  The counterintuitive fact that wave chaos appears in the bending spectrum of
free rectangular thin plates is presented. After extensive numerical
simulations, varying the ratio between the length of its sides, it is shown
that (i) frequency levels belonging to different symmetry classes cross each
other and (ii) for levels within the same symmetry sector, only avoided
crossings appear. The consequence of anticrossings is studied by calculating
the distributions of the ratio of consecutive level spacings for each symmetry
class. The resulting ratio distribution disagrees with the expected Poissonian
result. They are then compared with some well-known transition distributions
between Poisson and the Gaussian orthogonal random matrix ensemble. It is found
that the distribution of the ratio of consecutive level spacings agrees with
the prediction of the Rosenzweig-Porter model. Also, the normal-mode vibration
amplitudes are found experimentally on aluminum plates, before and after an
avoided crossing for symmetrical-symmetrical, symmetrical-antisymmetrical, and
antisymmetrical-symmetrical classes. The measured modes show an excellent
agreement with our numerical predictions. The expected Poissonian distribution
is recovered for the simply supported rectangular plate.
