Periodic orbit analysis of the Helmholtz equation in two-dimensional enclosures

Abstract

This thesis examines how periodic orbits may be used in acoustics to understand solutions of the Helmholtz equation. A review of the links between ray and wave mechanics is given including WKBJ (Wentzel, Kramers, Brillouin and Jeffreys) and EBK (Einstein, Brillouin, Keller) methods. It is also noted that some mode shapes in chaotic enclosures are scarred by the short periodic orbits. This motivates the proposal of the Mode Scar Hypothesis and the Mode Resonance Function Hypothesis.The trace formula, which is a sum over periodic orbits, approximates the level density for an acoustic enclosure. The trace formula in the concentric annulus domain is derived using a formulation for enclosures with continuous symmetry by Creagh and Littlejohn [1]. Results for the variance of the difference between the true and average mode counts are obtained. A technique called short periodic orbit theory (SPOT) for the approximation of mode shapes devised by Babic and Buldyrev [2] and Vergini [3] is given. SPOT is extended to impedance boundary conditions. SPOT is implemented in the quarter stadium, quadrupole, circle and eccentric annulus enclosures with Dirichlet, Neumann and impedance boundary conditions. Concave enclosures with Dirichlet or Neumann boundary conditions were best approximated using SPOT.A design loop for enclosures is proposed using the periodic orbit ideas given. A model problem is used to provide insight into the effectiveness of these methods. It was found that it was not possible to breakdown all mode shapes in the eccentric annulus into contributions from short periodic orbits.

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ORCID for M.C.M. Wright: orcid.org/0000-0001-9393-4918

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