Nodal statistics of billiards and boundary conditions

Wave dynamics in chaotic systems are typically modelled by using Gaussian random waves which consist of superpositions of plane waves travelling in different directions but with the same wavelength. Nodal statistics were derived for the Gaussian random waves in a plane, satisfying the Helmholtz equation and mixed boundary conditions which bridge Dirichlet and Neumann types. We find that effects of the boundary persist infinitely far from the boundary. As the boundary condition changes from the Dirichlet to Neumann type, nodal line structures migrate from the boundary, and can be described analytically In this presentation, those analytical results obtained before are compared with numerical calculations for eigenfunctions of chaotic billiards. We find very good agreement between the analytical and numerical results, confirming that Gaussian random waves are good mathematical models for waves in chaotic systems.