Contents -- Preface -- 1. Ray and Wave Propagation -- 1.1 Underwater Sound Channel -- 1.2 Basic Equations -- 1.2.1 Helmholtz equation -- 1.2.2 Parabolic equation -- 1.3 Geometrical Optics Approximations and Optical- Mechanical Analogy. The Hamiltonian Formalism -- 1.3.1 Eikonal and transport equations -- 1.3.2 Momentum-position variables -- 1.3.3 Action-angle variables in a range-independent waveguide -- 1.3.3.1 Canonical transformation -- 1.3.3.2 Ray path in the unperturbed waveguide -- 1.3.3.3 Canonical transformation in the form of Fourier series -- 1.3.4 Action-angle variables in a range-dependent waveguide -- 1.3.4.1 Slow range variation of the sound speed field -- 1.3.4.2 Weak variation of the sound speed field -- 1.3.5 Geometrical optics for the Helmholtz equation -- 1.3.5.1 Eikonal and transport equations -- 1.3.5.2 Momentum-position variables -- 1.3.5.3 Action-angle variables -- 1.4 Ray Travel Times -- 1.4.1 Timefront -- 1.4.2 Travel time as a function of starting momentum -- 1.5 Range-Dependent Environments -- 1.5.1 Internal waves -- 1.5.2 Rossby waves -- 1.5.3 Currents -- 1.5.4 Eddies -- 1.6 Acoustic Ocean Tomography -- 1.7 Experiments on Long-Range Sound Propagation -- 1.7.1 The Heard Island Feasibility Test -- 1.7.2 Experiments with bottom-mounted sources -- 1.7.2.1 Downsloped bathymetry near a source -- 1.7.3 Acoustic Engineering Test -- 1.7.4 Alternate Source Test -- 1.7.5 Acoustic Thermometry of Ocean Climate -- 1.8 Summary -- 2. Ray Chaos -- 2.1 Hamiltonian Chaos -- 2.1.1 Dynamics of Hamiltonian systems -- 2.1.2 Statistical description of Hamiltonian chaos -- 2.2 Lyapunov Instability -- 2.3 Ray-Medium Resonance -- 2.4 Overlapping of Resonances -- 2.5 Vertical Resonance -- 2.5.1 Adiabatic approximation -- 2.5.2 Passage through a resonance -- 2.5.2.1 Scattering on a resonance -- 2.5.2.2 Capturing into a resonance.

2.5.3 Vertical resonance versus ray-medium resonance -- 2.5.4 Resonance-induced chaotic layer -- 2.5.5 Influence of vertical resonance on a timefront of a received pulse -- 2.6 Manifestation of Regular and Chaotic Ray Motion in Distributions of Ray Travel Times -- 2.6.1 Perturbed waveguide -- 2.6.2 Timefront -- 2.6.3 Amplitude of a pulse signal in plane "travel time-depth" -- 2.6.4 Gap between travel times of chaotic and regular rays -- 2.6.5 "Focusing" of ray travel times -- 2.6.6 Role of stickiness and chaotic jets in "focusing" of ray travel times -- 2.6.7 Smoothed intensity of pulse signal -- 2.7 Summary -- 3. Wave Chaos -- 3.1 The Problem of Wave Chaos -- 3.1.1 Ehrenfest time -- 3.1.2 Semiclassical propagator -- 3.1.3 Fidelity or overlap of wave fields -- 3.1.4 Dynamical localization -- 3.2 Normal Modes -- 3.2.1 Range-independent waveguide -- 3.2.1.1 Eigenfunctions and eigenvalues in the WKB approximation -- 3.2.1.2 Mode amplitudes -- 3.2.1.3 Brillouin waves -- 3.2.1.4 Matrix elements -- 3.2.1.5 Ray-mode relations -- 3.2.2 Range-dependent waveguide -- 3.2.2.1 Coupled mode equations: slow and strong range dependence -- 3.2.2.2 Coupled mode equations: weak range dependence -- 3.2.3 Normal modes corresponding to the Helmholtz equation -- 3.2.3.1 Range-independent waveguide -- 3.2.3.2 Mode amplitudes -- 3.2.3.3 Brillouin waves -- 3.2.3.4 Range-dependent waveguide -- 3.2.4 Ray-based description of mode amplitudes -- 3.2.4.1 Parabolic equation -- 3.2.4.2 Helmholtz equation -- 3.2.5 Floquet modes of a periodic waveguide -- 3.3 Mode Coupling Under Chaotic Conditions -- 3.3.1 Spatial mode-medium resonance -- 3.3.2 Mode medium resonance and Floquet modes -- 3.3.3 Manifestations of stable islands and chaotic sea in structures of the Floquet modes -- 3.3.4 Scarring -- 3.4 Influence of Fine-Scale Inhomogeneities on Wave Dynamics -- 3.4.1 Introductory remarks.

3.4.2 Mode coupling in the presence of a fine structure -- 3.4.3 Influence of vertical oscillations of a sound-speed perturbation on Floquet modes -- 3.4.3.1 λz = 2000 m. -- 3.4.3.2 λz = 1000 m. -- 3.4.3.3 λz = 500 m -- 3.4.3.4 λz = 200 m -- 3.5 Summary -- 4. Chaotic Phenomena in Random Environment -- 4.1 Ray Chaos in a Random Medium -- 4.1.1 Statistical description of chaotic ray dynamics -- 4.1.2 Environmentalmodel -- 4.1.3 Fokker-Planck equation for action -- 4.1.4 Evaluation of diffusivity B -- 4.1.5 Wiener process approximation -- 4.1.6 Probability density functions of ray parameters -- 4.1.6.1 Rays starting from a small area in the phase space -- 4.1.6.2 Averaging over ray starting parameters -- 4.1.6.3 Rays escaping a point source -- 4.1.7 Coarse-grained Wigner function -- 4.1.7.1 Ray-based description of the Wigner function -- 4.1.7.2 Evaluation of the coarse-grained Wigner function in the Wiener process approximation -- 4.1.8 Specific Poincaré map for a randomlyinhomogeneous waveguide -- 4.2 Travel Times of Chaotic Rays -- 4.2.1 Timefront in a randomly inhomogeneous waveguide -- 4.2.2 Travel time differences -- 4.2.2.1 Two rays in a range-independent waveguide -- 4.2.2.2 Ray in a range-independent waveguide and ray in a rangedependent waveguide -- 4.2.3 Statistics of ray travel times -- 4.2.3.1 Variations of ray travel times caused by internal waves -- 4.2.3.2 Spread and bias of fuzzy segments -- 4.2.3.3 Pulse signal at the observation point -- 4.2.3.4 Stability of the timefront and the Fermat's principle -- 4.2.3.5 Travel time as a function of launch angle -- 4.3 Modal Structure of the Sound Field in a Waveguide with RandomInhomogeneities -- 4.3.1 Redistribution of acoustic energy between normal modes under conditions of ray chaos -- 4.3.2 Transient wave field -- 4.3.2.1 Arrival time of an instantaneous frequency.

4.3.2.2 Mode pulses in a range-independent waveguide -- 4.3.2.3 Mode pulses in the presence of internal waves -- 4.3.2.4 Bias and spread of a mode pulse -- 4.4 Wave Beam in an Ocean Acoustic Waveguide -- 4.4.1 Ray-based description of a wave beam -- 4.4.1.1 Directivity pattern in a free space -- 4.4.1.2 Ray-based description of the wave field excited by a vertical antenna in a waveguide -- 4.4.1.3 Applicability conditions of the stationary phase technique -- 4.4.1.4 A weakly divergent beam -- 4.4.2 Wave beam and its modal structure in the presence of randominternal waves -- 4.5 Arrival Times of Sound Pulses in the Presence of Internal Waves and a Mesoscale Inhomogeneity -- 4.5.1 Variation of the timefront in the presence of synoptic eddy -- 4.5.2 The use of a vertical receiving array for measuring travel time delays -- 4.6 Summary -- 5. Glossary of Some Concepts and Notations in Hamiltonian Chaos Theory -- Bernoulli Shift -- Bifurcation -- Cantorus -- Dynamical Trap -- Fractal -- Hamiltonian Chaos -- Hamiltonian Dynamics -- Heteroclinic (Homoclinic) Structure -- Horseshoe Mapping -- Invariant Torus -- Island of Stability -- Kolmogorov-Arnold-Moser Theorem and KAM Torus -- Lyapunov Exponent -- Manifold -- Measure in Dynamical Systems -- Melnikov's Method -- Nonlinear Resonance -- Phase Space -- Poincaré Map -- Separatrix -- Stable and Unstable Motion -- Stability Points -- Trajectories -- Appendix A Models of a Waveguide -- A.1 CanonicalMunk Profile -- A.2 Linear Profile -- A.3 Bilinear Profile -- A.4 Biexponential Profile -- Appendix B Derivation of Ray Equations -- Appendix C Solution of the Transport Equation -- Appendix D Stationary Phase Technique -- Bibliography -- Index.