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. 
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. 
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 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.
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 wavesof a frequency determined by the path-length allowed by the cavity walls (known to acousticians as “room modes”) .
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 , 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 diffractionprocesses 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.
The radiated sound is much more coherent than for the longer wavelength in the first example.
Figure 9: High frequency wave pattern
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. 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.
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.
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.
Impedance is the measure of how easily energy can make its way through the interface between two different media.
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.
Using the equation where c is the speed of sound and L is the cavity length
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.
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.
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.
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.