#
Evolution of Extreme Waves and Resonances

Volume I

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## Book Description

The theory of waves is generalized on cases of strongly nonlinear waves, multivalued waves, and particle–waves. The appearance of these waves in various continuous media and physical fields is explained by resonances and nonlinearity effects. Extreme waves emerging in different artificial and natural systems from atom scale to the Universe are explored. Vast amounts of experimental data and comparisons of them with the results of the developed theory are presented.

The book was written for graduate students as well as for researchers and engineers in the fields of geophysics, nonlinear wave studies, cosmology, physical oceanography, and ocean and coastal engineering. It is designed as a professional reference for those working in the wave analysis and modeling fields.

## Table of Contents

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

## Author(s)

### Biography

Shamil U. Galiev obtained his Ph.D. degree in Mathematics and Physics from Leningrad University in 1971, and, later, a full doctorate (ScD) in Engineering Mechanics from the Academy of Science of Ukraine (1978). He worked in the Academy of Science of former Soviet Union as a researcher, senior researcher, and department chair from 1965 to 1995. From 1984 to 1989, he served as a Professor of Theoretical Mechanics in the Kiev Technical University, Ukraine. Since 1996, he has served as Professor, Honorary Academic of the University of Auckland, New Zealand. Dr. Galiev has published approximately 90 scientific publications, and he is the author of seven books devoted to different complex wave phenomena. From 1965-2014 he has studied different engineering problems connected with dynamics and strength of submarines, rocket systems, and target/projectile (laser beam) systems. Some of these results were published in books and papers. During 1998-2017, he conducted extensive research and publication in the area of strongly nonlinear effects connected with catastrophic earthquakes, giant ocean waves and waves in nonlinear scalar fields. Overall, Dr. Galiev’s research has covered many areas of engineering, mechanics, physics, and mathematics.