The Ideal String

If we force the masses to move only up and down and restrict them to small displacements, we don t actually have to solve the acceleration of each mass as a function of the spring forces pushing them up and down. There is a simple differential equation that completely describes the motions of the ideal string. Appendix B gives the derivation of this equation describing the ideal string, which is 

Fignre 9.2. Mass-spring string network with frictionless guide rods to restrict motion to one dimension. 

The two delay lines taken together are called a waveguide filter. The sum of the contents of the two delay lines is the displacement of the string, and the difference of the contents of the two delay lines is the velocity of the string (check this yourself using the methods and equations given in Appendix B). If we wish to pluck the string, we simply need to load one-half 

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It is interesting to compare the digital waveguide simulation technique to the recursion produced by the finite difference approximation (FDA) applied to the wave equation. Recall from (10.10) that the time update recursion for the ideal string digitized via the... 

FDA of the Ideal String. Substituting the FDA into the wave equation gives... 

Figure 10.5 Transverse force propagation in the ideal string.

At very low frequencies, or for very small k, we return to the non-stiff case. At very high frequencies, or for very large k, we approach the ideal bar in which stiffness is the only restoring force. At intermediate frequencies, between the ideal string and bar, the stiffness contribution can be treated as a correction term [Cremer, 1984]. This... 

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... 

Just as we proved the D Alembert solution to the ideal string (Section B. 1.2), we know that a solution to Equation C.9 for the acoustic tube is ... 

Consider first the series junction of N waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of A ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance Rj (s), as depicted in Fig. 10.11 for TV= 4. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., V(s) = C v for velocity waves and F(s) = C / for force waves, where jC denotes the Laplace transform. In the discrete-time case, we use the z transform instead, but otherwise the story is identical. 

The wave equation for the ideal (lossless, linear, flexible) vibrating string, depicted in ... 

The gas dissipates in a couple d hours. Ideally, you diould place the trap so you can at least see the taut string whn you open the door. If it isn t taut you should hold your breath and go in and open some windows and get out bdore breathing. At the same time, turn on a ton or air conditioner if you have one. 

Predictably, most systems that produce sound are more complex than the ideal mass/spring/damper system. And of course, most sounds are more complex than a simple damped exponential sinusoid. Mathematical expressions of the physical forces (thus the accelerations) can be written for nearly any system, but solving such equations is often difftcult or impossible. Some systems have simple enough properties and geometries to allow an exact solution to be written out for their vibrational behavior. A string under tension is one such system, and it is evaluated in great detail in Chapter 12 and Appendix A. For... 

This chapter will focus on one-dimensional physical models, and on the techniques known as waveguide filters for constracting very simple models that yield expressive soimd synthesis. First weTl look at the simple ideal string, and then refine the model to make it more reahstic and flexible. At the end we will have developed models that are capable of simnlating an interesting variety of one-dimensional vibrating objects, inclnding stiff stmctures such as bars and beams. 

In a string under tension, the ends are restricted from moving (that s how we make the tension). The conditions of non-movement at the ends are called boundary conditions. The boundary condition solution to an ideal string of length L terminated at both ends is ... 

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