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Acoustic Figures
A. D. Jackson
7 May 2007
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Ernst Florenz Friedrich Chladni (1756-1827)
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Some of Chaldni’s original acoustic figures
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Hans Christian Ørsted (1777-1851)
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Kongens Nytorv, 4-5 September 1807
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These dust piles fascinated Faraday
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Sophie Germain (1776 – 1831)
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• Germain primes, [p,q] if p is prime and q=(2p+1) is also prime.
• Substantial contributions to Fermat’s last theorem.
• A correct description of acoustic resonances in thinplates. She received Napoleon’s prize on her 3rd attempt. One kilo of pure gold!
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Michael Faraday (1791-1867)
Ørsted’s dust piles inspiredthe discovery of electromagneticinduction.
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Charles Wheatstone (1802 - 1875)
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Heusler, Müller, Altland, Braun, Haake,“Periodic-Orbit Theory of Level Correlations”arXiv:nlin/0610053
“We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmidtconjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadrupletsof sets of periodic orbits. The asymptotic expansions of both the non-oscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summationof the series reproduces the exact correlator of random-matrix theory.”
… a less general but simplerpicture might be useful.
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a=1 a=1.2 a=100
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Nearest-neighbor distributions for the cardioid family:
spectrum of N(always RMT)
spectrum of H(Poisson to RMT)
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“Random” billiards:
Nearest-neighbor distributions for random billiards: (Note that spectrum of N is always given by RMT.)
Gaussian distributed (RMT for all t > 0)
Poisson distributed
Since spectral correlations of N are always RMT, the change inthe statistics of H can only be due to the support of this spectrum. (There is nothing else!)
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…It’s time for some acoustic coffee!