We derive an explicit expression for the coupling constants of individual e
igenstates of a closed billiard that is opened by attaching a waveguide. Th
e Wigner time delay and the resonance positions resulting from the coupling
constants are compared to an exact numerical calculation. Deviations can b
e attributed to evanescent modes in the waveguide and to the finite number
of eigenstates taken into account. The influence of the shape of the billia
rd and of the boundary conditions at the mouth of the waveguide are also di
scussed. Finally we show that the mean value of the dimensionless coupling
constants tends to the critical value when the eigenstates of the billiard
follow random-matrix theory.