We have looked in some detail at the vibration behaviour of stringed instrument bodies: but how important are all these details for the sound of the instrument? It is time to turn our models into something we can listen to. For the violin, that will involve understanding what happens when you bow a string, which will take us into the tricky territory of non-linear systems. We will put this off, until chapter ?. But for plucked-string instruments like the guitar and the banjo we can get a long way with linear ideas.

What happens when you pluck a string? The simplest description is that the player pulls the string to one side (with a plectrum, fingernail or the flesh of a finger), then lets go. The string is then left to its own devices, to vibrate freely and to feed some of its energy into the instrument body. When pulled aside, the string will take up a triangular shape as sketched in Fig. 1. The player can choose the plucking point along the string, and the orientation of the pluck. Otherwise, the only thing likely to vary is the sharpness of the corner: a small, sharp plectrum will make a very sharp corner, the flesh of a finger will make a more rounded one.

The very simplest model is to assume a perfectly sharp corner, and an ideal string without damping. We already know the mode shapes and natural frequencies for this ideal string from section 3.1.1. For this simple system there are three ways we can find the string motion following the pluck. They are all useful in their different ways so we will briefly summarise all three.

First, we can say that after the string has been released, the ensuing free motion must consist of a linear combination of the modes, each vibrating at its own natural frequency. This statement is, after all, true of *any* free motion of *any* linear system. All we need to do is find the amplitude and phase of each modal contribution. This can be done by expressing the initial triangular shape as a Fourier series: the slightly messy mathematical details are given in the next link.

One virtue of this approach is that it does not rely on the particular assumed shape of the pluck: it could be applied to any other initial conditions, such as a rounded corner from a finger-flesh pluck. The method would be followed in the same way, based on calculating the Fourier series representation of the initial shape. The effect of rounding off the sharp corner would be to reduce the amplitudes of the higher Fourier components. You need short wavelengths, corresponding to high-frequency modes, to represent a sharp corner, but a rounded corner will have a natural cutoff at the wavelength matching the curvature of the corner, above which the Fourier coefficients will become small. The result is a familiar one: the sound of a finger-flesh pluck is less bright than a plectrum pluck because it contains less energy at high frequency.

We can learn something interesting about the sound of plucked strings from the result of the calculation in the previous link. We can’t calculate ‘sound’ as such from this model string with rigidly fixed ends, but we can calculate the waveform of force exerted by the vibrating string on the bridge. This is the driving force for body vibration, and so is closely related to the sound that an instrument would make. The amplitude of the $n$th harmonic of this force signal is proportional to the mode shape at the pluck point, divided by the harmonic number: in other words, to $|\sin(n \pi a/L)/n|$ where $a$ is the plucking point on a string of length $L$. This amplitude is plotted in Fig. 2, against harmonic number $n$ and relative pluck position $a/L$.

This plot reveals several interesting features. For every plucking position, the fundamental ($n=1$) has the largest amplitude. But the value of that fundamental amplitude, for a fixed plucking force, is higher when plucking far from the bridge, and gets progressively lower as the pluck point gets nearer to the bridge. Every guitarist is familiar with the change in loudness and tone quality that results from plucking closer to the bridge: a synthesised example of the effect will be given at the end of this section.

There is another factor contributing to this change of sound when the plucking point is varied. Figure 2 shows that the variation of amplitude with $n$ is different for each plucking point, leading to the set of curving ridges which dominates the plot. The origin of this effect is easy to understand. Look first at the line of points on the far side of the plot, for plucking at $a/L=0.5$, in other words at the mid-point of the string. The line is a zigzag, with every second amplitude falling to zero. The reason is that whenever $n$ is an even number, the corresponding mode shape has a nodal point in the centre. It is impossible to excite a mode of any system by driving at a nodal point. So a mid-point pluck can only excite the odd harmonics, $n=1,3,5,…$. This argument extends in a simple way to other plucking positions. If you pluck at the 1/3 point, every third harmonic will be missing. At the 1/4 point, every fourth will be missing, and so on. This leads directly to the pattern of curving ridges and valleys.

The second approach to calculating the motion of our ideal string following a pluck is to notice that for this particular problem of a plucking force applied at a single point, we can short-cut some of the algebraic complication of the Fourier series approach by making use of a general formula for the *step response* of any linear system. We are applying a force at the chosen plucking point. That force has a constant value $F$ for times $t < 0$ but it then jumps down to zero for $t \ge 0$. The general formula for step response can be used to find the subsequent motion. The details are given in the previous link: reassuringly, it is found to agree with the result obtained by the Fourier series method.

A virtue of this step response method is that it can easily be extended to the case where the string is connected to the instrument body rather than rigidly fixed at both ends. Although Fig. 1 didn’t explicitly show any body motion, the sideways pull by the player will move the bridge a little, as well as moving the string. One way to solve the coupled problem of the string-body motion after the pluck is first to find the coupled modes of this combined system. Once we know those, the step response formula can be applied directly. We will come back to this shortly.

The third approach to finding the pluck response is very different. Rather than using modes, we can take advantage of a general solution to the differential equation governing the motion of an ideal string, due to the 18th-century French mathematician Jean le Rond d’Alembert. This approach uses travelling waves: d’Alembert showed that any possible free motion of a string can be described by a combination of a wave travelling leftwards at speed $c$, and another wave travelling rightwards at the same speed. These waves can have any shape whatsoever, but to satisfy the particular starting conditions of our plucked string we need both waves to have a triangular shape. The next link gives details of d’Alembert’s solution.

Figure 3 shows an animation of the result. At the top, the two separate waves are seen, travelling in opposite directions. You could picture these waves as travelling on a string which extends off to infinity in both directions beyond the actual vibrating string. We are viewing that infinite string through a window matching the length of our actual string, so that waves appear from one side, travel across the window, and vanish off the other side. The lower plot shows the actual string motion, which is the sum of the two waves. Both ends of the string are stationary, as they are supposed to be. If you watch carefully, you will see that the red and blue travelling waves always have equal and opposite magnitudes at the two end of the string, so that when you add them together they always cancel.

Now we have a good idea how an ideal string behaves following a perfect pluck. However, this is not enough for the purpose of making an accurate synthesis of the motion of a real string on a real instrument. Several factors have been omitted in this simple analysis, and we need to put at least some of them back.

First, real strings do not have natural frequencies that are precisely harmonic multiples of the fundamental frequency. There are two reasons. One is to do with coupling to the body of the instrument, which we deal with shortly. The other arises because any real string has some *bending stiffness*. This problem is easily dealt with, as described in the next link. We simply need to combine the behaviour of two systems we already know about, the string and a bending beam. Then we can use Rayleigh’s principle to get a good approximation for the natural frequencies allowing for bending stiffness.

The result is that the frequency of the $n$th overtone (we must now stop calling them ‘harmonics’) is no longer simply an integer multiple of the fundamental. Instead, $\omega_n \approx n \omega_1 (1+\alpha n^2)$ where $\alpha$ is a constant depending on the properties of the particular string (see the previous link for details). The overtones are spaced progressively wider apart than the harmonic series of the ideal string. The stronger the influence of the string’s bending stiffness, the bigger the value of $\alpha$ and the stronger is this *inharmonicity* effect. It has an important perceptual consequence, which we can guess from the discussion of tuned percussion instruments in chapter 3. The more accurately harmonic the frequencies are, the more precise is the resulting sense of pitch. So, for example, thin harpsichord strings will have lower bending stiffness and so be more accurately harmonic that thicker piano strings, which is one reason why harpsichordists pay so much more attention to tuning and to tempering systems.

The effects of inharmonicity due to bending stiffness can be reduced by using over-wrapped strings. To design a string of given length with a low pitch but which does not become too slack, the string has to be heavy. The simplest way to achieve that is to choose a thicker gauge of string, but this will increase the bending stiffness. Wrapping metal wire round a core of metal or nylon fibres allows a heavier string to be made without the penalty of higher bending stiffness. This is the reason behind the familiar appearance of sets of piano strings or guitar strings: the high strings are solid monofilaments of one material or another, but the lower strings are over-wrapped.

The modified natural frequencies can be inserted directly into a modal-based expression for the pluck response, such as the expression given by the step response method. However, the effect of bending stiffness invalidates the d’Alembert travelling-wave method. The wave speed is no longer a single constant value; instead it varies with frequency. Waves with high frequency, or equivalently with short wavelengths, travel faster than those with lower frequency or longer wavelength. The result is that the shape of the travelling triangular waves in Fig. 3 will change with time, because the different frequency components making up the shape travel at different speeds. The motion immediately after the pluck will still resemble Fig. 3, but it will gradually diverge from it as the effects of bending stiffness build up.

But there is a more important reason that the motion of a real string will rapidly deviate from the simple animation in Fig. 3. In the plot the motion went on for ever, but of course in a real string it will die away as energy is lost by the string. Furthermore, the different modes of the string will decay at different rates. Usually, the high-frequency components decay more rapidly than the lower-frequency ones, so that the motion will look increasingly smooth and rounded as time goes on. Eventually, it will become more or less sinusoidal because only the fundamental mode has significant amplitude. This frequency-dependent decay pattern is an important part of the audio “signature” of different instruments, and of different strings or different notes on a given instrument. Some sound examples to illustrate this will be given in sections 5.5 and 5.6.

There are three main mechanisms for energy loss from a vibrating string. Energy loss is additive, so the total loss factor for a given mode of the string is the sum of three separate loss factors associated with the three mechanisms. One we have already seen, in section 5.1.2. Energy is lost through the bridge to the body of the instrument, and the associated loss factor is determined by the product of the string impedance and the real part of the bridge admittance of the body. This source of energy loss will be included automatically, when we couple the string to the body to perform a synthesis.

The other two energy loss mechanisms are associated directly with the string motion itself, regardless of coupling to the body, and they need to be included explicitly in a model of the string dynamics. One source of energy loss is within the material of the string. The cyclical deformation of the string during the vibration will involve some energy loss. For a monofilament string, this loss is linked to the bending stiffness just discussed. The Young’s modulus of the string is in reality a complex number: the real part is the usual elastic modulus, while the imaginary part describes energy loss during cyclical deformation. This *complex modulus* can be written in the form $E(1+i \eta_E)$, where $\eta_E$ is a dimensionless quantity. The effective loss factor for the string, to describe this loss mechanism, is then $\eta_{bend} = 2\alpha n^2 \eta_E $ in terms of the inharmonicity parameter $\alpha$ introduced above: see the next link for details.

For a metal string, as on a piano or steel-string guitar, $\eta_E$ is usually very small, perhaps of the order of $10^{-3}$. But for a nylon or gut string it will be bigger, typically a few percent. For a multi-layer wrapped string, additional energy loss can occur as a result of friction between the windings. The energy loss in that case will not be represented very accurately by the complex modulus model: friction is intrinsically a nonlinear process, whereas the complex modulus representation is based on linear theory. Furthermore, on an instrument like the guitar where the strings are in direct contact with the player’s fingers, this loss mechanism is likely to vary with time. Contaminants from the fingers gradually build up between the windings of the string, increasing the energy dissipation rate. This seems to be the main reason for the difference in sound between new and old guitar strings [1].

The third mechanism for energy loss from a vibrating string is to do with the air surrounding the string: as the string moves, the associated air flow over its surface leads to losses from viscosity. A loss factor $\eta_{air}$ to account for this can be estimated using a classical result for flow round a circular cylinder: the details are given in the next link.

We now have all the main ingredients for a realistic model of a vibrating string, and we are ready to couple the string to the instrument body and synthesise a version of the sound of plucking the string. One possible method of synthesis has already been mentioned: if we can describe the body motion in terms of its modes, we could compute the coupled modes of the string/body system, then use the general formula for step response to create a pluck sound. This approach is certainly possible, and we will show examples later, but there are some details that are rather fiddly to get right. We will not discuss the method further here: see [2] for more discussion.

There is an alternative synthesis method that is faster and in many ways superior. This involves working in the frequency domain, and using an inverse FFT at the final stage to create the required time-varying transient response. This method is attractive because the string and the body are coupled together at a single point, and there is a very simple method which can be applied to any such problem: if two systems have input admittances $Y_1$ and $Y_2$ and they are then rigidly connected at the points at which these admittances are defined, then the coupled system has an admittance $Y_{coup}$ at that point satisfying

$$\dfrac{1}{Y_{coup}}=\dfrac{1}{Y_1}+\dfrac{1}{Y_2} . \tag{1}$$

This result expresses the fact that at the coupling point, the two subsystems have equal velocities while the total applied force is the sum of the forces applied to the two separate subsystems.

The body is naturally described in terms of its admittance, as already described. By calculating the corresponding admittance at the end of a string, which is derived in the next link, eq. (1) can be applied immediately to give the admittance of the strung instrument at the bridge.

To use this method to derive a pluck response, use can be made of a general *reciprocal theorem* for linear vibration response. This states that if you apply a force at some point on a system, and observe the motion at another point, then you would get *exactly the same response* if you observed at the first point, and applied the force at the second point. This reciprocal theorem is often used when making measurements: it allows you to swap your excitation point and your measurement point if that makes the experimental details easier. The proof of this result has actually already been given: it follows directly from eq. (11) of section 2.2.5. This formula for the frequency response function is completely symmetric between the forcing point (labelled $j$) and the measurement point (labelled $k$ ).

We wish to find the body vibration which results from a step function of force applied at a given position on the string. Reciprocally, we can consider applying the force at the bridge, and calculate the resulting motion at the relevant point on the string. To solve this reciprocal problem, we simply have to multiply two frequency response functions together. The first is the coupled admittance $Y_{coup}$ from eq. (1), which gives the velocity at the bridge when a force is applied there. The second is the dimensionless transfer function between a given displacement applied at one end of a string and the corresponding displacement at the point where the pluck is to be applied. This second transfer function is also derived in the next link.

Performing these operations and then applying an inverse FFT, we can create a time history representing the velocity waveform at the bridge following an ideal pluck at the chosen point on the string. There are two final ingredients to be considered, to make the model a little more realistic. First, we might want to allow for a pluck that is somewhat rounded rather than perfectly sharp. We saw earlier that the effect of a rounded pluck is to reduce the high-frequency content of the string motion, and hence of the sound of the instrument. We can represent this effect in a simple way by applying a suitable low-pass filter to the computed frequency response, before doing the inverse FFT.

The final ingredient involves thinking more carefully about the output of the synthesis. Really, we would like to compute the sound radiated by the vibrating instrument body. To do this accurately is a very complicated matter, as we glimpsed in Chapter 4. We will want to come back to the question of representing radiated sound, but for the moment we can try to find a simple way to synthesise something that gives a reasonably convincing impression of “sounding like a guitar” (or whichever instrument be are considering). A pragmatic approach will be used: empirically, the sound of bridge velocity is a little dull, while the sound of bridge acceleration is a bit too bright. In the frequency domain, the difference between these two things is a factor of $(i \omega)$. It is easy to move between these two extremes by applying a factor $(i \omega)^\beta$ where $\beta$ is some value lying between 0 and 1 which we can choose based on listening to some sound examples.

Examples of the effect of varying $\beta$ will be shown in section 5.5, together with many other sound examples computed by this approach. As a taster, we give a few preliminary examples here. The three sounds below were all synthesised using the frequency domain method just described. A brief snatch from an anonymous lute piece called “The English Huntsuppe” is “played” on a classical guitar. Parameter values for a typical set of nylon strings were used, together with the measured bridge admittance of a guitar. All the details of the modelling can be found in reference [1]. The middle example, Sound 2, uses the measured body admittance directly. Sounds 1 and 3 are the result of doing some “virtual woodwork” in the computer to modify the admittance in a controlled manner.

First, do these sound fairly credibly like guitars? Then, how would you describe the difference between the three sounds? Can you guess what has been changed? The answer is that in Sound 1, all the natural frequencies of the body have been reduced by one semitone, whereas in Sound 3 they have been increased by one semitone. Nothing else was changed. If you want to think of a physical change to a guitar that would have roughly the same effect, then you can picture Sound 1 as coming from a guitar with a slightly larger or thinner top plate, and Sound 3 from one with a slightly smaller or thicker top plate. As we showed in section 3.2.4, the area and the thickness of a plate both affect the modal density.

Earlier in this section, we saw in Fig. 2 that the plucking point on the string has a strong influence on the frequency spectrum of an ideal plucked string. The next pair of sound examples illustrate this effect via synthesised fragments of another lute piece, “Queen Elizabeth’s Galliard” by John Dowland. All the model parameters are the same as for Sound 2, except that the plucking point has been varied. In Sound 4, the strings are plucked 150 mm from the bridge, while in Sound 5 they are plucked 20 mm from the bridge. The open string length for this “guitar” was 650 mm, but as in a real instrument the actual vibrating length for each note varies depending on which fret is used to play the note. The difference of sound between these two examples is encouragingly familiar from real playing.

[1] J. Woodhouse, E.K.Y. Manuel, L.A. Smith and C. Fritz. “Perceptual thresholds for acoustical guitar models”. *Acta Acustica united with Acustica* **98**, 475-486, (2012). DOI 10.3813/AAA.918531

[2] J. Woodhouse “On the synthesis of guitar plucks”. *Acta Acustica united with Acustica* **90**, 928–944 (2004).