connecting spring between the top and back, changing the soundhole geometry will change the resonant characteristics of the guitar soundbody as a whole. The same applies to the braces reinforcing to top and back plates, opening up the possibility of tuning the resonant response of the instrument, with each one of the three affecting both of the others.
One of the complexities of coupling a number of resonators together is that each one affects the resonant frequency of the others. The resonant peaks tend to “repel” each other, so that the initial resonant peaks move further apart when two resonators are coupled.
This will also be true for the three coupled resonators pictured in the 3DOF model.
You can see this effect by comparing the spectral response of a guitar with the soundhole open (ie, a coupled response) and with it closed (uncoupled).
Figure 1: Response of guitar soundbox with soundhole open and closed (coupled and uncoupled)
The red line peak at a bit below 180Hz shows the response of the guitar top in isolation from the air-cavity which is constrained from vibrating by the closed soundhole.
The blue line shows how the top and the air-cavity respond when they are coupled together. The most obvious feature is the strong peak at around 95Hz, which is the main air-cavity response that cannot be present when the soundhole is blocked.
The peaks on the spectrum that are present in both coupled and uncoupled states, such as the one at just less than 180Hz, are soundboard responses that are not dependent on coupling for their existence.
The rigid experimental cavity deliberately damped out any vibration in the top and back plates, isolating the air-cavity as the single uncoupled resonator. This means that the results from the experiment will be different from those found from a real guitar simply because coupling is not present.
Keeping that in mind, the question to consider is the extent to which the rigid cavity data can be used to help in the design and building of a real guitar.
It’s unlikely that the resonant character of a real guitar will coincide exactly with the rigid cavity, so the answer to the question is likely to be found in ecognizable patterns rather than any matching of numerical values.
Here is a summary of the results from the rigid cavity experiment relating to round soundholes:
Round soundholes radiate most strongly in the band from 100 to 600Hz, and hardly at all in the higher 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
The greater the radius, the better the sound radiation becomes in Octave 3 (165 to 330Hz), central to the scale range of the guitar
The Helmholtz resonance varies with soundhole radius, suggesting that the air-cavity coupling can be varied by changing the soundhole size
So these are the points that now need to be verified for a real guitar. For the purpose of instrument design, Point 4 is perhaps the most important because it has most bearing on using soundhole size to control the coupled resonance of the instrument. It is only the Helmholtz resonance that changes in frequency as the soundhole changes size.
We know that the response of the real soundbox will be more complex than the rigid cavity because the top and back plates will be free to add their resonant frequency series to the total response.
The experiment to validate the rigid cavity findings is based on the assumption that if there is a good degree of agreement between the behavior of the rigid cavity and a real soundbox for round soundholes, then the further findings about high P:A soundslots will also hold.
With this in mind, the experiment required removing the neck of an old guitar and exciting the soundbox with a chirp signal in the same way the rigid cavity experiment was done. However the duration of the chirp was increased to 90s to allow time for each resonance to start and build its strength.
The rigid box was excited by putting the signal in through a speaker mounted in the top plate. To keep the real soundboard in place and free to vibrate – the whole purpose of the experiment – an input port was cut in the side of the lower bout and the speaker attached there.
Figure 2: Live cavity input port
Figure 3: Live cavity speaker input
The resonant response was measured in the same way by a microphone above the soundhole at a height of 80cm, and again just above the hole to pick up the Helmholtz resonance.
Figure 4: Live cavity experiment setup (one side of acoustic hood removed for clarity)
A square hole to receive the drop-in soundholes was cut into the soundboard upper bout and its edges reinforced. This allowed quick changeover of the holes between runs.
Figure 5: Drop-in soundhole
As with the rigid cavity, the response of each box/hole combination was recorded three times and averaged.
And, as with the rigid cavity procedure, the baseline for comparison was provided by testing the soundbox with the soundhole blocked. The baseline data provided two pieces of information:
a measurement of the sound travelling directly from the driving speaker to the microphone without mediation through the soundbox
a measure of the uncoupled response of the top and back plates
The first of these can be subtracted from each live result quite simply.
The second is less simple since the effect of coupling the top and back by opening the hole changes the frequency of their resonant peaks slightly. This is an added complication over the rigid box experiment in which there were no coupling issues.
The results that follow lend support to the validity of transferring conclusions from the rigid cavity to a live cavity.
The first plot shows again the response of round soundholes in the rigid cavity:
Figure 6: Response of round soundholes in rigid cavity
The next plot shows the reponse of the same round holes when placed in the live cavity:
Figure 7: Response of round soundholes in live cavity
The first impression is that there is a big difference between rigid and live.
The rigid cavity response is, rather surprisingly to me at first, richer than that from the live cavity. Thinking it over, another way of saying “less rich” is “more selective”. The live cavity is better at selecting out frequencies, which I think is exactly what it should be doing in order to sound like a guitar.
There are also these similarities between the two cavities:
a band of resonances between 100 and 300Hz
a second band of responses between 350 and 550Hz
One feature showing in the live cavity data is a small response from 100 to 130Hz, which seems to be the Helmholtz resonance. The following graph shows this in greater detail:
Figure 8: Low frequency response of round soundholes in live cavity
You may remember from Part 2 that the modified Helmholtz equation derived by Gore to predict the Helmholtz resonance frequency of a guitar soundbox is:
where R is the soundhole radius, S the soundhole area, V the cavity volume and α is a factor derived from experimental data for a particular guitar shape. For a Dreadnought Gore and Gilet measure αas 1.63.
This modified equation equation predicts the following values for fH:
Figure 9: Helmholtz resonant frequency prediction plotted on live cavity response
The predictions do not match the measured values particularly well – if they did, for example the green vertical dotted line should coincide with the green R54.9 peak.
This result is of some interest, but does not really support the validity of the rigid cavity experiment.
COMPARISON OF RADIATIVE POWER OF RIGID AND LIVE CAVITIES
Although the HR figures are inconclusive, measurements of total radiated power show a strong similarity between the rigid and the live cavities. Although the numbers differ , the linear relationship between soundhole area and relative radiated power is strong in each case.
Figure 10: Relative radiated power for round soundholes – rigid cavity
Figure 11: Relative radiated power for round soundholes – live cavity
Both cavities show a high-correlation linear relationship between radiative power and soundhole area. This strongly suggests equivalence between the rigid and the live cavities.
A second validity test compares the frequency distribution for each hole for the rigid and live cavities.
Figure 12: Rigid cavity round soundhole response by octave – % of total for each hole
Figure 13: Live cavity round soundhole response by octave – % of total for each hole
The radiative power graphs by percentage are by no means identical. The rigid cavity bass response (purple bars) seems generally better than the live cavity, but the live cavity outperforms the rigid one in the 82-165Hz range.
However, in general the pattern shown in the response for each soundhole is similar between the rigid cavity and the live cavity:
the 50-82Hz response decreases as the soundhole gets bigger
the 82-165Hz response increases across the range, though not as consistently for the live cavity
the 165-330Hz response stays roughly the same across the range.
This suggests that the rigid cavity responses are a reasonable match with the live cavity ones, adding support to the validity of the experiment, for all its technical limitations.
Any significant variation between the results for the rigid cavity and the live cavity would suggest that the rigid cavity results bore no relation to a real guitar, invalidating the experiment entirely. The lessons from the rigid cavity experiments would then be of no practical use for guitar makers, who deal in live resonant cavities.
As it is, the results show that the responses of the two cavities share similarities, enough to conclude that it is reasonable to assume the rigid cavity results can be applied to live cavities in broad terms at least.
Contemporary Acoustic Guitar Designfrom 2-10 to 2-23
The reality is more complicated again, because the sides need to be factored in as a connection between the top and back separate from the connection provided by the air-cavity. Massive side panels or bolt-in weights can have a positive effect on guitar response as Gore and Gilet show (Contemporary Acoustic GuitarDesign2-28), but that is beyond the scope of this experiment.
Notice that the frequency of each of these peaks shifts slightly after coupling, showing the “repulsion effect”.
The reason for this assumption is a purely practical one: to avoid building a series of soundboxes. The same soundbox was used with “drop-in” round soundholes as used for the rigid box. It was impractical to modify the test soundbox in such a way that it would be possible to drop in slots without drastically reducing the live area of the soundboard. The expense of making purpose-built soundboxes matching the properties of a real guitar ruled out testing slots at this stage.
I had expected to see a stronger signal from the soundboard with the live cavity, since freeing the soundboard is the major difference between the rigid and live cavities. I expected the soundboard signal to show a set of resonant peaks associated with the different vibrational modes (see Appendix 8). The signal may be there, but it would take a much higher level of analysis to reveal it.
 This equation predicts that the Helmholtz resonance fHwill change in the following ways:
increase as the speed of sound increases
increase as the soundhole gets larger
decrease as the soundbox volume gets larger
decrease as the α factor gets larger
The live cavity used was a little larger than the rigid cavity because of the availability of an old guitar soundbox, so it would be unlikely for the actual numbers to agree – the nature of the relationship is the important thing for making a judgement about equivalence between the two.
This is the eighth 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.
Here is a summary of my findings so far:
If using round soundholes, the larger the area of the hole the better it is at radiating sound produced inside the soundbox, which represents about 30% of the total sound generated by a guitar.
Helmholtz resonance can be detected in the signal coming from a soundhole, but only very close to it. The significance of Helmholtz resonance in forming the bass response of an instrument is its function as an important coupling between the other main resonators – the soundboard and the back/sides system.
There is a strong relationship between the radiative performance of a soundhole of given area and the ratio of its perimeter to its area (P:A ratio). If this ratio is greater than 0.1 there is a very marked increase in radiation from a soundbox. A high P:A ratio can be achieved by longer, narrower apertures called soundslots. The classic f-hole in violin family instruments is an example.
The finding is supported by the 2015 Royal Society paper The evolution of air resonance power efficiency in the violin and its ancestors (see Part 1)
A soundslot of equal area to a 50mm soundhole must be quite long to have a reasonably high perimeter : area ratio – for example an 18mm wide slot must be 436mm long. The only practical way to achieve this without badly compromising the strength and resonant characteristics of the soundboard is to make two slots instead of one. We’re back to the violin approach, or looking to the archtop jazz guitar of the twenties and thirties, made to play with big bands before electric pickups became available.
However, placing the f-holes in the lower bout as in the violin or jazz guitar is not the only approach possible, and is not even desirable in a flat top guitar. Putting the slots around the edge of the upper bout has the advantage of leaving the lower bout unpierced so it can resonate in the same way as a standard guitar. In addition, removing the round hole with reinforced edges from the middle of the upper bout should free that area up to vibrate as well.
If true, would that be a good thing? Normally the guitar soundboard vibrates as a plate with fixed edges, producing a defined set of vibrational modes that make a guitar sound like a guitar. The extra live area in the top bout defined by the slots will effectively have free edges, introducing an unknown element into the mix.
So the upper bout two-slot system introduces a new element: half of the soundboard now has free edges. Without building such an instrument, it’s hard to predict what this will mean. If the upper bout becomes live, then the soundboard will become effectively larger and therefore produce more sound – a good thing for several reasons. However, the free edge of the new live area will cause a change in how the whole soundboard responds. Will it still sound like a guitar?
There’s only one way to find out, of course. The question for now is whether changing to slots is worthwhile at all in terms of better efficiency.
The graph below shows how two round soundholes compare with the two-slot geometry, one having the same area (R41.1) having the same total area as the slots, and one a “standard 50mm radius soundhole).
Figure 1: Performance of double slot geometry compared to two round soundholes
The double slot DSLOT1 is very clearly superior to the round hole of equal area (in green) and the “standard” size 50mm soundhole (in blue).
The next graph shows how the total radiated power compares:
Figure 2: Comparison of total radiated power from 50 to 1,000Hz
Keep in mind that these are as always in this paper comparative results only, and the actual figures on the y-axis show the differences only, and are not absolute values.
Here is the same result expressed as a percentage, with the “standard” round soundhole with 50mm radius pegged at 100%
Figure 3: Comparative performance of double slot geometry compared to standard round soundhole
The main point is how much more efficient the two-slot geometry is – an 80% improvement over the standard soundhole even with 70% of the area (7885mm2compared to 5542mm2for the double slot). Unfortunately a double slot with equal area is not really achievable.
But are there any differences in how the two geometries select out the frequencies that they radiate?
The graph below breaks down the response into octave bands. The bands are:
FREQUENCY BAND (Hz)
GUITAR FRETBOARD RANGE
82 – 165
STRING 6 FRET 0 TO12
165 – 330
STRING 4 FRET 2 TO 14
330 – 660
STRING1 FRET 0 TO 12
660 – 1319
STRING 1 ABOVE FRET 12
Figure 4: Comparison of octave band response between round soundholes and double slot
The main features are:
There is little to choose between the three geometries at low frequencies
The two-slot geometry performs by far the most strongly in Octave 3 (165 to 330Hz)
None of the three perform well in the highest frequency band, although the double slot marginally outperforms the round holes
There is strong evidence that soundslots outperform round soundholes of the same area by a significant margin, and not just at air-cavity resonance frequencies
The P:A ratio needs to be above 0.1 for this to be so
It is likely that applying this approach to a guitar will result in an overall higher projected volume of sound, possibly by up to 80% of the total 30% soundhole projection
The overall effect on a guitar’s timbre is hard to predict because of the different vibrational modes available to the soundboard due to some free edges
Differences in timbre between instruments are largely caused by different top bracing systems and different qualities in the soundwood used for the top plate. These variables result in different weightings between the defined set of vibrational modes. In my opinion, the material used for the sides and back of a soundbox has very little influence on timbre, popular opinion notwithstanding