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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 thin plates. She received Napoleon’s prize on her 3 rd attempt. One kilo of pure gold!

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Michael Faraday (1791-1867) Ørsted’s dust piles inspired the discovery of electromagnetic induction.

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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-Schmidt conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of periodic orbits. The asymptotic expansions of both the non-oscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.” … a less general but simpler picture might be useful.

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a=1a=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 in the 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!

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