  We study experimentally nodal domains of wave functions (electric field
distributions) lying in the regime of Shnirelman ergodicity in the chaotic
microwave half-circular ray-splitting rough billiard. For this aim the wave
functions Psi_N of the billiard were measured up to the level number N=415. We
show that in the regime of Shnirelman ergodicity (N>208) wave functions of the
chaotic half-circular microwave ray-splitting rough billiard are extended over
the whole energy surface and the amplitude distributions are Gaussian. For such
ergodic wave functions the dependence of the number of nodal domains aleph_N on
the level number N was found. We show that in the limit N->infty the least
squares fit of the experimental data yields aleph_N/N = 0.063 +- 0.023 that is
close to the theoretical prediction aleph_N/N = 0.062. We demonstrate that for
higher level numbers N = 215-415 the variance of the mean number of nodal
domains sigma^2_N/ N is scattered around the theoretical limit sigma^2_N /N =
0.05. We also found that the distribution of the areas s of nodal domains has
power behavior n_s ~ s^{-tau}, where the scaling exponent is equal to tau =
2.14 +- 0.12. This result is in a good agreement with the prediction of
percolation theory.
