The jungles of randomness by Ivars Peterson

Author:Ivars Peterson
Language: eng
Format: epub

Computer simulations of vibrations of a membrane shaped like a fractal snowflake reveal the first harmonic (top) and second harmonic (bottom). (Michel L. Lapidus, J. W. Neuberger, Robert J. Renka, and Cheryl A.

Griffith)

The colorful images generated by Griffith vividly illustrate the dramatically frilled edges of the waveforms created on fractal-bounded membranes. Such fractal tambourines produce normal modes wrinkled in ways that reflect the influence of the membrane's boundary (see color plates 3 and 4). It also becomes evident that if one were to construct a fractal tambourine, it would have a distinctive sound. Its crinkled edge would muffle, even deaden, the tones, giving the tambourine's sound a subdued, muted flavor.

From a mathematical point of view, Griffith's numerical results couldn't convincingly depict all the nuances of a fractal membrane's vibrational behavior. As in a physical experiment, a computer can only approximate a fractal; it cannot render it in every detail. The resulting

pictures must be interpreted carefully, and the underlying mathematics formulated and worked out. So, the studies continue.

Investigating the vibration of drums with fractal boundaries and drums with punctured membranes (flat surfaces perforated by an infinite array of holes) could, in the end, lead to a better understanding of such physical processes as the diffusion of oil through sand and the passage of waves through rubble.

One intriguing facet of fractal drums is what they may say about the apparent prevalence of irregular, fractal-like forms in nature, from crazily indented coastlines to the intricate branching of air passages in the human lung. For example, irregular (nearly fractal) coastlines may be more common than smooth ones because their shapes are fundamentally more stable. This would certainly account for the efficacy of disordered heaps of variously sized rocks as breakwaters. Other fractal structures may prove efficient in other settings (see chapter 8).

We venture now from the smoothness of individual waves and their interactions with edges, both even and irregular, to the complexity of their collective behavior and of the vibrating objects that generate them. We move from the harmony of pure tones to the randomness of noise.

How does the irregularity that characterizes noise arise from simple vibrations? One of the first experiments to show how pure tones generate noise occurred in 1952 at Gottingen, the city in Germany where Lorentz gave his lecture on electromagnetic waves in a box. When a liquid is irradiated with sound waves of high intensity, the liquid may rupture to form bubbles or cavities. This phenomenon is called acoustic cavitation and is typically accompanied by an intense emission of noise.

What the researchers at Gottingen found particularly interesting was how this emitted sound changes as the intensity of the pure tone that initiates the process slowly increases, starting from a low level. At first, the sound coming out of the liquid matches that of the input frequency. As the intensity increases, a second frequency abruptly becomes apparent, at half the original frequency. Then, as the intensity increases further, the frequency is halved again, so four different tones come out of the liquid.

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