### Appendix 1.1: Standard Audiogram

*From Pietro Righini*—*Giuseppe Ugo Righini.*

### Appendix 1.2: Sound Pressure Level (or SPL)

SPL (

*Sound Pressure Level*) is the parameter commonly used to denote the power of a sound, as defined by the formula

$$SPL = 20\,Log_{10} \left( {\frac{\text{Acoustic Pressure }}{{20 \times 10^{ - 6} }}} \right)$$

In this formula

*Acoustic Pressure* is the physical value of the pressure generated by the source of the sound, and it is measured in Pascal (1 Pa = 1 N/m^{2}).

The constant 20 × 10^{−6} Pa indicates the hearing threshold, or the acoustic pressure corresponding to the faintest audible sound to which, based on this formulation, sounds generated by a specific source are referred.

The acoustic pressure level, as defined by this formula, is measured in decibels (*dB*).

The function

\(y = Log_{10} \left( x \right)\), which is applied in the SPL formula, is the

*logarithmic representation* of a generic value (

*x*). To understand how this function transforms the value

*x* into its logarithmic representation

\(y = Log_{10} \left( x \right)\) see the following table where, beside some values of the

*x*, we find the values of its logarithmic representation

*y* and the corresponding values in dB.

If we reported onto an A4 sheet the values of a variable *x* ranging between 1 and 100,000 on a linear scale, the smaller values would be so crowded in the initial part of the scale as to be hardly distinguishable. Otherwise, we should use an extremely extended, hence unmanageable scale.

The logarithmic representation contracts the greater values, so that they can be reported on a scale that clearly displays both the minimum and the maximum values of the parameter *x,* and their mutual relationship. The representation in dB enlarges the scale of the logarithmic representation.

Please note that the SPL value in dB is zero when the acoustic pressure equals the hearing threshold pressure (20 × 10^{−6} Pa), which corresponds to the faintest audible sound level.

### Appendix 1.3: Initial Transient and Decreasing Transient

The observed representation of the tuning fork sound shows that the amplitude of the tone at 440 Hz firstly increases, then decreases over time. The first part is the *rising transient*; the next is the *decreasing transient.*

The mathematical formulation of the amplitude during the rising time is:

$$A(t) = A_{0} \sin (2 \pi f t)(1 - e^{{ - t / \tau_{s} }} )\quad (e = 2.71828)$$

where

*t* is the time variable.

*A*(*t*) is the instantaneous amplitude value.

*A*_{0} is the asymptotic value that would be reached by amplitude after a very long time.

*f* is the tone frequency (440 Hz in this case).

*τ*_{s} is the *rising time constant* of the transient.

The curve connecting the sinusoid peaks (the *envelope*) is a growing exponential curve that starts from the initial time (*t* = 0) and tends to *A*_{0}, a value that would be reached after a very long time. This exponential curve is determined by *τ*_{s}, that is the value of the *rising time constant*.

After a time

*t*_{1}, when amplitude reaches the value

*A*_{1}, and the effect of the excitation is over, the decreasing transient begins. This is defined by the following formula:

$$A(t) = A_{1} \,\sin \,(2 \pi \,f\, t)(e^{{ - (t - t_{1} )/\tau_{d} }} )$$

Now the envelope starts from *A*_{1} and tends to zero, the decreasing time constant being *τ*_{d}. Notice the different formulation of the exponential function.

The two time constants

*τ*_{s} and

*τ*_{d} dominate the progression of the envelope curves in the rising and decreasing stages, and are related to the physical characteristics of the source (hence not related to excitation nor to tone frequency):

the rising time constant *τ*_{s} depends on the source *inertia*, i.e. on how promptly the system reacts. It is generally very short (in our case is *τ*_{s} = 35 ms).

On the contrary, the decreasing time constant depends on how quickly the energy stored in consequence of the excitation is dissipated, due to the internal viscous losses and to sound radiation towards the surrounding environment. It is generally much longer than the rising time (in our case *τ*_{d} = 1.4 s).

### Appendix 1.4: Fourier’s Theorem Formulation

Fourier’s Theorem tells us that a periodic signal (as those produced by a plucked string) can be approximately assessed by summing up sinusoidal functions at multiple frequency of the fundamental:

$$\begin{aligned} f\left( t \right) &= \frac{{A_{0} }}{2} + A_{1} \cos \omega \,t + B_{1} \,\sin \,\omega \, t + A_{2} \,\cos \,2\omega \,t + B_{2} \,\sin \,2\omega \,t \\ &\quad +\, A_{3} \cos \,3\omega \,t + B_{3} \,\sin \,3\omega \,t + \cdots \end{aligned}$$

The constant *A*_{0} is proportional to the continuous component of the signal (when present).

The constants *A*_{n} and *B*_{n} (or the *Fourier coefficients*) are the *amounts* that each sinusoid at the frequency \(f\left( t \right) = \frac{n\,\omega \,t}{\pi }\) contribute to the whole *in amplitude* and *phase*.

If we transfer onto a diagram the amplitude and phase of each of the sinusoids resulting from the decomposition of the original signal *f*(*t*), we get the *frequency representation*, or *spectrum*, of the signal.

The expression above shows the time representation *f*(*t*) of the periodic signal and the constants *A*_{n} and *B*_{n} that allow construction of the signal spectrum. Consequently, this expression is—in a sense—the bridge linking the *time domain* to the *frequency domain.*

Under certain conditions, it is possible to extend the Fourier’s Theorem formulation to non-strictly periodic signals, as those representing the guitar resonator response. However, in this case the sinusoidal components of the spectrum will not be integer multiples of a fundamental. Later on in the text we will investigate the relationship between spectrum, resonances, and vibration modes of the resonator.

Different algorithms based on the Fourier series are available for constructing the spectrum of a signal. The algorithm of the Fast Fourier Transform (FFT), published in 1965, is probably the most powerful tool we have today for the analysis of signals.

We implemented this computing method in our software, and we used the consequent results to analyse the response of the examined systems.

For details on implementation, please refer to specialised texts.

### Appendix 1.5: Center Values of Octaves and Thirds of Octave Bands.