  The proplem of sound propagation in an oceanic waveguide is considered.
Scattering on random inhomogeneity of the waveguide leads to wave chaos. Chaos
reveals itself in spectral properties of the finite-range evolution operator
(FREO). FREO describes transformation of a wavefield in course of propagation
along a finite segment of a waveguide. We study transition to chaos by tracking
variations in spectral statistics with increasing length of the segment.
Analysis of the FREO is accompanied with ray calculations using the one-step
Poincar\'e map which is the classical counterpart of the FREO. Underwater sound
channel in the Sea of Japan is taken for an example. Several methods of
spectral analysis are utilized. In particular, we approximate level spacing
statistics by means of the Berry-Robnik and Brody distributions, explore the
spectrum using the procedure elaborated by A. Relano with coworkers (Relano et
al, Phys. Rev. Lett., 2002; Relano, Phys. Rev. Lett., 2008), and analyze modal
expansions of the eigenfunctions. We show that the analysis of FREO
eigenfunctions is more informative than the analysis of eigenvalue statistics.
It is found that near-axial sound propagation in the Sea of Japan preserves
stability even over distances of hundreds kilometers. This phenomenon is
associated with the presence of a shearless torus in the classical phase space.
Increasing of acoustic wavelength degrades scattering, resulting in recovery of
localization near periodic orbits of the one-step Poincar\'e map. Relying upon
the formal analogy between wave and quantum chaos, we suggest that the concept
of FREO, supported by classical calculations via the one-step Poincar\'e map,
can be efficiently applied for studying chaos-induced decoherence in quantum
systems.
