Ray and Wave Chaos in Ocean Acoustics

The following sections are included:

• To the memory of George Zaslavsky

• Preface

• Contents

The following sections are included:

• Underwater Sound Channel

• Basic Equations
  • Helmholtz equation
  • Parabolic equation

• Geometrical Optics Approximations and Optical-Mechanical Analogy. The Hamiltonian Formalism
  • Eikonal and transport equations
  • Momentum-position variables
  • Action-angle variables in a range-independent waveguide
    • Canonical transformation
    • Ray path in the unperturbed waveguide
    • Canonical transformation in the form of Fourier series
  • Action-angle variables in a range-dependent waveguide
    • Slow range variation of the sound speed field
    • Weak variation of the sound speed field
  • Geometrical optics for the Helmholtz equation
    • Eikonal and transport equations
    • Momentum-position variables
    • Action-angle variables

• Ray Travel Times
  • Timefront
  • Travel time as a function of starting momentum

• Range-Dependent Environments
  • Internal waves
  • Rossby waves
  • Currents
  • Eddies

• Acoustic Ocean Tomography

• Experiments on Long-Range Sound Propagation
  • The Heard Island Feasibility Test
  • Experiments with bottom-mounted sources
    • Downsloped bathymetry near a source
  • Acoustic Engineering Test
  • Alternate Source Test
  • Acoustic Thermometry of Ocean Climate

• Summary

The following sections are included:

• Hamiltonian Chaos
  • Dynamics of Hamiltonian systems
  • Statistical description of Hamiltonian chaos

• Lyapunov Instability

• Ray-Medium Resonance

• Overlapping of Resonances

• Vertical Resonance
  • Adiabatic approximation
  • Passage through a resonance
    • Scattering on a resonance
    • Capturing into a resonance
  • Vertical resonance versus ray-medium resonance
  • Resonance-induced chaotic layer
  • Influence of vertical resonance on a timefront of a received pulse

• Manifestation of Regular and Chaotic Ray Motion in Distributions of Ray Travel Times
  • Perturbed waveguide
  • Timefront
  • Amplitude of a pulse signal in plane “travel time–depth”
  • Gap between travel times of chaotic and regular rays
  • “Focusing” of ray travel times
  • Role of stickiness and chaotic jets in “focusing” of ray travel times
  • Smoothed intensity of pulse signal

• Summary

The following sections are included:

• The Problem of Wave Chaos
  • Ehrenfest time
  • Semiclassical propagator
  • Fidelity or overlap of wave fields
  • Dynamical localization

• Normal Modes
  • Range-independent waveguide
    • Eigenfunctions and eigenvalues in the WKB approximation
    • Mode amplitudes
    • Brillouin waves
    • Matrix elements
    • Ray-mode relations
  • Range-dependent waveguide
    • Coupled mode equations: slow and strong range dependence
    • Coupled mode equations: weak range dependence
  • Normal modes corresponding to the Helmholtz equation
    • Range-independent waveguide
    • Mode amplitudes
    • Brillouin waves
    • Range-dependent waveguide
  • Ray-based description of mode amplitudes
    • Parabolic equation
    • Helmholtz equation
  • Floquet modes of a periodic waveguide

• Mode Coupling Under Chaotic Conditions
  • Spatial mode-medium resonance
  • Mode medium resonance and Floquet modes
  • Manifestations of stable islands and chaotic sea in structures of the Floquet modes
  • Scarring

• Influence of Fine-Scale Inhomogeneities on Wave Dynamics
  • Introductory remarks
  • Mode coupling in the presence of a fine structure
  • Influence of vertical oscillations of a sound-speed perturbation on Floquet modes
    • λ_z = 2000 m
    • λ_z = 1000 m
    • λ_z = 500 m
    • Λ_z = 200 m

• Summary

The following sections are included:

• Ray Chaos in a Random Medium
  • Statistical description of chaotic ray dynamics
  • Environmental model
  • Fokker-Planck equation for action
  • Evaluation of diffusivity B
  • Wiener process approximation
  • Probability density functions of ray parameters
    • Rays starting from a small area in the phase space
    • Averaging over ray starting parameters
    • Rays escaping a point source
  • Coarse-grained Wigner function
    • Ray-based description of the Wigner function
    • Evaluation of the coarse-grained Wigner function in the Wiener process approximation
  • Specific Poincaré map for a randomly-inhomogeneous waveguide

• Travel Times of Chaotic Rays
  • Timefront in a randomly inhomogeneous waveguide
  • Travel time differences
    • Two rays in a range-independent waveguide
    • Ray in a range-independent waveguide and ray in a range-dependent waveguide
  • Statistics of ray travel times
    • Variations of ray travel times caused by internal waves
    • Spread and bias of fuzzy segments
    • Pulse signal at the observation point
    • Stability of the timefront and the Fermat's principle
    • Travel time as a function of launch angle

• Modal Structure of the Sound Field in a Waveguide with Random Inhomogeneities
  • Redistribution of acoustic energy between normal modes under conditions of ray chaos
  • Transient wave field
    • Arrival time of an instantaneous frequency
    • Mode pulses in a range-independent waveguide
    • Mode pulses in the presence of internal waves
    • Bias and spread of a mode pulse

• Wave Beam in an Ocean Acoustic Waveguide
  • Ray-based description of a wave beam
    • Directivity pattern in a free space
    • Ray-based description of the wave field excited by a vertical antenna in a waveguide
    • Applicability conditions of the stationary phase technique
    • A weakly divergent beam
  • Wave beam and its modal structure in the presence of random internal waves

• Arrival Times of Sound Pulses in the Presence of Internal Waves and a Mesoscale Inhomogeneity
  • Variation of the timefront in the presence of synoptic eddy
  • The use of a vertical receiving array for measuring travel time delays

• Summary

The following sections are included:

• 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

• Poincarée Map

• Separatrix

• Stable and Unstable Motion

• Stability Points

• Trajectories

The following sections are included:

• Appendix A Models of a Waveguide
  • Canonical Munk Profile
  • Linear Profile
  • Bilinear Profile
  • Biexponential Profile

• Appendix B Derivation of Ray Equations

• Appendix C Solution of the Transport Equation

• Appendix D Stationary Phase Technique

• Bibliography

• Index