Why You Hear What You Hear by Eric J. Heller

Author:Eric J. Heller
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2013-04-14T16:00:00+00:00

Figure 15.1 A string network.

There is no reason the strings in a network all have to lie in the same plane; the network could occupy a volume, further increasing the possibilities and the number of (certainly inharmonic) partials.

15.2 Stretched Membranes

We encountered stretched membranes in section 8.10 as a natural extension of the construction of a string from beads and filament under tension, by simply building a two-dimensional rectangular array of beads connected by filaments (see figure 8.10 and, in this regard, figure 7.36).

Like a string, a thin membrane has little resistance of its own to bending and so derives its resistance to deformation from the tension that is applied at its extremities—for example, a rim in the case of a drum. Even if the rim is a simple shape (circles, ovals, squares, and the like, all lying in the same plane), the partials are not harmonically related. The frequencies of the particals depend on the tension and the shape of the rim. In the next section, we consider the consequences of a thicker membrane (a plate or shell). The additional thickness causes profound changes, making plate and shell vibrations a separate subject.

The boundary around which the membrane is stretched is the analog of the two points between which a string is stretched. With a few notable exceptions (the tabla for example manages a more harmonic relation of its modes by membrane density variations), almost all instruments use a membrane uniformly stretched in all directions and uniform in thickness (mass density). The equations and the solutions governing the motion and modes of such a membrane are well-known.

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