Acoustic Figures

A. D. Jackson

7 May 2007

Ernst Florenz Friedrich Chladni (1756-1827)

Some of Chaldni’s original acoustic figures

Hans Christian Ørsted (1777-1851)

Kongens Nytorv, 4-5 September 1807

These dust piles fascinated Faraday

Sophie Germain (1776 – 1831)

• 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!

Michael Faraday (1791-1867)

Ørsted’s dust piles inspiredthe discovery of electromagneticinduction.

Charles Wheatstone (1802 - 1875)

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.

a=1 a=1.2 a=100

Nearest-neighbor distributions for the cardioid family:

spectrum of N(always RMT)

spectrum of H(Poisson to RMT)

“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!)

…It’s time for some acoustic coffee!