Nodal-line densities of chaotic quantum billiard modes satisfying mixed boundary conditions

Statistics of nodal lines for eigenmodes u in the stadium are computed, and compared with previously derived formulae for monochromatic boundary-adapted Gaussian random waves in the plane. These modes and random waves satisfy the Helmholtz equation, and mixed boundary conditions in which a linear combination of u and its normal derivative must vanish. For the density of nodal lines and the excess density of nodal lines arising from the boundary, the Gaussian model accurately describes the statistics of the billiard eigenfunctions.