### Appendix 5.1: The Resonator Model

In the next scheme the fundamental resonator components (soundboard, back, Helmholtz resonator) are represented in such a manner as to highlight both their physical structure and their interaction.

Back and soundboard are represented in this model as simple oscillators featuring mass, stiffness, and loss coefficient. The soundhole is shaped like an element defined by a mass of air and a loss coefficient that, along with the elasticity of the air in the body, define the Helmholtz resonator.

In developing the model equations we adopted the *electro*-*acoustic analogy*, that allows to relate acoustic parameters (mass, stiffness, loss factors, vibrating surfaces) to electrical parameters (inductance, capacitance, resistance). According to this analogy, the force applied on the acoustic system corresponds to the voltage applied on the equivalent electrical circuit, while the velocity in the acoustic system corresponds to the current in the electrical circuit. Furthermore, capacitance corresponds to stiffness, inductance to mass, electric resistance to loss coefficient. The electrical parameters involved in the model also depend on the *vibrating surfaces* of the resonator components (soundboard, soundhole, back) that are relevant to sound radiation.

This is not the only possible description of the phenomena involved in the guitar resonator. We opted for this one because it allows to apply techniques that are normally employed in the study of electrical circuits, and so to extend them to acoustic systems.

According to this electro-acoustic analogy, the relation between

*force* applied and velocity

*V*_{p} in the application point (usually the bridge) is the

*acoustic impedance Z*_{r} of the resonator at the bridge:

$$Z_{r} = \frac{Force\;applied\;on\;the\;bridge}{Velocity\;at\;the\;bridge}$$

This impedance is a function of the frequency: it is very low at resonance conditions (where velocity is maximum) and very high at antiresonances (where velocity is minimum).

Generally, the resonator response property is not expressed as

*impedance* Z

_{r} but rather as

*mobility*, which is the reciprocal of impedance, so

$$Mobility = \frac{1}{{Z_{r} }} = \frac{Velocity\;at\;the\;bridge}{Force\;applied\;on\;the\;bridge}$$

The advantage of the representation in terms of mobility instead than in terms of impedance is that, on the mobility diagram, the resonances assume very large amplitudes, while the antiresonances show very low amplitudes. This is in fact a more intuitive representation, the one we used in reporting the resonator measurements and in tracing a graphical description of them.

We recall the expression of the

*reflection coefficient ρ* introduced in Chap.

2:

$$\rho = \frac{{R_{c} - Z_{r} }}{{R_{c} + Z_{r} }}$$

where

*R*_{c} is the

*characteristic resistance* of the string, defined as:

$$R_{c} = \frac{Longitudinal\;Tension}{Waves\;travelling\;speed} = \sqrt {\mu T}$$

At resonant frequencies (where the resonator impedance is low) the wave that propagates along the string, from the nut towards the bridge, is mostly absorbed and exploited by the resonator; the available energy is returned as sound pressure through the oscillation of the resonator surfaces (soundboard, back, soundhole), while the reflection is too scarce to support the oscillation of the string that, as a consequence, quickly damps down.

At antiresonances (where the resonator impedance is high) the incoming wave at the bridge reflects almost completely without energy absorption on the resonator part, so the oscillation amplitude of the vibrating surfaces is small.

The force in the previous scheme is the *force that the string actually applies on the resonator, net of reflection*.

By the previous scheme we understand that the force received by the soundboard in the application point (usually the bridge) is used in part to excite the soundboard and in part to excite the other resonator components (air, soundhole, back). The fraction exploited by the soundboard determines its displacement velocity *V*_{p} in the application point, while the residual fraction determines the displacement velocity *V*_{b} (of the air in the soundhole), *V*_{f} (of the back) and *V*_{a} (of the air in the body).

So the three effective radiant surfaces of the resonator (soundboard, soundhole, back) move respectively at velocity *V*_{p}, *V*_{b}, and *V*_{f.} Each of them oscillates an *air volume* whose oscillating energy depends on both the extent of the working surface and the oscillating velocity of the air volume itself.

The motion of each of these air volumes brings about in the environment a sound pressure that is individually generated by each of the resonator radiant surfaces. At a certain distance from the body, the *sound field* generated by the instrument is the result of a combination between amplitudes and phases of the sound pressures due to the various vibrating surfaces.

When the surfaces that define the resonator oscillate in the surrounding air, they meet a *radiation resistance*, that is the resistance of the air to be set into vibration. This resistance causes a loss of oscillating energy, hence a damping of the oscillation in the vibrating surfaces. In the model it is merged into a single resistance parameter with the losses due to viscous friction in the material.

On the previous scheme we notice that the resonator soundhole is defined by the mass of air it contains and by the loss coefficient that—as already observed—also takes into account its radiation resistance. The impedance of the element that represents the soundhole grows with the frequency, since the mass of air (the inductance in the equivalent electrical circuit) tends to oppose the quick variations of the force applied (the voltage at the ends of the equivalent electro-acoustic circuit). Accordingly, as the frequency increases, the velocity of the air in the soundhole *V*_{b} falls progressively down to zero. When this velocity is very slow or—in case—nil, the soundhole, as far as acoustics is concerned, is virtually closed. In the frequency range that involves the basic resonances and the first resonances of the back, the soundhole is working and offers an effectual contribution to sound radiation while, beyond this limit, its role gradually comes to an end.

If we impose a very high stiffness value on the back in the preceding scheme, we get a situation where the back is perfectly rigid. It can be helpful, during construction of the instrument, to couple provisionally the soundboard—possibly glued to the frame—with a rigid back. This procedure can help in optimizing the soundboard when still accessible and modifiable, i.e. not yet secured with an elastic back. Yet in some instruments the back is kept intentionally rigid. The ‘rigid back’ scheme enables studying these kind of instruments as well.

On the previous scheme we can see that back and soundboard are *structurally* similar, being both shaped like simple oscillators characterized by mass, stiffness, loss coefficient, and vibrating surface. This analogy calls for the examination of a case where back and soundboard mutually *exchange their roles*, according to the next scheme.

Here the force is applied on the back (usually at the level of the bridge) and no more to the soundboard as before. The values of the distinctive parameters of the model do not vary, in comparison with the previous case, since they are related to the physical nature of the resonator components. Conversely, the model response changes: in this situation, the resonances of the back stand out above all. This setting allows in fact a study of the contribution of the back to the global resonator response, and an optimization of the placement of the resonances of the back with respect to those of the soundboard.

The last scheme concerns the case when the soundhole is closed and the back is rigid (or deemed so). In this situation the soundboard operates on the air in the body, counteracting its elasticity. The force applied on the soundboard is exploited partly in exciting the soundboard, partly in oscillating the air. Air and soundboard oscillate at the same velocity. Clearly, there is no airflow from the body interior towards the outside, so the sound emission is entirely due to the soundboard.

This simplified system features a single resonance at a frequency we will call **F**_{p}. As already observed, we need to know the **F**_{p} to assess the other resonator parameters: chiefly the vibrating mass and, secondly, the vibrating surface and the coupling coefficient.

### Appendix 5.2: Verification of the Model Outcomes

The exactness of a model must always be verified by comparison between experimental results and evaluations that the model itself provides. In other words, we must make sure that the model outcomes are sufficiently reliable.

For this purpose we refer to formerly mentioned data, obtained by measurements executed on a reference guitar, that we recall here for the reader’s convenience:

**F**_{1} (air resonance) = 93 Hz,

**F**_{2} (soundboard resonance) = 213 Hz,

**F**_{h} (Helmholtz resonance) = 129 Hz.

**F**_{p} (covered soundhole resonance) = 196 Hz.

In addition, the value

**F**_{p0} of the soundboard natural frequency:

that was available during the instrument construction. In many cases (e.g. finished instruments) a direct measurement of this parameter is obviously not possible; as an alternative, the **F**_{p0} can be obtained by estimation via the model.

The first fundamental relation we employ after development of the model equations is:

$$F_{p}^{2}\, (estimated) = F_{1}^{2} + F_{2}^{2} - F_{h}^{2} \quad so\quad F_{p} (estimated) = \sqrt { \left( {F_{1}^{2} + F_{2}^{2} - F_{h}^{2} } \right)}$$

This equation allows to *estimate* the covered soundhole resonant frequency of the resonator when we know the two basic resonances (*F*_{1} and *F*_{2}) and the Helmholtz resonance *F*_{h}, and also allows to assess the inaccuracy of the model by means of the known (*measured*) *F*_{p}.

Setting these data into the equation above we get *F*_{p} (*estimated*) = 193.3 Hz, in front of a *measured F*_{p} = 196 Hz (covered soundhole measurement on the reference guitar).

The error between the estimation and the experimental value is equal to 2.7 Hz: a tolerable discrepancy, considering that we are comparing experimental data with calculated ones.

The preceding formula is congruent with the theory of coupled resonators we have formerly cited, whereby *the sum of the squares of the resonant frequencies in the coupled system is equal to the sum of the squares of the resonant frequencies in the uncoupled system*. As for the resonator, the frequencies of the coupled system are the two basic resonances **F**_{1} and **F**_{2}, while the frequencies of the uncoupled system are **F**_{p} and **F**_{h}.

The second fundamental equation taken from the model is

$$F_{p0} (estimated) = \frac{{F_{1} F_{2} }}{{F_{h} }}$$

This equation allows to estimate the natural frequency of the fastened soundboard, when we know the two basic resonances **F**_{1} and **F**_{2} and the Helmholtz resonance **F**_{h}.

Setting the given values of **F**_{1}, **F**_{2}, and **F**_{h} into the equation we get the model estimation of the soundboard natural frequency: **F**_{p0} (*estimated*) = 153.6 Hz. The experimental value measured on the reference guitar is 155 Hz. Once again, the error between the model estimation and the experimental value is very small (about 1.4 Hz).

The values estimated through the model, either regarding the covered soundhole resonance **F**_{p} or the soundboard natural frequency **F**_{p0}, are very close to the experimental findings. This means that the model is correct, apart from expectable, little differences between calculated and experimental results. These differences are due to approximations introduced in order to describe the real functioning of the guitar resonator in a simplified and computable manner.

However, the model is sufficiently reliable for our purpose of evaluating the quality parameters in an instrument finished or under construction.