This is the second in a series of ten posts summarising the results of an experiment I carried out during 2017 and 2018 to try to increase my understanding of resonance in acoustic guitars, and in particular how the design of soundholes could be improved.
The first post of this series put forward a simple model to help understand what happens when an acoustic instrument is played.
The model represents the soundbox as an interaction between three soundbox elements, the soundboard, the air contained inside the soundbox, and the sides/back of the box:
Figure 1: the three mass/spring model for a soundbox
Any up and down movement in the soundboard transfers into the whole connected system via the springs, and causes all three elements to vibrate – each has its own natural frequency, so they “fight it out” and find a way of responding that works for all three.
Now, Part 2 here we come…
In this post we’re building a theoretical knowledge of guitar acoustics, focusing on Helmholtz resonance.
PART 2 is all about a fundamental type of resonance that powers the acoustic response of a guitar. You may quite reasonably choose to skip this post and move on the the next few, but you will probably need to come back to it later to fully grasp how a guitar works.
One part of a real airbody’s response is to “slosh” in and out of the soundhole, pumped back and forth by the vibrating soundboard. This “sloshing” or “breathing” response is called Helmholtz resonance.
A more familiar example of Helmholtz resonance happens in a car when one of the back windows is lowered while driving. The air in the car vibrates in and out through the window at a frequency determined by the size of the car interior and the amount the window is open (see Appendix 7). The signal that excites the airbody in the car is the turbulent airflow rushing past the window opening.
An important thing to keep in mind is that Helmholtz resonance is an air pressure variation inside a rigid container – in a car, you can feel the rapid pressure changes in your ears because they’re quite low in frequency.
Figure 2: the car as a Helmholtz resonator
This spectrum shows a strong response peak at 15Hz. Human audition can’t experience frequencies below 20Hz as continuous tones, so the car’s Helmholtz resonance is an unpleasant pressure cyclic pressure variation.
Helmholtz resonance in a car is somewhat like blowing over the top of a bottle to make it sound a note. The air in the bottle has its own particular natural frequency at which it will vibrate in and out through the neck. If this frequency is present in the airflow over the neck, the air in the bottle will feel it and respond to it by resonating. (See Appendix 9 for more detail about air resonances in bottles and pipes.)
You can perhaps also begin to see the similarity between what happens in the car and what happens in a guitar soundbox. In the case of the guitar, the driving impulse is provided by the strings vibrating and moving the soundboard up and down.
So it’s no surprise that discussions of the air-cavity/soundhole response revolve around Helmholtz resonance (which I’ll refer to as HR for the rest of this paper).
Helmholtz Resonance takes place when air in a container with a small opening in it is excited by an external signal of some kind. A guitar soundbox excited by the strings is an example.
THE HELMHOLTZ EQUATION
Professor von Helmholtz himself derived an equation to predict the resonant frequency for a rigid spherical flask with a short neck (see diagram below).
(c = speed of sound; S = area of the neck aperture; V = volume of resonator body; L = length of neck)
His experiments led to this equation that predicts the frequency the flask selects out for resonance
The Helmholtz equation predicts that the wider the flask’s neck (giving a larger area S), the higher the resonant frequency will be. This will be important when we see what part Helmholtz resonance plays in a guitar soundbox.
The larger the volume of the sphere V, the lower the frequency – an inverse relationship. This is the same for the neck length L as well.
You will have noted right away that a though a guitar soundbox isn’t spherical, and the soundhole doesn’t have a hollow neck projecting from it , it isa resonant cavity with an aperture.
Figure 3: the Helmholtz resonator ( http://newt.phys.unsw.edu.au/jw/Helmholtz.html)
Helmholtz thought of the air contained in the bottle’s neck as separate from the air in the cavity: he used a mass/spring model where the neck air is the mass, and the airbody inside the bottle is the spring.
The model predicts the fundamental Helmholtz frequency very well.
There is one complication to consider. The effect of the resonance actually extends a small distance outside the neck, so a factor called the “end effect” needs to be added in to the basic equation before it will predict with full accuracy. It adjusts the size and mass of the “neck plug” to make the model more realistic.
The UNSW website has this to say:
The extra length that should be added to the geometrical length of the neck is typically (and very approximately) of 0.6 times the radius at the outside end, and one radius at the inside end.
In my own spherical Helmholtz resonator experiment I used the UNSW figure to apply the effective lengths of the different necks I used, and it resulted in a very good match between calculation and measurement.
I made a 128mm internal diameter Helmholtz resonator (I used a small globe of the world) with an adjustable-length neck (I sawed bits off as I went…) to confirm the equation.
I blew across the neck to excite the air-cavity, digitally captured the response and analysed it with Audacityas described in the Experimental method section below. This is the response of the spherical airbody with a 13mm long neck:
Figure 4: measured response of 128mm diameter Helmholtz resonator with 13mm neck
This is the spectral response of the resonator with a 13mm long neck. There is a clear series of resonance peaks between 0 and 1000Hz – my first surprise. Why? Well, the Helmholtz model predicts only one resonance, not a series of them. The spring/mass model used in deriving the simple equation can have only one solution, and it has no place for a harmonic series .
Here is the same data, showing a closeup the response from 0 to 600Hz:
Figure 5: harmonic series produced by 128mm diameter Helmholtz resonator with 13mm neck
The resonant peaks occur at 117.2Hz, 237.3Hz, 354.5Hz, 471.7Hz, and 585.9Hz. These are very close to whole multiples of the 117.2Hz fundamental, so clearly represent a harmonic series. 
But how well does the frequency of the first harmonic compare with the model? The next graph shows the data for the fundamental frequency for the whole set of different resonator neck lengths, next to the model’s prediction.
Calculations from the Helmholtz equation used the UNSW recommendation for end effect correction:
Effective neck length = 0.6R + R + L
HOW WELL DOES THE EQUATION WORK?
I do recognise that unless you’re as much of a Physics tragic as me, you don’t really care. Anyway, these are the fundamental frequencies I found for the spherical resonator with a range of different neck lengths (H) from 13mm to 58mm:
Figure 6: Helmholtz resonance fundamentals for 128mm spherical cavity
The predictions match the calculated values very well.
Figure 7: Response of 128mm diameter Helmholtz resonator compared to theory
The variation between the measured results and the calculated model predictions ranges from 1 to 6% – quite good agreement.
Gore and Gilet (Contemporary Acoustic Guitar, Vol 1 1-29) are correct when they say that:
…The Helmholtz effect is important because it largely determines the low frequency response of a guitar.
But while developing analytical models for guitar response, Gore and Gilet warn that the resonant frequency for a real guitar cavity is not straightforward to calculate.
With the guitar, there are several complications: a guitar isn’t spherical; it doesn’t have a neck stuck on the soundhole; it isn’t rigid; and the airbody is only one out of three resonators that make up a soundbox.
So Helmholtz’s simple model may not seem to have much chance of working very well, but two of the four issues are reasonably easy to deal with.
Firstly, my experimental guitar-shaped resonant cavity is designed to be very rigid.
There is indeed no bottle neck attached to the soundhole , but the physics recognizes the “end effect” that takes into account the fact that air beyond each end of the pipe participates in the resonance, making a kind of invisible “neck”.
In the case of a guitar, “neck length” is the thickness of the soundboard at the soundhole, plus a fudge factor (oh, did I say that?) for end effects..
SUMMARY OF PART 2
Part 2 is about Helmholtz resonance, a vibrational mode of the airbody contained in a soundbox. This is introductory theory underpinning the experiment on how soundhole geometry can affect the resonant response of a guitar.
You may need to come back to this later to fully understand what the importance is – it took me several months before I could truly grasp why it is important.
In the next post we look into the process of measuring the effect of changing soundhole geometry on the response of a guitar soundbox. As far as I know, this is new research for guitar acoustics.
Here’s a picture of the rigid guitar-shaped cavity I used for my experiments:
Figure 8: the experimental rigid guitar-shaped cavity
This is made to be heavy and rigid. You can see that the space between the cavity walls and the box is packed with sand, and in operation the top is heavily damped by lead weights.
The effect in terms of the mass/spring model is to clamp the soundboard and sides/back so they can’t move – that way, the only response I measure will be from the airbody.
Figure 9: The experimental equipment effectively “clamps” the soundbox so only the airbody can resonate
…but what if it did? Perhaps this could be an easy way of tuning the soundbox resonances.
Many oscillators – like guitar strings – can resonate at a series of different frequencies, characterized by multiplying the fundamental frequency by 2, 3, 4,…,n – this is called a harmonic series
While of interest, I didn’t find any evidence that there is a similar harmonic series in the airbody response of a guitar soundbox, so I’ll leave it at that.
But what if there were…? Just a thought.