  We present the results of experimental study of nodal domains of wave
functions (electric field distributions) lying in the regime of Shnirelman
ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains
are regions where a wave function has a definite sign. The wave functions Psi_N
of the rough billiard were measured up to the level number N=435. In this way
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 a least squares fit of the
experimental data reveals the asymptotic number of nodal domains aleph_N/N =
0.058 +- 0.006 that is close to the theoretical prediction aleph_N/N +- 0.062.
We also found that the distributions of the areas s of nodal domains and their
perimeters l have power behaviors n_s ~ s^{-tau} and n_l ~ l^{-tau'}, where
scaling exponents are equal to \tau = 1.99 +- 0.14 and \tau'=2.13 +- 0.23,
respectively. These results are in a good agreement with the predictions of
percolation theory. Finally, we demonstrate that for higher level numbers N =
220-435 the signed area distribution oscillates around the theoretical limit
Sigma_{A} = 0.0386 N^{-1}.
