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**Modeling of Extreme Waves in Technology and Nature** is a two-volume set, comprising *Evolution of Extreme Waves and Resonances* (Volume I) and *Extreme Waves and Shock-Excited Processes in Structures and Space Objects* (Volume II).

The theory of waves is generalized on cases of extreme waves. The formation and propagation of extreme waves of various physical and mechanical nature (surface, elastoplastic, fracture, thermal, evaporation) in liquid and solid media, and in structural elements contacting with bubbly and cryogenic liquids are considered analytically and numerically. The occurrence of tsunamis, giant ocean waves, turbulence, and different particle-waves is described as resonant natural phenomena.

Nonstationary and periodic waves are considered using models of continuum. The change in the state of matter is taken into account using wide-range determining equations.

The desire for the simplest and at the same time general description of extreme wave phenomena that takes the reader to the latest achievements of science is the main thing that characterizes this book and is revolutionary for wave theory. A description of a huge number of observations, experimental data, and calculations is also given.

Chapter 1. Models of continuum

1.1. The system of equations of mechanics continuous medium

1.2. State (constitutive) equations for elastic and elastic-plastic bodies

1.3. The equations of motion and the wide range equations of state of an inviscid fluid

1.4. Simplest example of fracture of media within rarefaction zones

1.4.1. The state equation for bubbly liquid

1.4.2. Fracture (cold boiling) of water during seaquakes

1.4.3. Model of fracture (cold boiling) of bubbly liquid

1.5. Models of moment and momentless shells

1.5.1. Shallow shells and the Kirchhoff - Love hypotheses

1.5.2. The Timoshenko theory of thin shells and momentless shells

Chapter 2. The dynamic destruction of some materials in tension waves

2.1. Models of dynamic failure of solid media

2.1.1. Phenomenological approach

2.1.2. Microstructural approach

2.2. Models of interacting voids (bubbles, pores)

2.3. Pores on porous materials

2.4. Mathematical model of materials containing pores

Chapter 3. Models of dynamic failure of weakly-cohesived media (WCM)

3.1. Introduction

3.1.1. Examples of gassy material properties

3.1.2. Behavior of weakly-cohesive geomaterials within of extreme waves

3.2. Modelling of gassy media

3.2.1. State equation for mixture of condensed matter/gas

3.2.2. Strongly nonlinear model of the state equation for gassy media

3.2.3. The Tait-like form of the state equation

3.2.4. Wave equations for gassy materials

3.3. Effects of bubble oscillations on the one-dimensional governing equations

3.3.1. Differential form of the state equation

3.3.2. The strongly nonlinear wave equation for bubbly media

3.4. Linear acoustics of bubbly media

3.4.1. Three speed wave equations

3.4.2. Two speed wave equations

3.4.3. One-speed wave equations

3.4.4. Influence of viscous properties on the sound speed of magma-like media

3.5. Examples of observable extreme waves of WCM

3.5.1. Mount St Helens eruption

3.5.2. The volcano Santiaguito eruptions

3.6. Nonlinear acoustic of bubble media

3.6.1. Low frequency waves: Boussinesq and long wave equations

3.6.2. High frequency waves: Klein-Gordon and Schrödinger equations

3.7. Strongly nonlinear Airy-type equations and remarks to the Chapters 1-3

Chapter 4. Lagrangian description of surface water waves

4.1. The Lagrangian form of the hydrodynamics equations: the balance equations, boundary conditions, and a strongly nonlinear basic equation

4.1.1. Balance and state equations

4.1.2. Boundary conditions

4.1.3. A basic expression for the pressure and a basic strongly nonlinear wave equation

4.2. 2D strongly nonlinear wave equations for a viscous liquid

4.2.1. The vertical displacement assumption

4.2.2. The 2D Airy-type wave equation

4.2.3. The generation of the Green-Naghdi-type equation

4.3. A basic depth-averaged 1D model using a power approximation

4.3.1. The strongly nonlinear wave equation

4.3.2. Three-speed variants of the strongly nonlinear wave equation

4.3.3. Resonant interaction of the gravity and capillary effects in a surface wave

4.3.4. Effects of the dispersion

4.3.5. Examples of nonlinear wave equations

4.4. Nonlinear equations for gravity waves over the finite-depth ocean

4.4.1. Moderate depth

4.4.2. The gravity waves over the deep ocean

4. 5. Models and basic equations for long waves

4.6. Bottom friction and governing equations for long extreme waves

4. 7. Airy- type equations for capillary waves and remarks to the Chapter 4

Chapter 5. Euler’s figures and extreme waves: examples, equations and unified solutions

5.1. Example of Euler's elastica figures

5.2. Examples of fundamental nonlinear wave equations

5.3. The nonlinear Klein-Gordon equation and wide spectre of its solutions

5.3.1 The one dimensional version and one hand travelling waves

5.3.2. Exact solutions of the nonlinear Klein-Gordon equation

5.3.3. The sine-Gordon equation: approximate and exact elastica-like wave solutions

5.4. Cubic nonlinear equations describing elastica-like waves

5.5. Elastica-like waves: singularities, unstabilities, resonant generation

5. 5. 1. Singularities as fields of the Euler’s elastic figures generation

5. 5. 2. Instabilities and generation of the Euler’s elastica figures

5. 5. 3. 'Dangerous' dividers and self-excitation of the transresonant waves

5. 6. Simple methods for a description of elastica-like waves

5. 6. 1. Modelling of unidirectional elasica-like waves

5. 6. 2. The model equation for Faraday waves and Euler’s figures

5.7. Nonlinear effects on transresonant evolution of Euler figures into particle-waves

References

PART II. Waves in finite resonators

Chapter 6. Generalisation of the d’Alembert’s solution for nonlinear long waves

6.1. Resonance of travelling surface waves (site resonance)

6.2. Extreme waves in finite resonators

6. 2. 1. Resonance waves in a gas filling closed tube

6. 2. 2. Resonant amplification of seismic waves in natural resonators

6. 2. 3. Topographic effect: extreme dynamics of Tarzana hill

6. 3. The d' Alembert- type nonlinear resonant solutions: deformable coordinates

6.3.1. The singular solution of the nonlinear wave equation

6.3.2. The solutions of the wave equation without the singularity with time

6.3.3. Some particular cases of the general solution (6.22)

6.4. The d' Alembert- type nonlinear resonant solutions: undeformable coordinates

6.4.1. The singular solution of the nonlinear wave equations

6. 4. 2. Resonant (unsingular in time) solutions of the wave equation

6. 4. 3. Special cases of the resonant (unsingular with time ) solution

6. 4. 4. Illustration to the theory: the site resonance of waves in a long channel

6. 5. Theory of free oscillations of nonlinear wave in resonators

6. 5. 1. Theory of free strongly nonlinear wave in resonators

6. 5. 2. Comparison of theoretical results

6. 6. Conclusion on this Chapter

Chapter 7. Extreme resonant waves: a quadratic nonlinear theory

7.1. An example of a boundary problem and the equation determining resonant plane waves

7.1.1. Very small effects of nonlinearity, viscosity and dispersion

7.1.2. The dispersion effect on linear oscillations

7.1.3. Fully linear analysis

7.2. Linear resonance

7. 2. 1. Effect of the nonlinearity

7. 2. 2. Waves excited very near band boundaries of resonant band

7. 2. 3. Effect of viscosity

7. 3. Solutions within and near the shock structure

7.4. Resonant wave structure: effect of dispersion

7. 5. Quadratic resonances

7. 5. 1. Results of calculations and discussion

7.6. Forced vibrations of a nonlinear elastic layer

Chapter 8. Extreme resonant waves: a cubic nonlinear theory

8. 1. Cubically nonlinear effect for closed resonators

8. 1. 1. Results of calculations: pure cubic nonlinear effect

8. 1. 2. Results of calculations: joint cubic and quadratic nonlinear effect

8. 1. 3. Instant collapse of waves near resonant band end

8. 1. 4. Linear and cubic-nonlinear standing waves in resonators

8. 1. 5. Resonant particles, drops, jets, surface craters and bubbles

8. 2. A half-open resonator

8. 2.1. Basic relations

8. 2.2. Governing equation

8.3 Scenarios of transresonant evolution and comparisons with experiments

8. 4. Effects of cavitation in liquid on its oscillations in resonators

Chapter 9. Spherical resonant waves

9.1. Examples and effects of extreme amplification of spherical waves

9. 2. Nonlinear spherical waves in solids

9.2.1. Nonlinear acoustics of the homogeneous viscoelastic solid body

9. 2.2. Approximate general solution

9. 2.3. Boundary problem, basic relations and extreme resonant waves

9.2.4. Analogy with the plane wave, results of calculations and discussion

9.3. Extreme waves in spherical resonators filling gas or liquid

9.3.1. Governing equation and its general solution

9.3.2. Boundary conditions and basic equation for gas sphere

9. 3.3. Structure and trans-resonant evolution of oscillating waves

9.3.3.1. First scenario (C -B)

9.3.3.2. Second scenario (C = -B)

9.3.4. Discussion

9. 4. Localisation of resonant spherical waves in spherical layer

Сhapter 10. Extreme Faraday waves

10. 1. Extreme vertical dynamics of weakly-cohesive materials

10. 1.1. Loosening of surface layers due to strongly-nonlinear wave phenomena

10.2 . Main ideas of the research

10. 3. Modelling experiments as standing waves

10.4. Modelling of counterintuitive waves as travelling waves

10. 4. 1. Modeling of the Kolesnichenko's experiments

10. 4. 2. Modelling of experiments of Bredmose et al.

10. 5. Strongly nonlinear waves and ripples

10.5. 1. Experiments of Lei Jiang et al. and discussion of them

10. 5. 2. Deep water model

10. 6. Solitons, oscillons and formation of surface patterns

10.7. Theory and patterns of nonlinear Faraday waves

10. 7.1 Basic equations and relations

10. 7.2. Modeling of certain experimental data

10.7.3. Two-dimensional patterns

10. 7.4 Historical comments and key result

References

PART III. Extreme ocean waves, resonances and phenomena

Chapter 11. Long waves, Green's law and topographical resonance

11.1. Surface ocean waves and vessels

11.2. Observations of the extreme waves

11.3. Long solitary waves

11. 4. KdV-type, Burgers-type, Gardner-type and Camassa-Holm-type equations for the case of the slowly-variable depth

11. 5. Model solutions and the Green law for solitary wave

11. 6. Examples of coastal evolution of the solitary wave

11. 7. Generalizations of the Green’s law

11. 8. Tests for generalisated Green’s law

11. 8. 1. The evolution of harmonical waves above topographies

11. 8. 2. The evolution of a solitary wave over trapezium topographies

11. 8. 3. Waves in the channel with a semicircular topographies

11. 9. Topographic resonances and the Euler’s elastica

Chapter 12. Modelling of the tsunami described by Charles Darwin and coastal waves

12.1. Darwin’s description of tsunamis generated by coastal earthquakes

12.2. Coastal evolution of tsunami

12.2.1. Effect of the bottom slope

12. 2. 2. The ocean ebb in front of a tsunami

12. 2. 3. Effect of the bottom friction

12 .3. Theory of tsunami: basic relations

12.4. Scenarios of the coastal evolution of tsunami

12.4.1. Cubic nonlinear scenarios

12.4.2. Quadratic nonlinear scenario

12. 5. Cubic nonlinear effects: overturning and breaking of waves

Chapter 13. Theory of extreme (rogue, catastrophic) ocean waves

13. 1. Oceanic heterogeneities and the occurrence of extreme waves

13. 2. Model of shallow waves

13. 2. 1. Simulation of a “hole in the sea” met by the tanker “Taganrogsky Zaliv”

13. 2. 2. Simulation of typical extreme ocean waves as shallow waves

13.3. Solitary ocean waves

13. 4. Nonlinear dispersive relation and extreme waves

13. 4.1. The weakly nonlinear interaction of many small amplitude ocean waves

13. 4.2. The cubic nonlinear interaction of ocean waves and extreme waves formation

13. 5. Resonant nature of extreme harmonic wave

Chapter 14. Wind -induced waves and wind-wave resonance

14.1. Effects of wind and current

14.2. Modeling the effect of wind on the waves

14. 3. Relationships and equations for wind waves in shallow and deep water

14. 4. Wave equations for unidirectional wind waves

14.5. The transresonance evolution of coastal wind waves

Chapter 15. Transresonant evolution of Euler’s figures into vortices

15.1. Vortices in the resonant tubes

15.2. Resonance vortex generation

15.3. Simulation of the Richtmyer-Meshkov instability results

15.4. Cubic nonlinearity and evolution of waves into vortices

15.5. Remarks to extreme water waves (Parts I-II)

References

Part IV. Counterintuitive behaviour CIB of structural elements after impact loads

Chapter 16. Experimental data

16. 1. Introduction and method of impact loading

16. 2. CIB of circular plates: results and discussion

16. 3. CIB of rectangular plates and shallow caps

16.3.1. Discussion of CIB of shallow caps

16.3.2. Cap/ permeable membrane system

16.3.3. CIB of panels

Chapter 17. CIB of plates and shallow shells: theory and calculations

17. 1. Distinctive features of CIB of plates and shallow shells

17.1. 1. Investigation techniques

17. 1.2. Results and discussion: plates , spherical caps and cylindrical panels

17. 2. Influences of atmosphere and cavitation on CIB

17. 2. 1. Theoretical models

17. 2. 2. Calculation details

17. 2. 3. Results and discussion

References

PART V. Extreme waves and structural elements

Chapter 18. Extreme effects and waves in impact loaded hydrodeformable systerms

18.1. Introduction

18.2. Underwater explosions and the cavitation wave: experiments

18. 3. Experimental studies of formation and propagation of the cavitation waves

18. 3.1. Elastic plate/underwater wave interaction

18. 3.2. Elastoplastic plate/underwater wave interaction

18.4. Extreme underwater wave and plate interaction

18. 4. 1. Effects of deformability

18. 4. 2. Effects of cavitation on the plate surface

18. 4. 3. Effects of cavitation in the liquid volume on the plate-liquid interaction

18. 4. 4. Effects of plasticity

18. 5. Modelling of extreme wave cavitation and cool boiling in tanks

18. 5. 1. Impact loading of tank

18. 5. 2. Impact loading of liquid in tank

Chapter 19. Shells and cavitation (cool boiling) waves

19. 1. Interaction of a cylindrical shell with shock wave in liquid

19. 2. Extreme waves in cylindrical elastic container

19. 2. 1. Effects of cavitation and cool boiling on the interation of shells

19. 2. 2. Features of bubble dynamics and their effect on shells

19. 3. Extreme wave phenomena in the hydro - gas-elastic system

19. 4. Effects of boiling of liquids within rarefaction waves on the transient deformation of hydroelastic systems

19. 5. A method of solving transient three-dimensional problems of hydroelasticity for cavitating and boiling liquids

19.5.1. Governing equations

19.5.2. Numerical method

19.5.3. Results and discussion

Chapter 20. Interaction of extreme underwater waves with structures

20.1. Fracture and cavitation waves in thin plate/underwater explosion system

20.2. Fracture and cavitation waves in plate/underwater explosion system

20.3. Generation of cavitation waves after tank bottom buckling

20. 4. Transient interaction of a stiffened spherical dome with underwater shock waves

20. 4. 1. The problem and method of solution

20. 4. 2. Numeric method of problem solution

20. 4. 3. Results of calculations

20. 5. Extreme amplification of waves at vicinity of the stiffening rib

References

PART VI. Extreme waves excited by impact of heat, radiation or mass

Chapter 21. Formation and amplification of heat waves

21.1. Linear analysis. Influence of hyperbolicity

21.2. Forming and amplifing of nonlinear heat waves

21.3. Strongly nonlinearity of thermodynamic function as a cause of formation of cooling shock wave

Chapter 22. Extreme waves excited by radiation

22.1. Impulsive deformation and destruction of bodies at temperatures below the melting point

22.1.1. Thermoelastic waves excited by long-wave radiation

22.1.2. Thermo-elastic waves excited by short-wave radiation

22.1.3. Stress and fracture waves in metals during rapid bulk heating

22.1.4. Optimization of the outer laser–induced spalling

22.2. Effects of melting of material under impulse loading

22.2.1. Mathematical model of fracture under thermal force loading

22.2.2. Algorithm and results

22.3. Modelling of fracture, melting, vaporization and phase transition

22. 3.1. Calculations: effects of temperature

22.3.2. Calculations: effects of vaporization

22.3.3. Calculations: effect of vaporization on spalling

22.4. Two dimensional fracture and evaporation

22.5. Fracture of solid by radiation pulses as a method of ensuring safety in space

22.5. 1. Introduction

22.5. 2. Mathematical formulation of the problem

22.5. 3. Calculation results and comparison with experiments

22.5. 4. Special features of fracture by spalling

22.5. 5. Efficiency of laser fracture

22.5.6. Discussion and conclusion

Chapter. 23. The melting waves in front of a massive perforator

23.1. Experimental investigation

23. 2. Numerical modeling.

23. 3. Results of the calculation and discussion

References

Part VII. Modelling of particle-wave, slit experiments and the origin of the Universe

Chapter 24. Resonances, Euler figures and particle-waves

24.1. Scalar fields and Euler figures

24.1.1 Own nonlinear oscillations of a scalar field in a resonator

24.1.2. The simplest model of the evolution of Euler’s figures into periodical particle-wave

24.2. Some data of exciting experiments with layers of liqud

24. 3. Stable oscillations of particle-wave configurations

24.4. Schrödinger and Klein-Gordon equations

24.5. Strongly localised nonlinear sphere-like waves and wave packets

24.6. Wave trajectories, wave packets and discussion

Chapter 25. Nonlinear quantum waves in the light of recent slit experiments

25.1. Introduction

25. 2. Experiments using different kind of "slits" and the beginning of the discussion

25.3. Explanations and discussion of the experimental results

25.4. Casimir’s effect

25.5. Thin metal layer and plasmons as the synchronizators

25.6. Testing of thought experiments

25.7. Main thought experiment

25. 8. Resonant dynamics of particle-wave, vacuum and Universe

Chapter 26. Resonant models of origin of particles and the Universe due to quantum perturbations of scalar fields

26.1. Basic equation and relations

26.2. Basic solutions. Dynamic and quantum effects

26.3. Two-dimensional maps of landscapes of the field

26.4. Description of quantum perturbations

26.4.1. Quantum perturbations and free nonlinear oscillations in the potential well

26.4.2. Oscilations of scalar field, granular layer and the Bose-Einstein condensate

26.4.3. Simple model of the origin of the particles: mathematics and imaginations

26.5. Modelling of quantum actions: theory

26.6. Modelling of quantum actions: calculations

References

### 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 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 did 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.