  We show that using the concept of the two-dimensional level number N_{|} one
can experimentally study of the nodal domains in a three-dimensional (3D)
microwave chaotic rough billiard with the translational symmetry. Nodal domains
are regions where a wave function has a definite sign. We found the dependence
of the number of nodal domains aleph_{N_{|}} lying on the cross-sectional
planes of the cavity on the two-dimensional level number N_{|}. We demonstrate
that in the limit N_{|} -> infinity the least squares fit of the experimental
data reveals the asymptotic ratio aleph_{N_{|}}/N_{|} = 0.059 +- 0.029 that is
close to the theoretical prediction aleph_{N_{|}}/N_{|} = 0.062. This result is
in good agreement with the predictions of percolation theory.
