Evolution of Extreme Waves and Resonances-Volume I

PART I. Basic equations and ideas
Chapter 1 Lagrangian description of surface water waves 
1.1. The Lagrangian form of the hydrodynamics equations: the balance equations, boundary conditions, and a strongly nonlinear basic equation 
1.1.1. Balance and state equations
1.1.2. Boundary conditions
1.1.3. A basic expression for the pressure and a basic strongly nonlinear wave equation  
1.2. 2D strongly nonlinear wave equations for a viscous liquid
1.2.1. The vertical displacement assumption
1.2.2. The 2D Airy-type wave equation
1.2.3. The generation of the Green-Naghdi-type equation  
1.3. A basic depth-averaged 1D model using a power approximation 
1.3.1. The strongly nonlinear wave equation
1.3.2. Three-speed variants of the strongly nonlinear wave equation  
1.3.3. Resonant interaction of the gravity and capillary effects in a surface wave
1.3.4. Effects of the dispersion
1.3.5. Examples of nonlinear wave equations 
1.4. Nonlinear equations for gravity waves over the finite-depth ocean
1.4.1. Moderate depth 
1.4.2. The gravity waves over the deep ocean
1.5. Models and basic equations for long waves
1.6. Bottom friction and governing equations for long extreme waves
1.7. Airy- type equations for capillary waves and remarks to the Chapter 4
Chapter 2 Euler’s figures and extreme waves: examples, equations and unified solutions
2.1. Example of Euler`s elastica figures 
2.2. Examples of fundamental nonlinear wave equations
2.3. The nonlinear Klein-Gordon equation and wide spectre of its solutions
2.3.1. The one-dimensional version and one hand travelling waves
2.3.2. Exact solutions of the nonlinear Klein-Gordon equation
2.3.3. The sine-Gordon equation: approximate and exact elastica-like wave solutions
2.4. Cubic nonlinear equations describing elastica-like waves 
2.5. Elastica-like waves: singularities, unstabilities, resonant generation
2.5.1. Singularities as fields of the Euler’s elastic figures generation
2.5.2. Instabilities and generation of the Euler’s elastica figures 
2.5.3. `Dangerous` dividers and self-excitation of the transresonant waves
2.6. Simple methods for a description of elastica-like waves
2.6.1. Modelling of unidirectional elasica-like waves
2.6.2. The model equation for Faraday waves and Euler’s figures
2.7. Nonlinear effects on transresonant evolution of Euler figures into particle-waves
References
PART II. Waves in finite resonators
Chapter 3  Generalisation of the d’Alembert’s solution for nonlinear long waves
3.1. Resonance of travelling surface waves (site resonance)
3.2. Extreme waves in finite resonators
3.2.1. Resonance waves in a gas filling closed tube
3.2.2. Resonant amplification of seismic waves in natural resonators
3.2.3. Topographic effect: extreme dynamics of Tarzana hill
3.3. The d` Alembert- type nonlinear resonant solutions: deformable coordinates 
3.3.1. The singular solution of the nonlinear wave equation
3.3.2. The solutions of the wave equation without the singularity with time
3.3.3. Some particular cases of the general solution (3.22) 
3.4. The d` Alembert- type nonlinear resonant solutions: undeformable coordinates 
3.4.1. The singular solution of the nonlinear wave equations
3.4.2.  Resonant (unsingular in time) solutions of the wave equation
3.4.3. Special cases of the resonant (unsingular with time) solution
3.4.4. Illustration to the theory: the site resonance of waves in a long channel
3.5. Theory of free oscillations of nonlinear wave in resonators
3.5.1. Theory of free strongly nonlinear wave in resonators
3.5.2. Comparison of theoretical results
3.6. Conclusion on this Chapter
Chapter 4 Extreme resonant waves: a quadratic nonlinear theory
4.1. An example of a boundary problem and the equation determining resonant plane waves
4.1.1. Very small effects of nonlinearity, viscosity and dispersion
4.1.2. The dispersion effect on linear oscillations
4.1.3. Fully linear analysis 
4.2. Linear resonance
4.2.1. Effect of the nonlinearity
4.2.2. Waves excited very near band boundaries of resonant band
4.2.3. Effect of viscosity
4.3. Solutions within and near the shock structure
4.4. Resonant wave structure: effect of dispersion
4.5. Quadratic resonances
4.5.1. Results of calculations and discussion
4.6. Forced vibrations of a nonlinear elastic layer
Chapter 5.  Extreme resonant waves: a cubic nonlinear theory
5.1. Cubically nonlinear effect for closed resonators   
5.1.1. Results of calculations: pure cubic nonlinear effect
5.1.2. Results of calculations: joint cubic and quadratic nonlinear effect
5.1.3. Instant collapse of waves near resonant band end
5.1.4. Linear and cubic-nonlinear standing waves in resonators
5.1.5. Resonant particles, drops, jets, surface craters and bubbles 
5.2. A half-open resonator
5.2.1. Basic relations
5.2.2. Governing equation
5.3 Scenarios of transresonant evolution and comparisons with experiments 
5.4. Effects of cavitation in liquid on its oscillations in resonators
Chapter 6  Spherical resonant waves
6.1. Examples and effects of extreme amplification of spherical waves
6.2. Nonlinear spherical waves in solids
6.2.1. Nonlinear acoustics of the homogeneous viscoelastic solid body
6.2.2. Approximate general solution 
6.2.3. Boundary problem, basic relations and extreme resonant waves
6.2.4. Analogy with the plane wave, results of calculations and discussion 
6.3. Extreme waves in spherical resonators filling gas or liquid
6.3.1. Governing equation and its general solution
6.3.2. Boundary conditions and basic equation for gas sphere
6.3.3. Structure and trans-resonant evolution of oscillating waves
6.3.3.1. First scenario (C -B)
6.3.3.2. Second scenario (C = -B)
6.3.4. Discussion
6.4. Localisation of resonant spherical waves in spherical layer
Chapter 7  Extreme Faraday waves
7.1. Extreme vertical dynamics of weakly-cohesive materials 
7.1.1. Loosening of surface layers due to strongly-nonlinear wave phenomena  
7.2. Main ideas of the research
7.3. Modelling experiments as standing waves
7.4. Modelling of counterintuitive waves as travelling waves
7.4.1. Modeling of the Kolesnichenko`s experiments
7.4.2. Modelling of experiments of Bredmose et al.
7.5. Strongly nonlinear waves and ripples
7.5.1. Experiments of Lei Jiang et al.  and discussion of them
7.5.2.  Deep water model
7.6. Solitons, oscillons and formation of surface patterns
7.7. Theory and patterns of nonlinear Faraday waves
7.7.1 Basic equations and relations
7.7.2. Modeling of certain experimental data 
7.7.3. Two-dimensional patterns
7.7.4 Historical comments and key result
References
PART III. Extreme ocean waves and resonant phenomena
Chapter 8  Long waves, Green`s law and topographical resonance
8.1. Surface ocean waves and vessels  
8.2. Observations of the extreme waves
8.3. Long solitary waves  
8.4. KdV-type, Burgers-type, Gardner-type and Camassa-Holm-type equations for the case of the slowly-variable depth
8.5. Model solutions and the Green law for solitary wave
8.6. Examples of coastal evolution of the solitary wave
8.7. Generalizations of the Green’s law 
8.8. Tests for generalised Green’s law 
8.8.1. The evolution of harmonical waves above topographies 
8.8.2. The evolution of a solitary wave over trapezium topographies
8.8.3. Waves in the channel with a semicircular topographies
8.9. Topographic resonances and the Euler’s elastica
Chapter 9  Modelling of the tsunami described by Charles Darwin and coastal waves
9.1. Darwin’s description of tsunamis generated by coastal earthquakes  
9.2. Coastal evolution of tsunami
9.2.1. Effect of the bottom slope
9.2.2. The ocean ebb in front of a tsunami
9.2.3. Effect of the bottom friction 
9.3. Theory of tsunami: basic relations 
9.4. Scenarios of the coastal evolution of tsunami 
9.4.1. Cubic nonlinear scenarios
9.4.2. Quadratic nonlinear scenario
9.5. Cubic nonlinear effects: overturning and breaking of waves
Chapter 10.  Theory of extreme (rogue, catastrophic) ocean waves
10.1. Oceanic heterogeneities and the occurrence of extreme waves
10.2. Model of shallow waves
10.2.1. Simulation of a “hole in the sea” met by the tanker “Taganrogsky Zaliv”
10.2.2. Simulation of typical extreme ocean waves as shallow waves
10.3. Solitary ocean waves
10.4. Nonlinear dispersive relation and extreme waves
10.4.1. The weakly nonlinear interaction of many small amplitude ocean waves 
10.4.2. The cubic nonlinear interaction of ocean waves and extreme waves formation
10.5. Resonant nature of extreme harmonic wave  
Chapter 11.  Wind-induced waves and wind-wave resonance
11.1. Effects of wind and current 
11.2. Modeling the effect of wind on the waves
11.3. Relationships and equations for wind waves in shallow and deep water
11.4. Wave equations for unidirectional wind waves
11.5. The transresonance evolution of coastal wind waves
Chapter 12.  Transresonant evolution of Euler’s figures into vortices
12.1. Vortices in the resonant tubes
12.2. Resonance vortex generation
12.3. Simulation of the Richtmyer-Meshkov instability results
12.4. Cubic nonlinearity and evolution of waves into vortices
12.5. Remarks to extreme water waves (Parts I-III)  
References
PART IV. Modelling of particle-waves, slit experiments and the extreme waves in scalar fields
Chapter 13.  Resonances, Euler figures, and particle-waves
13.1. Scalar fields and Euler figures
13.1.1 Own nonlinear oscillations of a scalar field in a resonator
13.1.2. The simplest model of the evolution of Euler’s figures into periodical particle-wave
13.2. Some data of exciting experiments with layers of liqud
13.3. Stable oscillations of particle-wave configurations
13.4. Schrödinger and Klein-Gordon equations
13.5. Strongly localised nonlinear sphere-like waves and wave packets
13.6. Wave trajectories, wave packets and discussion
Chapter 14.  Nonlinear quantum waves in the light of recent slit experiments
14.1. Introduction
14.2. Experiments using different kind of \"slits\" and the beginning of the discussion
14.3. Explanations and discussion of the experimental results
14.4. Casimir’s effect
14.5. Thin metal layer and plasmons as the synchronizators
14.6. Testing of thought experiments
14.7. Main thought experiment
14.8. Resonant dynamics of particle-wave, vacuum and Universe
Chapter 15.  Resonant models of origin of particles and the Universe due to quantum perturbations of scalar fields
15.1. Basic equation and relations
15.2. Basic solutions. Dynamic and quantum effects
15.3. Two-dimensional maps of landscapes of the field
15.4. Description of quantum perturbations
15.4.1. Quantum perturbations and free nonlinear oscillations in the potential well
15.4.2. Oscilations of scalar field, granular layer and the Bose-Einstein condensate
15.4.3. Simple model of the origin of the particles: mathematics and imaginations
15.5. Modelling of quantum actions: theory
15.6. Modelling of quantum actions: calculations
References