THE 171/172 PROJECT – Part 1

Over the last few years I have been doing research into acoustic guitar resonance, as I’ve reported elsewhere in this blog (see Guitar resonance and soundhole geometry sections).

The research was prompted by an article published in the Proceedings of the Royal Society in 2015 called The evolution of air resonance power efficiency in the violin and its ancestors

http://rspa.royalsocietypublishing.org/content/471/2175/20140905.full

This paper traces and explains the physics behind the evolution of violin family soundhole over 8 centuries leading up to the classical f-hole design we’re familiar with.

I wondered whether the same principles could be used to improve to the modern flat top steel string guitar, so I decided to find out experimentally.

The report of what I found is in the Guitar resonance and soundhole geometry sections on this site.

The conclusion I reached is that, as the Royal Society paper suggested, acoustic instruments with long thin soundholes radiate more effectively than traditional round soundholes. The critical factor is the ratio of the soundhole perimeter to its area – the P:A ratio. My research showed significant benefits where the P:A ratio is larger than 0.1m-1 as shown in the diagram below. 

The purple data point shows the total radiative power of a 50mm diameter round soundhole compared to a set of different P:A ratio soundslots.

Diagram 1: Radiative power of high P:A sounslots compared to a traditional round soundhole

It’s clear that above a P:A ratio 0f 0.1 the high P:A slots strongly outperform the traditional round soundhole.

From now on, I’ll refer to high P:A holes as soundslots.

My experiments also showed that the area of a soundhole is important for good projection, something as a guitar builder I had wondered about for a while. As with a lot of acoustic guitar lore, there are many opinions but not much real data.

Diagram 2: Radiative power of different sized round soundholes

The data shows that the total radiative power of a soundhole increases more or less linearly with the soundhole radius. Bass (purple) and treble (red) response doesn’t vary that much over the frequency range 82 to 1,000Hz, so the increase with area is due to much stronger response in the mid-frequency range (green) as the soundhole size increases. 

Larger soundholes are better than smaller ones, it seems, at least until the hole begins to encroach too much on the live area of the soundboard. It would be interesting to know at what point that happens.

But if the area of a soundslot is to be equivalent to a 50mm diameter round hole, the slot needs to be divided in two to be fitted into the top bout of a guitar.

It may be possible to decrease the area of the soundslots in future as the actual performance becomes known, but it made sense to start out with a standard size.

The next picture shows the design I came up with on the right:

Diagram 3: The design of guitars 171 and 172, showing the traditional round soundhole in 171 compared to the soundslots of 172

The experimental data below shows the performance of the double slot design compared with two typically-sized round soundholes (radius 41 and 50mm):

Diagram 4: Total radiative power of high P:A double slot (DSLOT1) compared to two different sizes of round soundhole.

My experiments predict that the double soundslot design will show an increase in radiative power over the 50mm round soundhole of about 80%. This is a rather startling result, but echoes the findings of the Royal Society paper.

Getting into more detail, the diagram below shows that a double soundslot (DSLOT1) strongly outperforms the round soundholes in the 165-330Hz (note E3 to E4) range.

Diagram 5: Performance across 4 octave bands of double soundslot (red) compared to traditional round soundholes (green and blue)

Is the double soundslot design a new idea? Yes and no.

Archtop jazz guitars were built with f-holes before magnetic pickups were available so they would be loud enough to play with swing orchestras. The soundboards of these instruments were usually carved out of a thick wood blank, like a violin top plate, and the f-holes were placed in the same position as a violin relative to the bridge.

The timbre of the archtops was different to the accepted steel-string sound of Martin and Gibson acoustics. The violin family is different to the guitar in that instruments are bowed, giving a steady high-energy input that can sustain a note for as long as the stick-slip action of the bow can be maintained. 

Guitars, on the other hand, are plucked instruments with varying but always small ability to sustain an audible note. The archtop jazz guitars needed a rapid, chopped playing style to keep up the volume they needed to be heard over the top of an orchestra. That’s a very different thing to what is needed for the parlour environment in which flat top guitars are usually used.

And the f-hole design isn’t a very suitable arrangement for a flat top guitar with a thin soundboard, because it interferes with the bracing needed to resist string tension. Like the violin family, the carved arched soundboard took much of the string tension load without the need for much extra bracing.

More importantly, f-holes would also change the vibrational modes of the guitar’s lower bout because they rely on the soundboard being firmly connected to the sides of the soundbox. The entire lower bout of a flat top guitar is crucial for the instrument to produce the timbre expected from a modern guitar, and to shift enough air to produce a reasonable volume of sound.

The rationale for the double soundslot flat top design is to increase the responsiveness of the acoustic guitar in its commonly-used context: either as a purely acoustic instrument, or with the use of a microphone in concert.

It seems there would be little benefit to the soundslot design for use with a built-in pickup, but that remains to be seen.

THE AIMS OF THE 171/172 PROJECT

Having gathered the experimental data, the obvious next step is to make a guitar with a high P:A ratio soundhole, both to confirm the experimental result and find out what the effect of the soundhole geometry will have on the guitar’s overall sound. A control instrument with the traditional round soundhole but otherwise identical would also be necessary as a comparison.

The building of two identical-as-possible guitars, one with a round soundhole and one with soundslots is what I am calling the 171/172 Project, referring to the serial numbers of the two guitars to be built.

The two instruments need to have the same area of soundhole. The round hole is 45mm in radius, giving it an area of 6,362mm2, so the high P:A slot has to have the same area. A length of 425mm and a width of 15mm achieves this, but is quite hard to fit into the guitar top without compromising the lower bout tonally and structurally.

The solution was to fit two slots around the edge of the upper bout on either side of the neck. 

This should be structurally sound, and has an extra advantage. The normal reinforcement that goes in around a centrally-placed round soundhole tends to deaden the response of the upper bout. With two slots placed at the edge, there is greater potential for the upper bout to remain live over a greater area than with the round hole. What effect this will have tonally is difficult to predict – which is one reason for building the instrument to find out.

This picture shows the design I arrived at for the soundslot geometry for instrument 172:

Diagram 6: 172 soundboard showing carbon-balsa-carbon lattice bracing and soundslots

Diagram 7: 171 soundboard showing carbon-balsa-carbon lattice bracing and round soundhole

I decided the two test instruments would be nylon sting neoclassical guitars. 

Not having any great attachment to traditional design and build, I decided to incorporate some other features in the two guitars:

  • Using a truss rod in the neck for adjustment of playability and better sustain
  • The use of a crude form of side mass-loading (in this case, a 6mm layer of bamboo lining the side of the soundbox (see Gore Contemporary Acoustic Guitar Design and Build Vol 1, 2.3.12.2))
  • Using a steel-string style headboard and tuners rather than the classical pierced style (which I find a pain to build and use)
  • Using a carbon fibre and balsa laminate for the top bracing, in the form of a lattice rather than the traditional fan bracing

Once the two guitars are complete, I will be investigating their resonance characteristics by tap testing to measure differences is projecting power and timbre.

Timbre, of course, is the more important of the two. How the vibrational modes of the soundboard will form with the new design is unpredictable, particularly with the free edges to the slots in the upper bout. 

Guitar resonance and soundhole theory – Part 14: GUITAR SOUNDBOARD VIBRATIONAL MODES

PART 14: VIBRATIONAL MODES OF GUITAR SOUNDBOARD

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”[1].

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:

This diagram comes from:

https://www.researchgate.net/publication/319683486_On_the_Fully_Automation_of_the_Vibrating_String_Experiment

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.[2]

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

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. [3]

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[4]

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.


FOOTNOTES


[1]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.

[2]These points of maximum displacement from rest in either direction are called “antinodes”.

[3] 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.

[4]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.

Guitar resonance and soundhole geometry – Part 13: YOUR CAR AS A HELMHOLTZ RESONATOR

APPENDIX 6: THE CAR AS HELMHOLTZ RESONATOR

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.

Guitar resonance and soundhole geometry – Part 12: EVALUATION OF THE PERFORMANCE OF THE ACOUSTIC HOOD

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.

Guitar resonance and soundhole geometry – Part 11: THE ACOUSTIC UNITS USED IN THE EXPERIMENT AND WHY

PART 11: ACOUSTIC UNITS

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:

β (dB)I (W/m2)
01.000E-12
101.000E-11
201.000E-10
301.000E-09
401.000E-08
501.000E-07
601.000E-06
701.000E-05
801.000E-04
901.000E-03
1001.000E-02
1101.000E-01
1201.000E+00

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.

Guitar resonance and soundhole geometry – Part 10: EXPERIMENTAL EQUIPMENT USED

PART 10: EQUIPMENT USED

CREATING AND ANALYSING THE AUDIO SIGNAL

  • 60sec chirp signal generated using Audacity 2.0.5 (Generate/chirp)
  • Cavity response recorded using Audacity 2.0.5
  • Frequency spectrum of response produced by Audacity Fast Fourier Transform (Analyse/Plot spectrum)

INJECTING CHIRP SIGNAL INTO CAVITY

  • Chirp signal played from Classic iPod connected by audio cable to Digitech stereo 18W rms amplifier
  • CW2190 4” shielded woofer (70Hz-7kHz response) speaker injects signal into cavity

RECORDING CAVITY RESPONSE

  • Cavity response was recorded by Samson Meteor digital condenser microphone

Guitar resonance and soundhole geometry – Part 9: EXPERIMENTAL VALIDATION

PART 9: EXPERIMENTAL VALIDATION

How well do the results from a rigid cavity compare with a real guitar soundbox?

DIFFERENCES BETWEEEN A RIGID CAVITY AND A REAL GUITAR

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[1] treats the soundbox as three coupled resonators: the top plate, the air-cavity, and the back plate.[2](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.[3]

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:

  1. Round soundholes radiate most strongly in the band from 100 to 600Hz, and hardly at all in the higher frequencies
  2. 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
  3. The greater the radius, the better the sound radiation becomes in Octave 3 (165 to 330Hz), central to the scale range of the guitar
  4. 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 VALIDATION EXPERIMENT

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.[4]

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:

  1. a measurement of the sound travelling directly from the driving speaker to the microphone without mediation through the soundbox
  2. 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[5]. 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


ANALYSIS OF HR NUMBERS

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 [7], 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.


VALIDATION EXPERIMENT CONCLUSIONS


  • 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.

FOOTNOTES

[1]Contemporary Acoustic Guitar Designfrom 2-10 to 2-23

[2]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.

[3]Notice that the frequency of each of these peaks shifts slightly after coupling, showing the “repulsion effect”.

[4]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.

[5]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.

[6] This equation predicts that the Helmholtz resonance fHwill change in the following ways:

  1. increase as the speed of sound increases
  2. increase as the soundhole gets larger
  3. decrease as the soundbox volume gets larger
  4. decrease as the α factor gets larger

[7]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.