Guitar resonance and soundhole geometry – Part 1: A MODEL FOR HOW ACOUSTIC GUITARS WORK

SOME BASICS ABOUT HOW ACOUSTIC GUITARS WORK

This is the first 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.

The questions I ask and try to answer in this series of posts on guitar soundhole performance are suggested by my own experience as a guitar maker, and my physics-based research into how acoustic instruments actually produce sound.

The questions are:

o  What effect does soundhole geometry (size, shape, and placement) have on how a guitar sounds? (see Part 4)

o  What is the source of the sound that comes from the soundhole of a guitar?

o  If soundslots do perform better than round holes, is there any disadvantage in dividing the slot into two segments? (Designing a single long and narrow soundslot with area equivalent to the traditional 50mm radius soundhole is difficult without compromising soundboard strength.) 

o  Do soundslots (holes with a high perimeter to area ratio) perform better than round soundholes of the same area? [1]

Given my plan to use a rigid box for my experiment, how well do experimental results derived from a rigid cavity translate to a real, flexible guitar soundbox?

 

AN ASSUMPTION…

The assumption underlying this paper is this: 

Guitar making need not be a matter of reproducing traditional geometries and methods.

Guitar makers can be designers and innovators if they understand better how the instrument works. New materials are worth investigating (such as bamboo and carbon-balsa-carbon laminates for bracing). 

So are different soundbox geometries.

…AND ITS DANGERS

I can best express the dangers in my assumption by saying that I have come to see an acoustic guitar as a box with strings that makes sounds our brains can recognise as “guitar”. “Guitar” equates to a particular sound, not to a particular box.

There’s nothing wrong with an instrument sounding unusual, but there comes a point when you have to come up with a new name for the sound because it’s no longer “guitar”. Electric guitars were unacceptable for a while before people were willing to expand their definition of “guitar”.[2]

INTRODUCTION

The traditional shape for a guitar soundhole is circular, with a radius of about 50mm.

As with all traditional design solutions, it is worth asking whether round soundholes of this size have been proved by trial and error to be the best design, or whether contemporary designers simply accept tradition and no longer wonder whether better designs are possible.

An article published by Royal Society Publishing Proceedings Ain 2015 called

The evolution of air resonance power efficiency in the violin and its ancestors(Hadi T Nia, Ankita D Jain, Yuming Liu , Mohammad-Reza Alam, Roman Barnas, Nicholas C Makris)

The authors trace the evolution of violin-type instrument soundholes from the 10th to the 18th century, from the initial round shape to the familiar classical f-hole. 

The Royal Society paper argues that violin-style f-holes are in fact considerably better sound radiators than round soundholes.

The paper is wide-ranging, so here are the main points from my point of view:

By determining the acoustic conductance of arbitrarily shaped sound holes, it is found that air flow at the perimeter rather than the broader sound-hole area dominates acoustic conductance (my emphasis)

….As a result of the former, it is found that as sound-hole geometry of the violin’s ancestors slowly evolved over centuries from simple circles to complex f-holes, the ratio of inefficient, acoustically inactive to total sound-hole area was decimated, roughly doubling air-resonance power efficiency (my emphasis).

F-hole length then slowly increased by roughly 30% across two centuries in the renowned workshops of Amati, Stradivari and Guarneri, favouring instruments with higher air-resonance power, through a corresponding power increase of roughly 60%…

Here’s a quick summary of what I find interesting as a guitar maker and designer:

  • It turns out that air can move in and out through a soundhole more effectively if the hole is longer and narrower than if it is shorter and rounder. This will particularly influence the instrument’s low frequency response.
  • The air contained in the soundbox of an acoustic instrument contributes to the overall sound by vibrating in a simple way. When the strings make the soundboard vibrate, the air inside responds by expanding and contracting in and out through the soundhole. This is sometimes called the “breathing mode” [3], because the soundboard moving up and down acts just like the diaphragm at the bottom of our chest cavity driving air in and out of our lungs.[4]
  • The reason for this is that most of the air movement happens at the edges of the hole rather than towards the middle. Compared to the air movement at the edges, the centre is acoustically inactive. [5]
  • Making a soundhole longer and narrower reduces the size of the acoustically dead central area of a soundhole and can “roughly double air resonance power efficiency”.

We need to carefully note that the RS article refers specifically to low frequency sound radiation from the violin air cavity only. The frequencies involved are around 100Hz – the low E string on a guitar sounds at 82Hz. A violin radiates much more complex sound from other surfaces than that generated by the air-cavity alone, as does a guitar.[6]

As an added bonus, the results of the resonance experiments I have carried out show that the scope for re-designing guitar soundholes in fact goes beyond just control of the low frequency air-cavity response.

Let’s ask a practical question to start with. What happens if you block off the soundhole of a guitar?

This is what [7]:

Figure 1: Change in response of a Jumbo guitar when its soundhole is blocked

The blue line is the tap response with the soundhole open. It’s clear that the open hole response of this guitar is strongest at 92Hz (very close to F#2). But this response is choked off when the soundhole is blocked, as shown by the red line. 

The measurements show that the radiated power of the sound coming from the guitar over a wide frequency range drops by about 30% when you close off the soundhole (0.027W/m2open drops to 0.019W/m2closed).[8]

Here is the result of an identical experiment with a violin:

Figure 2: Change in response of a violin when the f holes are blocked

The air-cavity response for the much smaller violin body is at 271Hz (about C#3), and is drastically choked off when the f holes are closed.

Although the responses of the two instruments are different, they share the same air-cavity resonance mechanism as part of their bass tone production. There’s reason to believe, then, that guitar designers can profit from the knowledge violin makers have arrived at by evolution over a number of centuries.

So air-cavity responserefers to the ability of the air contained in a stringed instrument’s soundbox to resonate at certain frequencies.

HOW IMPORTANT IS THE AIR CAVITY RESPONSE?

We’ve established that when listening to an acoustic guitar, about 30% of the sound you hear comes from the soundhole.

As I will show later, only a very small portion of this total 30% comes from the air-cavity itself vibrating. This may seem confusing right now, but it’s important to keep it in mind – I will explain. For the moment we’ll leave the question of where the rest of that 30% comes from.

This series of posts attempts to explain why a resonance that contributes such a small part of the overall projected sound is so important, and how the geometry of the soundhole can influence it.

The other roughly 70% of the sound you hear from a guitar comes from the vibrating soundboard pushing on the air next to it. Pressure waves travel out into the environment from the moving board surface. These are more complex and about twice as strong as the soundhole signal [9].

Soundboard signals are multipolarbecause the board has several different modes of vibration available to it (monopole, cross dipole, long dipole etc; see Appendix 8andContemporary Acoustic Guitar1-74).[10]

GUIDES FOR INNOVATION

To avoid trial and error, some scientific background is important to understanding what might be fruitful lines to work along for innovation to be successful.

The best source I have found for understanding the principles is Contemporary Acoustic Guitar Design Volume 1by Trevor Gore and Gerard Gilet. Its mathematical modeling approach can be a little intimidating, but you can skip the maths if you want and concentrate on their conclusions. 

Much of what is in this report is inspired and informed by their work.

RESONANCE AND RESONATORS – SOME BASICS

If you haven’t come across the term resonance in a technical sense before, Appendix 1 summarises most of what you need to know to understand the majority of this report. Here are some basics:

A simple resonator is something that vibrates in a predictable way after it has been displaced from a stable position. A pendulum is an example, as is a guitar string or a drumhead.

Resonance happens when a resonator responds to exactly-timed external pushes that cause the resonator to continue or build up its motion rather than coming back to its rest position.

SIMPLE PENDULUM

In the case of real-world examples, often an audio resonator will respond to impulses of one (or more likely several related) frequency that it selects out of a complex sonic environment. [11]

Think of a child’s swing – when you give it a push it swings away from the push, slows, stops and then moves back toward you. If you leave it alone it keeps swinging but eventually comes to a stop at its original stable position.

The important requirement for something to vibrate around a central position is a restoring force

In the case of a swing, the restoring force is gravity, or the weight of the swing. As the swing moves away from its stable rest position, its weight begins to pull it back. The swing slows down, stops, and then accelerates back toward the centre. When it arrives back at the central position it is moving at its fastest and overshoots, slowing again as it moves out towards the other extreme of the cycle.

RESONANCE

If you want to keep a vibration going for longer, or even build it up, everyday experience shows us that you need an external force to push at just the right moment in each swing – it demands a particular frequency of push before it responds, and it also demands that you push at the right moment in its cycle. [12]

Think of pushing a kid on the swing to build up the amplitude.

For the purposes of this paper, it is the need for a particular frequency of driving impulse to get a response is one of the most important characteristics of a resonator.

In the case of an acoustic guitar, my experiment focuses on the resonance of the air contained inside the soundbox – the airbody.

The airbody is one out of the three most important resonators that make up a guitar.

HOW CAN A FLABBY LUMP OF AIR RESONATE?

This question is crucial to understanding my experiment and its results.

If it’s contained in a soundbox, air can “slosh” in and out through the soundhole because it’s a fluid. It will slosh with a particular frequency determined by the size of the box and the size and shape of the soundhole.

In a real instrument the vibration of the soundboard pushes the air in and out through the soundhole.

When the soundboard moves down it pushes air out, and when moving up draws it back in. Even without going into more detail at this point, it’s clear that there’s a very strong connection between soundboard resonance and airbody resonance.

In my experiment, the airbody is driven to resonate not by a vibrating soundboard but by a loudspeaker pumping a pressure wave into a guitar-shaped cavity. When the pressure in the cavity is high, air is pushed out through the soundhole, then drawn back in when the pressure is low. Details of the experiment are in Part 3

COUPLED RESONATORS

As Gore and Gilet show [13], acoustic guitars can be modeled quite accurately by thinking of the soundbox as three coupled resonators: 

  • the soundboard
  • the air body contained by the soundbox 
  • the sides/back of the box. 

Picture these three components as three masses connected by springs:

Figure 3: Three coupled resonators: an analogy for an acoustic guitar soundbox

If you pull the top mass down and let it go (as in tapping the soundboard of a guitar), you disturb all three and they will all necessarily start to vibrate. The three resonators will “fight it out” between themselves until they come up with a compromise that will allow them all to vibrate in harmony with each other.

They are coupled and they form a single system, although each one keeps its own resonant characteristics.

If you now change the nature of one of the resonators, the others must also change in response because they are coupled. For example, if you were to change the mass or the stiffness of the soundboard, the resonant frequency of all three would change in response.

The fundamental principle of conservation of energyis very important here. The energy you put into the system by moving the top mass and stretching the springs is spread among all three as they begin to vibrate. 

Resonance does not create energy. In a guitar the only energy the soundbox has to work with is the amount given to the strings when you pluck or strum them – the job of the instrument is to turn this rather small amount of mechanical energy into sound energy as efficiently as it can.

With the three mass/spring model of the guitar soundbox, what happens if you block off the soundhole? That would be like clamping the centre mass so it can no longer vibrate:

Figure 4: Three mass model with the soundhole blocked

Clamping the central mass changes the system drastically. Both the soundboard and the side/back elements revert to their own resonant characteristics – it’s no longer a coupled system. [14]

Keep in mind that for a system to vibrate it must have both mass (inertia) and elasticity (more strictly, a force that acts to return the resonator to its stable position). Both force and mass determine its behavior.

Anything that affects one element in a system will affect the others. In the case of the three coupled resonators that make up a soundbox, a change to one will affect the sound of an instrument as a whole.

This series of posts concentrates entirely on the characteristics of:

  • the airbody’s mass – determined mainly by the size of the soundbox 
  • the airbody’s elasticity – determined mainly by the soundhole.

Blocking off the soundhole has the effect of clamping the second of the three resonators –called decoupling.

You can see the effect of the decoupling in the graph below. This is the spectral signature of a Jumbo sized guitar, produced by tapping, and there is a lot of information about the instrument contained in it. [15]

Figure 5: The effect of blocking the soundhole of a guitar

It’s well worth another look at this spectrum now that we know more about what’s happening when you block a guitar’s soundhole.

The blue line shows the response to tap testing with the soundhole open, and the red with it blocked off. Blocking the soundhole stops the air-cavity vibrating, and hence removes the main connection between the top and the back – hence coupled responseanduncoupled response

The air-cavity response (the strong peak at 91Hz on the blue line) is cut off when the soundhole is blocked.

The red/blue peak to its right at just under 176Hz is the main soundboard response, which is unchanged except for a slight frequency shift (from 176Hz open to 174Hz closed). Just to the right of that you can see the main back response at 204Hz, which also disappears when the soundhole is blocked. [16]

The challenge in designing and building a good guitar is to set the mass of the soundbox parts and their stiffness to make the best use of the energy being pumped in by the strings. 

The purpose of this report is to describe the effects of soundhole geometry on guitar resonance, beginning with the air-cavity response.

By blocking off the soundhole of an instrument it is possible to measure that the air-cavity/soundhole contributes about 30% of the projected sound, though this varies one instrument to another. The remaining 70% of the sound is projected by the vibrating surface of the soundboard. 

Clearly a doubling of the air-resonance power efficiency in a guitar would usefully increase its projection – taking the 30% figure as an example, a doubling would produce an extra 15% in total radiated power.[17]

The air-cavity resonance is most marked at low frequencies, so increasing its efficiency will also alter the instrument’s tonal balance toward the bass end. 

This offers a way of shaping the instrument’s sound at the design stage.

The experiments I am describing in the next sections of this paper are designed to investigate the airbody/soundhole response of a guitar body in isolation.

To do this the airbody has to be contained in a box that is too solid to easily vibrate. Looking at the triple mass/spring model in Figure 7, I set up the apparatus to be the equivalent of holding the top and bottom masses firmly in clamps to isolate the airbody response. This uncouples the system and allowed me to study the airbody response by its self as I changed the size and shape of the soundhole.

Figure 6: “Clamping” the soundboard and sides/back of box to isolate the airbody response

You may quite likely be wondering what the springs linking the airbody to the soundboard and the sides/back component represent in a real soundbox. This will be explained more in Part 2, using the concept of Helmholtz resonance.

SUMMARY

  • The experiment I am reporting on in this series is designed to give information to help in the design of the soundhole of a guitar
  • A grasp of the concept of resonance is important in reading this experimental report
  • About 30% of the sound coming from a guitar is radiated from the soundhole; however only a small part of this comes directly from airbody resonance
  • A good way to visualize the workings of an acoustic instrument’s soundbox is to think of it as three coupled oscillators: the soundboard, the airbody, and the sides/back
  • Air cavity resonance is determined by the size of the cavity and the properties of the soundhole
  • The importance of the air cavity resonance is in providing an important part of the coupling between the top and back/sides resonators
  • The experiment described used a rigid cavity to isolate the airbody response, allowing study of changes to the soundhole geometry

WHAT’S NEXT?

Part 2 in the series delves into soundbox resonance in more detail, and introduces the important concept of Helmholtz resonance.

FOOTNOTES


[1]  This question comes from reading a paper published by the Royal Society titled The evolution of air resonance power efficiency in the violin and its ancestors

[2]I personally love the idea of new sounds, but guitarists have legitimate expectations about what they expect from an instrument.So it comes down to the riddle of when a guitar is not a guitar.  In my opinion there is a certain amount of mythology about instrument sounds. For example, blind tests of highly experienced professional violinists show that picking a Stradivarius from supposedly lesser instruments is not as clear cut as we might think. Psychoacoustics plays a large part in it.

[3]See http://www.acs.psu.edu/drussell/guitars/hummingbird.htmlfor an animation of the breathing mode – more on this later in this series.

[4]You can make a guitar blow out a candle by pointing the soundhole at the flame and tapping the soundboard.

[5]The reason for can be found in hydrodynamic theory, often referred to as the Bernoulli effect

[6]“Monopolar” means that an instrument’s soundhole acts as a single point source that radiates uniformly outwards in all directions like the ripples from a pebble dropped in a pond – this is the simplest form of radiator. 

[7]This information comes from the tap method, which uses a small padded hammer to tap the instrument

[9]A hint about the radiation from the soundhole: the soundboard also projects sound intothe soundbox.

[8]These figures, like all thepower numbers in this paper are relative only, as I’ll explain later.

[10]Though it is not the focus of this paper, an excellent set of animations to show these more complex modes can be found at:

https://www.acs.psu.edu/drussell/guitars/hummingbird.html

[11] I will talk a lot about Helmholtz resonance in this paper. Before electronic equipment existed, Helmholtz used banks of audio resonators to analyse the spectral content of complex sounds, each one responding to a narrow, known, frequency band.

[12]The timing of the driving impulses is known as phase – in phase maintains or builds up resonance, where out of phasedamps it out.

[13]Contemporary Acoustic Guitar Vol 1, P 2-3

[14]It isn’t quite that simple in reality, because of course the top and sides and back are glued together, so they do remain mechanically connected together. Strangely, this connection is often less important for low frequency resonance than the airbody connection.

[15] The reason this graph looks different from Figure 1is that I have left the results in deciBels (dB) rather than convert to relative radiated power

[16]This isbecause I tapped the soundboard only. In its decoupled state, with no strong connection to the top, the back doesn’t respond strongly to the tap.

[17]This last point isn’t evident from Figure 6because the strength of the tap was inevitably different between the open and closed measurements. A “standard tap” would help here, and it illustrates one of the difficulties of trying to make direct comparisons between instruments using the tap method.

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