Plucked string

Fig. 2 Suing with fixed ends (a) the plucked string (b) the struck string.

As an example of die application of condition (i) above, consider the plucked string (see Fig. 2). The string is displaced at its midpoint by a distance h and released at r = 0. Thus, the initial conditions are... 

Source-filter Synthesis (Subtractive synthesis, Karplus-Strong Plucked string algorithm [Karplus and Strong, 1983])... 

This formula is reminiscent of the famous relationship that was discovered by Pythagoras to exist between the pitch of sound produced by the plucked string of a lyre and the effective length of the string. Based on this... 

I want to impress upon you the full set of implications of this analogy between sound and light, between the plucked string and the hydrogen... 

A plucked string showing displacements along the string at an arbitrary time. 

Figure 4.3. Top Plucked string. Center Possible sinusoidal modes of vibration of a center-plucked string. Bottom The even modes, which would not be excited by the center-plucked condition.

We will learn more about the plucked string and higher-dimensional vibrating systems such as bars, plates, membranes, etc., in later chapters. The point of introducing the example of the plucked-string system here was to motivate the notion that sinusoids can occur in systems more complex than just the simple mass/spring/damper. 

We saw in Section 4.2 that the plucked string supports certain spatial vibrations, called modes. These modes have a very special relationship in the case of the plucked string (and some other limited systems) in that their frequencies are all integer multiples of one basic sinusoid, called thefundamental. This special series of sinusoids is called a harmonic series, and lies at the basis of the Fourier series representation of shapes, waveforms, oscillations, etc. The Fourier series solves many types of problems, including physical problems with boundary constraints, but is also applicable to any shape or function. Any periodic waveform (repeating over and over again), can be transformed into a Fourier series, written as ... 

Figure 9.7. Fairly complete digital filter simulation of plucked-string system.

Figure 9.7 shows a relatively complete model of a plucked-string simulation system using digital filters. Output channels for pickup position and body radiation are provided separately. A solid-body electric guitar would have no direct radiation and only pickup output(s), while a purely acoustic guitar would have no pickup output, but possibly a family of directional filters to model body radiation in different directions. 

K. Karplus and A. Strong. Digital Synthesis of Plucked-String and Drum Timbres. Computer Music Journal 7(2) 43-55 (1983). 

D. A. Jaffe and J. O. Smith. Extensions of the Karplus-Strong plucked string algorithm. Computer Music Journal 7(2) 56-69 (1983). 

Figure 10.3. Plucked string with high initial displacement shows pitch bending due to displacement-dependent tension.

Figure 10.5. Plucked-string system with a discontinuous nonlinearity. High frequency components are introduced as the string comes into contact with the secondary termination.

Track 40] Plucked String and Drum Displacement-Modulated Pitch. [Track 41] Guitar Chatter Nonlinearity. 

In fact, if you find a nice long piece of pipe and slap the end, you ll get a sound very much like a simple damped plucked string. 

We noted that the ideal string equation and the ideal acoustic tube equation are essentially identical. Just as there are many refinements possible to the plucked-string model to make it more realistic, there are many possible improvements for the clarinet model. Replacing the simple reed model with a variable mass/spring/damper allows the modeling of a lip reed as is found in... 

In Chapter 4 we talked some about how the point of excitation of a plucked string determines which harmonics are present, and how strongly, in the spectrum. We also talked about this in Chapter 12 for square membranes. In these cases of systems with simple geometry, we can predict what the modes of the system should be, and then use Fourier methods to compute the amplitude of each of the modes. 

Do Equations C.8 and C.9 look familiar They should, because the form is exactly the same traveling wave equation we derived for the plucked string in Appendix B. Note that the area term has cancelled out in both traveling wave equations. 

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