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.

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