Periodic orbit techniques for mode shape approximation

Abstract

At high frequency, numerical methods for calculating mode shape functions in a waveguide are expensive. The effect of modifying the shape of a waveguide in an industrial design context may only be calculated on a case‐by‐case basis if numerical methods are used. Periodic orbits are closed, repeating ray paths. A significant body of research in quantum physics indicates that periodic orbits provide fundamental insight into mode theory, which can be applied to acoustics. A method to approximate mode shape functions around short periodic orbits proposed by Vergini [J. Phys. A: Math. Gen. 33, 4709–4716, (2000)] and Babic and Buldryrev [‘‘Short‐Wavelength Diffraction Theory’’ (Springer, Berlin) (1991)] is implemented in this paper. In particular, mode shape functions in nonintegrable waveguides are approximated. This method links the geometry of the waveguide to the mode shape functions. The technique is extended to waveguides that have partly chaotic ray dynamics and is applied to the oval and eccentric annul...