Strings and Bars

Physical modeling synthesis endeavors to model and solve the physics of sound-producing systems in order to synthesize sound. Unlike sinusoidal additive and modal synthesis (Chapter 4), or PCM sampling synthesis (Chapter 2), both of which can nse one powerM generic model for any sound, physical modeling reqnires a different model for each family of sound producing object. LPC (Chapter 8) is a spectral modeling techniqne, but also has physical interpretations in the one-dimensional ladder implementation. 

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. 

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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... 

The 0.7/0.3 ratio of dark energy to dark matter has fairly small error bars if you take all the observations seriously, so theorists began, in the new century (ApOl), to try to identify mechanisms that could lock in such a ratio (vs. zero or infinity) long enough for life to evolve and all. Some of the associated concepts are a Brans-Dicke (scalar-tensor) field, a false vacuum, a scalar field that locks onto negative pressure, and, of course, ideas connected with strings and branes. Up to this point, at least, most of the ideas first surfaced in Physical Review Letters, Physics Letters, and other reasonably respectable venues, in contrast to some of the more imaginative candidates for dark matter. This happy state of affairs probably won t last. 

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. 

The main complication arises when we consider the off-diagonal blocks. The blocks on the first off-diagonal have contributions from the odd-bar integrals and operators and the blocks on the second off-diagonal have contributions from integrals with two bars. The x operators in both cases can be reduced to expressions that contain X - and Xp operators. These operators connect A and B strings, and so we lose the benefit of the factorization of the determinants into A and B strings. 

Amber, or fossilized tree sap, is also made up of polymers. Pine trees contain a polymer called rosin. Violinists use rosin to make their bows slide more easily over the violins strings. Gymnasts also use rosin to improve their grips on uneven bars and other gymnastics equipment. Rosin is used in some kinds of soap, too. 

Figure 10.2 Transmission electron micrographs of (A) pCMV-beta and (B) trilau-rin-cored emulsion and their complexes at different weight ratios. (C) DNA-emul-sion (1 1). A few emulsion particles are found on the string of DNA. (D) DNA-emulsion (1 4). The emulsion and DNA are fully combined and form a chromatin-like structure. Scale bar = 100 nm.

Fig. 10 Cryo-TEM images of polyelectrolyte block copolymer micelles (PB-P2VPMeI) with unperturbed spherical corona (a), corona filaments (b), filament networks (c), and micellar strings. The scale bar is 50 nm [56]...

You will notice that the atomic mass of hydrogen appears in cell F2, but we do not quite have it right yet because the left parenthesis appears at the end of the string. This difficulty is easily fixed by typing — 1 at the end of the FIND function, which subtracts one from the character position of the left parenthesis to give the last character position of the atomic mass. Click on cell F2, then click in the formula bar at the end of the FIND function, and change the cell contents to the following ... 

FIGURE 9.4 Details of a conventional Frasch sulfur string. Superheated water (140-I60°C, 17 bar) enters outermost pipe, and air at I50-200°C and 34 bar enters the innermost pipe. 

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