![]() ![]() For every plucking position, the fundamental ($n=1$) has the largest amplitude. This plot reveals several interesting features. The plucking point is expressed as a relative fraction of the string length, so that a value 0.5 means that the string is plucked at its mid-point, while a small value means that the string is plucked very close to the bridge. The first 20 harmonic amplitudes for the waveform of force applied to the instrument bridge by an ideal plucked string, for different values of the plucking point. 2, against harmonic number $n$ and relative pluck position $a/L$ (where $L$ is the length of the string and $a$ is the plucking distance). ![]() 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, and the mode shapes are simply sinusoidal as plotted in Fig. This is the driving force for body vibration, and so is closely related to the sound that an instrument would make. ![]() 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. We can learn something interesting about the sound of plucked strings from the result of the calculation in the previous link. 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. 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 effect of rounding off the sharp corner would be to reduce the amplitudes of the higher Fourier components. The method would be followed in the same way, based on calculating the Fourier series representation of the initial shape. 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. 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. All we need to do is find the amplitude and phase of each modal contribution. This statement is, after all, true of any free motion of any linear system. They are all useful in their different ways so we will briefly summarise all three.įirst, 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. ![]() For this simple system there are three ways we can find the string motion following the pluck. We already know the mode shapes and natural frequencies for this ideal string from section 3.1.1. The very simplest model is to assume a perfectly sharp corner, and an ideal string without damping. Sketch of a string pluck: the string is pulled to this triangular shape by a force (the black arrow), then it is released to vibrate freely. 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 player can choose the plucking point along the string, and the orientation of the pluck. When pulled aside, the string will take up a triangular shape as sketched in Fig. The string is then left to its own devices, to vibrate freely and to feed some of its energy into the instrument body. 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. But for plucked-string instruments like the guitar and the banjo we can get a long way with linear ideas. 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 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. ![]()
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