Ray chaos and two‐dimensional wave fields

In a two‐dimensional wave field under irregular shape boundary conditions, the trajectory of a sound ray is unstable because of chaotic motion [M. V. Berry, Eur. J. Phys. 2, 91 (1981)]. For wave theory, the spacing distribution of the neighboring eigenvalues is not degenerated in irregular systems because of the repulsion effects, but follows a Wigner distribution. [S. W. McDonald and A. N. Kaufman, Phys. Rev. Lett.42 , 1189 (1979); R. H. Lyon, J. Acoust. Soc. Am. 45, 545 (1969)]. This article discusses the sound fields in 2‐D irregularly shaped boundaries, using both ray and wave theories. In ray theory, the reflection processes of billiard problems are considered to be like mapping processes on a logistic map, and the sensitivities to the initial conditions under several boundary conditions such as a stadium and an ellipse are investigated. It is shown that the reflection points at the boundary are sensitive to small changes in initial conditions, illustrating that changes in the reflection positions ha...