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Physical Audio Signal Processing
    Elementary String Instruments
       The Ideal Plucked String

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The Ideal Plucked String

The ideal plucked string is defined as an initial string displacement and a zero initial velocity distribution [325]. More generally, the initial displacement along the string $ y(0,x)$ and the initial velocity distribution $ {\dot y}(0,x)$, for all $ x$, fully determine the resulting motion in the absence of further excitation.

An example of the appearance of the traveling wave components and the resulting string shape shortly after plucking a doubly terminated string at a point one fourth along its length is shown in Fig. 4.6. The negative traveling-wave portions can be thought of as inverted reflections of the incident waves, or as doubly flipped ``images'' which are coming from the other side of the terminations.

Figure 4.6: A doubly terminated string, ``plucked'' at 1/4 its length.
\includegraphics[width=\twidth]{eps/f_t_waves_term}

An example of an initial ``pluck'' excitation in a digital waveguide string model is shown in Fig. 4.7. There is one fine point to note for the discrete-time case: We cannot admit a sharp corner in the string since that would have infinite bandwidth which would alias when sampled. Therefore, for the discrete-time case, we define the ideal pluck to consist of an arbitrary shape as in Fig. 4.7 lowpass filtered to less than half the sampling rate. Alternatively, we can simply require the initial displacement shape to be bandlimited to spatial frequencies less than $ f_s/2c$. Since all real strings have some degree of stiffness which prevents the formation of perfectly sharp corners, and since real plectra are never in contact with the string at only one point, and since the frequencies we do allow span the full range of human hearing, the bandlimited restriction is not limiting in any practical sense.

Figure 4.7: Initial conditions for the ideal plucked string. The initial contents of the sampled, traveling-wave delay lines are in effect plotted inside the delay-line boxes. The amplitude of each traveling-wave delay line is half the amplitude of the initial string displacement. The sum of the upper and lower delay lines gives the actual initial string displacement.
\includegraphics[width=\twidth]{eps/fidealpluck}

Note that acceleration (or curvature) waves are a simple choice for plucked string simulation, since the ideal pluck corresponds to an initial impulse in the delay lines at the pluck point. Of course, since we require a bandlimited excitation, the initial acceleration distribution will be replaced by the impulse response of the anti-aliasing filter chosen. If the anti-aliasing filter chosen is the ideal lowpass filter cutting off at $ f_s/2$, the initial acceleration $ a(0,x)\isdeftext {\ddot y}(0,x)$ for the ideal pluck becomes

$\displaystyle a(0,x) = \frac{A}{X}$sinc$\displaystyle \left(\frac{x-x_p}{X}\right)$