Contents

Cover

Title page

Copyright page

Preface

Introduction

Chapter 1: Equations of Hydrodynamics

1.1 Features of the Problems in the Formulation of Mathematical Physics
1.2 Classification of Linear Differential Equations with Partial Derivatives of the Second Order
1.3 Nonlinear Equations of Fluid Dynamics
1.4 Methods for Solving Nonlinear Equations
1.5 The Basic Laws of Hydrodynamics of an Ideal Fluid
1.6 Linear Equations of Hydrodynamic Waves
Conclusions

Chapter 2: Modeling of Wave Phenomena on the Shallow Water Surface

2.1 Waves on the Sea Surface
2.2 Review of Research on Surface Gravity Waves
2.3 Investigation of Surface Gravity Waves
2.4 Spatial modeling of Wave Phenomena on Shallow Water Surface
2.5 Actual Observations of Wave Phenomena on the Surface of Shallow Water
2.6 Ship Waves. “Reactive” Ducks of the Alexander Garden
Conclusions

Chapter 3: Modeling of Nonlinear Surface Gravity Waves in Shallow Water

3.1 Overview of Studies on Nonlinear Surface Gravity Waves in Shallow Water
3.2 Nonlinear Models of Surface Gravity Waves in Shallow Water
3.3 Solution of the Nonlinear Shallow Water Equation by the Method of Successive Approximations
3.4 Modeling the Propagation of Nonlinear Surface Gravity Waves in Shallow Water
3.5 Modeling the Refraction of Nonlinear Surface Gravity Waves
3.6 Modeling of Propagation and Refraction of Nonlinear Surface Gravity Waves Under Shallow Water Conditions with Account of Dispersion
Conclusions

Chapter 4: Numerical Simulation of Nonlinear Surface Gravity Waves in Shallow Water

4.1 Review of Studies on Computational Modeling of Surface Waves
4.2 Statement of the Problem
4.3 The Research of a Discrete Model
4.4 Results of Numerical Modeling based on Shallow Water Equations
4.5 Discussion and Comparison of Results
Conclusions

Chapter 5: Two-Dimensional Numerical Simulation of the Run-Up of Nonlinear Surface Gravity Waves

5.1 Statement of the Problem
5.2 Construction of a Discrete Finite-Volume Model
5.3 Discrete Model Research
5.4 Results of Two-Dimensional Numerical Modeling and Their Analysis
5.5 Discussion and Comparison of Results
Conclusions

Chapter 6: Three-Dimensional Numerical Modeling of the Runup of Nonlinear Surface Gravity Waves

6.1 Statement of the Problem. Boundary and Initial Conditions
6.2 Construction of a Discrete Model
6.3 The Construction of a Discrete Finite-Volume Model
6.4 Discrete Model Research
6.5 Results of Three-Dimensional Numerical Modeling and Their Analysis
6.6 Discussion and Comparison of Results
Conclusions

Conclusion

References

Index

End User License Agreement

List of Illustrations

Chapter 1

Figure 1.4.1 Convergent iterative processes.
Figure 1.4.2 The divergent iterative process.

Chapter 2

Figure 2.1.1 Transformation of the surface wave profile.
Figure 2.2.1 The trochoidal wave.
Figure 2.2.2 Stokes wave profile.
Figure 2.2.3 Trajectory of particle motion in wave.
Figure 2.3.1 Geometry of boundaries.
Figure 2.3.2 Dispersion ratio for shallow and deep water.
Figure 2.3.3 Dispersion dependence for gravitational-capillary waves.
Figure 2.4.1 Spatial model of the surface gravity wave at: f = 0.3Hz; λ = 33 m; H = 10 m; c = 9.9 m/s.
Figure 2.4.2 Spatial model of superposition of two surface gravity waves at: f = 0.3 Hz; λ = 33 m; H = 10 m; c = 9.9 m/s, θ = 0⁰ and 45⁰.
Figure 2.4.3 Spatial model of a packet of surface gravity waves at: f = 0,3; 0,33 Hz; λ = 33; 30 m; H = 10 m; c = 9.9 m/s, θ = 45⁰.
Figure 2.4.4 Spatial model of superposition of two packets of surface gravity waves at: f = 0.3; 0.33 Hz; λ = 33; 30 m; H = 10 m; c = 9.9 m/s, θ = 90⁰ and 45⁰.
Figure 2.4.5 Spatial model of the cape (250 × 250 × 10 m) according to the analytical expression z(x,y) = 0,0002xy × sin(0,00003xy).
Figure 2.4.6 The spatial model of the shore protrusion (250 × 250 × 10 m) according to the analytical expression z(x,y) = 0,0002yx². (250 × 250 × 10 m)
Figure 2.4.7 Spatial model of the bay (250 × 250 × 10 m) by analytical expression z(x,y) = 0,00017x² × (cos(0,05y) + 3)).
Figure 2.4.8 Spatial model of refraction of a packet of surface gravity waves on the cape at: f = 0.3; 0.33 Hz; λ = 33; 30 m; H = 12 m; c = 11 m/s, θ = 0⁰.
Figure 2.4.9 Spatial model of refraction of a packet of surface gravity waves on a ledge at: f = 0.3; 0.33 Hz; λ = 33; 30 m; H = 12 m; c = 11 m/s, θ = 45⁰.
Figure 2.4.10 Spatial model of refraction of a packet of surface gravity waves in a bay at: f = 0.3; 0.33 Hz; λ = 33; 30 m; H = 12 m; c = 11 m/s, θ = 90⁰.
Figure 2.4.11 Spatial model of refraction of two packets of surface gravity waves on the cape at: f = 0.3; 0.33, (0,4; 0,44) Hz; λ = 33; 30, (25, 22) m; H = 12 m; c = 11 m/s, θ = 90⁰ и 85⁰.
Figure 2.4.12 Spatial model of refraction of two packets of surface gravity waves on the cape at: f = 0.3; 0.33, (0.4; 0.44) Hz; λ = 33; 30, (25, 22) m; H = 12 m; c = 11 m/s, θ = 90⁰ и 60⁰.
Figure 2.4.13 Spatial model of refraction of three surface gravity waves on the cape at: f = 0.3; 0.33; 0.4 Hz; λ = 33; 30; 25 m; H = 12 m; c = 11 m/s, θ = 90⁰, 60⁰ and 30⁰.
Figure 2.4.14 Spatial model of refraction of three surface gravity waves in a bay at: f = 0.3; 0.33; 0.4 Hz; λ = 33; 30; 25 m; H = 12 m; c = 11 m/s, θ = 90⁰, 85⁰ and 80⁰.
Figure 2.4.15 Spatial model of refraction of three surface gravity waves on the ledge at: f = 0.3; 0.33; 0.4 Hz; λ = 33; 30; 25 m; H = 12 m; c = 11 m/s, θ = 90⁰, 45⁰ and 70⁰.
Figure 2.5.1 The wave structure on the water surface, formed from surface waves with an infinitesimal amplitude (Dyurso village, the Black Sea).
Figure 2.5.2 Periodic wave cells formed by the superposition of surface waves of different lengths, small ripples on the swell (Dyurso village).
Figure 2.5.3 Attenuation of a surface wave with an infinitesimal amplitude on the shore (Taganrog Bay, Azov Sea).
Figure 2.5.4 Refraction of surface waves in the bay conditions (Dyurso village).
Figure 2.5.5 Dispersal of wind waves along the coastline, oblique surf (Dyurso village).
Figure 2.6.1 Wedge-shaped wave tracks from floating ducks in the pond of the Alexander Garden (St. Petersburg).
Figure 2.6.2 Ship Waves.
Figure 2.6.3 Construction of wave elements.
Figure 2.6.4 The envelope of perturbations arising in consecutive moments.
Figure 2.6.5 Divergent waves from the swimmer. Speed of swimmer is less than the speed of divergent waves (Taganrog Bay).

Chapter 3

Figure 3.1.1 Tidal wave profile.
Figure 3.1.2 Stages of waves from the open seas to the shore.
Figure 3.2.1 Cnoidal waves.
Figure 3.4.1 The main characteristics of a surface gravity wave with parameters: f = 0.2 Hz; λ = 35 m; c = 7 m/s; H = 5 m; 2a/λ = 0.024; kH = 0.89; ε = 0.083; γ = 0.02.
Figure 3.4.2 Accumulation of amplitudes of secondary waves of a surface gravity wave with parameters: f = 0.2 Hz; λ = 35 m; H = 5 m; 2a/λ = 0.024; kH = 0.89; ε = 0.083; γ = 0.02.
Figure 3.4.3 Main characteristics of the surface gravity waves with parameters: f = 0.09 Hz; λ = 77.8 m; c = 7 m/s; H = 5 m; 2a/λ = 0.014; kH = 0.4; ε = 0.107; γ = 0.004.
Figure 3.4.4 Main characteristics of the surface gravity waves with parameters: f = 0.09 Hz; λ = 77.8 m; c = 7 m/s; H = 5 m; 2a/λ = 0.015; kH = 0.4; ε = 0.118; γ = 0.004.
Figure 3.4.5 Spatial model of the propagation of a surface gravity wave with parameters: f = 0.09 Hz; λ = 77.8 m; H = 5 m; 2a/λ = 0.014; ε = 0.107; γ = 0.004.
Figure 3.4.6 Spatial model of the propagation of a surface gravity wave with parameters: f = 0.045 Hz; λ = 155.6 m; H = 5 m; 2a/λ = 0.0088; ε = 0.136; γ = 0.001.
Figure 3.4.7 The influence of the fourth harmonic on the profile of the surface gravitational wave: (a) lags behind the main wave by π/2; (b) in-phase subtracted; (c) ahead of the main wave by π/2; (d) – is in-phase summed. Wave parameters: f = 0.09 Hz; λ = 77.8 m; c = 7 m/s; H = 5 m; 2a/λ = 0.014; kH = 0.4; ε = 0.107.
Figure 3.4.8 The steepening of the leading edge of the wave crest and the formation of a diving breaker (ejection of a jet from the crest with the entrapment of air) (Durso village).
Figure 3.4.9 The breaking of the crest of a wave of swell on the shore (Durso village).
Figure 3.5.1 Bottom lines according to analytic expressions: s1(x) = 0.0006x × sin(0.0001x); s2(x) = 2x × J(5, 0.0001x).
Figure 3.5.2 Geometry of the problem related to the propagation of a surface gravity wave along a sloping bottom in conditions of a bay.
Figure 3.5.3 The main characteristics of a surface gravity wave with initial parameters: f = 0.09 Hz; λ = 77.8 m; c = 7 m/s; 2a/λ = 0.014; kH = 0.4; ε = 0.107; γ = 0.004, bottom line based on the Bessel function.
Figure 3.5.4 The main characteristics of a surface gravity wave with initial parameters: f = 0.09 Hz; λ = 77.8 m; c = 7 m/s; 2a/λ = 0.014; kH = 0.4; ε = 0.107; γ = 0.004, bottom line based on the sine function.
Figure 3.5.5 Spatial model of the cape (10 km × 10 km × 25 m) by analytic expression z(x,y) = 0.0024(xy)^1/2×sin(2 × 10⁻⁸ xy).
Figure 3.5.6 Spatial model of the bay (10 km × 10 km × 15 m) by analytic expression z(x,y) = 3 × 10⁻⁸y²cos(0,0014x) + (3.4 × 10⁻⁴y)².
Figure 3.5.7 Three-dimensional model of refraction to the cape of a nonlinear surface gravity wave with initial parameters: f = 0.045 Hz; λ = 155.6 m; H = 5 m; 2a/λ = 0.0076; θ = 0⁰.
Figure 3.5.8 Three-dimensional model of refraction in the bay of a nonlinear surface gravity wave with initial parameters: f = 0.045 Hz; λ = 155.6 m; H = 5 m; 2a/λ = 0.0076; θ = 0⁰.
Figure 3.5.9 Three-dimensional model of refraction on the flat coastline of a nonlinear surface gravity wave with initial parameters: f = 0.045 Hz; λ = 155.6 m; H = 5 m; 2a/λ = 0.0076; θ = 45⁰.
Figure 3.5.10 The fragment of the three-dimensional refraction model on the cape of a nonlinear surface gravity wave with the approach angle θ = 45⁰.
Figure 3.6.1 The main characteristics of a surface gravitational wave with allowance for variance: f = 0.09 Hz; λ = 77.8 m; 2a/λ = 0.014; kH = 0.4; ε = 0.107; γ = 0.004.
Figure 3.6.2 Surface gravity wave profiles without taking into account (a) and taking into account (b) the dispersion: f = 0.2 Hz; λ = 35 m; 2a/λ = 0.024; kH = 0.9; ε = 0.08; γ = 0.02.
Figure 3.6.3 Surface gravity wave profiles without taking into account (a) and taking into account (b) the dispersion: f = 0.09 Hz; λ = 77.8 m; 2a/λ = 0.014; kH = 0.4; ε = 0.107; γ = 0.004
Figure 3.6.4 Surface gravity wave profiles without taking into account (a) and taking into account (b) the dispersion: f = 0.045 Hz; λ = 155.6 m; 2a/λ = 0.0088; kH = 0.2; ε = 0.136; γ = 0.001.
Figure 3.6.5 Spatial model of the propagation of a surface gravity wave with parameters: f = 0.09 Hz; λ = 77.8 m; H = 5 m; 2a/λ = 0.014; ε = 0.107; γ = 0.004.
Figure 3.6.6 Refraction profiles of the surface gravity wave without taking into account and taking into account the dispersion, initial parameters: f = 0.09 Hz; λ = 77.8 m; 2a/λ = 0.014; kH = 0.4; ε = 0.107; γ = 0.004.
Figure 3.6.7 Three-dimensional model of refraction to the cape of a nonlinear surface gravity wave with allowance for dispersion, initial parameters: f = 0.045 Hz; λ = 155.6 m; H = 5 m; 2a/λ = 0.0076; θ = 0⁰.

Chapter 4

Figure 4.1.1 Stages of breaking of surface waves in the conditions of the bay (Dyurso village, the Black Sea).
Figure 4.1.2 Stages of breaking of surface waves in the conditions of the bay (Dyurso village, the Black Sea).
Figure 4.4.1 Map of the depths of the Azov Sea.
Figure 4.4.2 Dependence of the surface gravity wave profile on the initial steepness: f = 0.045Hz; l = 155.6m; H = 5m; c = 7m/s; kH = 0.2; (a) 2a/l = 0.0024; ε = 0.04; (b) 2a/l = 0.0039; ε = 0.06; (c) 2a/l = 0.006; ε = 0.09.
Figure 4.4.3 Dependence of the velocity of particles and the function of the elevation of the surface wave on the depth: f = 0.045Hz; (a) λ = 120.5m; H = 3m; c = 5.4m/s; kH = 0.16; 2a/λ = 0.003; ε = 0.06; (b) λ = 98.4m; H = 2m; c = 4.4m/s; kH = 0.13;.2a/λ = 0.0037; ε = 0.09; (c) λ = 69.6M;. H = 1m; c = 3.1m/s; kH = 0.09; 2a/λ = 0.005; ε = 0.18; (d) λ = 49.2m; H = 0,5m; c = 2.2m/s; kH = 0.064; 2a/λ = 0.007; ε = 0.36.
Figure 4.4.4 Dependence of the surface gravity wave profile on depth: f= 0.09Hz; (a) λ = 34.8m; c = 3.1m/s; H = 1m; 2a/λ = 0.008; kH = 0.18; ε = 0.14; (b) λ = 24.6m; c = 2.2m/s; H = 0.5m; 2a/λ = 0.01; kH = 0.13; ε = 0.2.
Figure 4.4.5 Dependence of the surface gravity wave profile on depth: f = 0.0225Hz; (a) l = 196.8m; c = 4.4m/s; H = 2m; 2a/l = 0.0015; kH = 0.06; ε = 0.08; (b) l = 139.2m; c = 3.13m/s; H = 1m; 2a/l = 0.008; kH = 0.04; ε = 0.12.
Figure 4.4.6 Lines of the bottom according to analytic expressions: where n is the size of the grid:
Figure 4.4.7 Transformation of the profile of the surface gravity wave as it approaches the shore, initial wave parameters: f = 0.045Hz; l = 155.6m; H = 5m; c = 7m/s; kH = 0.2; 2a/l = 0.0023; ε = 0.036, the bottom rises linearly; mesh size n = 4000.
Figure 4.4.8 Transformation of the profile of the surface gravity wave as it approaches the shore, initial wave parameters: f = 0.045Hz; λ = 120.5m; H = 3m; c = 5.4m/s; kH = 0.16; 2a/λ = 0.003; ε = 0.06; (a) steepened bottom line - s4(x); (b) flat bottom line - s3(x).
Figure 4.5.1 Gradual distortion of the profile of the surface gravity wave (a): ε = 0.75; layer t/T = 0.3– nonlinear dispersion model [Shimozono, Sato, 2009]; (b): f = 0.045Hz; λ = 49.2m; H = 0.5m; c = 2.2m/s; ε = 0.4; t = 39.9s – model of nonlinear shallow water equations.
Figure 4.5.2 Calculated and experimental surface wave profiles (shore on the left): (2a/λ = 0.03, h/λ = 0.2), (a) at the beginning of the wave breaker (x/λ = 0,1); (b) above the wavebreaker (x/λ = 0,3); (c) above the wavebreaker (x/λ = 0,4); d) behind the breakwater (x/λ = 0,7).
Figure 4.5.3 Results of numerical calculation of surface wave profiles with different initial steepness values: (a) f = 0.0225Hz; l = 139.2m; H = 1m; 2a/l = 0.008; ε = 0.12; (b) f = 0.045 Hz; l = 155.6m; H = 5m; 2a/l = 0.006; ε = 0.09.
Figure 4.5.4 Surface wave profiles – numerical calculation.
Figure 4.5.5 Surface wave profiles are the semigraphical method, the main parameters of the surface gravity wave: f = 0.045Hz; l = 155.6m; H = 5m; c = 7m/s; kH = 0.2; 2a/l = 0.006; ε = 0.1.

Chapter 5

Figure 5.1.1 Geometry of the problem.
Figure 5.2.1 Location of the cell with adjacent nodes.
Figure 5.2.2 Location of nodes relative to cells.
Figure 5.4.1 Windows for constructing a bottom profile and visualizing the pressure field.
Figure 5.4.2 Lines of the bottom according to analytic expressions: s1(x) = ax + bx⁸; s2(x) = ax + bx^2– cx³;
Figure 5.4.3 Dynamics of changes in the profile of incident surface gravity wave on the coastal slope (gentle rise), the initial parameters of the wave: f = 0.39Hz; λ = 10m; c = 4m/s; H = 5m; a = 0.5m; kH = 3.14; ε = 0.1; time (from top to bottom) t = 1.6s; t = 3.6s; t = 6.8s.
Figure 5.4.4 A run-up on the coastal slope of surface gravity waves of different amplitude, the initial wave parameters: f = 0.39Hz; λ = 10m; c = 4m/s; H = 5m; kH = 3.14; time (from top to bottom) 1) t = 7s; a = 0.25m; ε = 0.05; 2) t = 6.8s; a = 0.5m; ε = 0.1; 3) t = 6.9s; a = 1m; ε = 0.2.
Figure 5.4.5 The run-up on the coastal slope of surface gravity waves of different lengths, the initial wave parameters: H = 5m; a = 0.5m; ε = 0.1; (from top to bottom),
Figure 5.4.6 A run-up on the coastal slopes of different depths of the surface gravity wave, the initial wave parameters, from top to bottom: 1) f = 0.28Hz; λ = 15m; c = 4.3m/s; H = 2.5m; a = 0.5m; kH = 1; ε = 0.4; t = 8s; 2) f = 0.32Hz; λ = 15m; c = 4.8m/s; H = 5m; a = 0.5m; kH = 2.1; ε = 0.1; t = 8s.
Figure 5.4.7 Dynamics of the change in the profile of the surface gravity wave incident on the coastal slope (linear rise), the initial parameters of the wave: f = 0.39Hz; λ = 10m; c = 4m/s; H = 5m; a = 0.5m; kH = 3.14; ε = 0.1; time (from top to bottom) t = 1.4s; t = 4.3s; t = 8.9s.
Figure 5.4.8 Dynamics of changes in the profile of a surface gravity wave that hits the coastal slope (steep rise), initial wave parameters: f = 0.39Hz; λ = 10m; c = 4m/s; H = 5m; a = 0.5m; kH = 3.14; ε = 0.1; time (from top to bottom) t = 2.9s; t = 5.1s; t = 6.2s.
Figure 5.4.9 The successive stages of superimposing a gravity wave incident and reflected from a steep coastal slope, the initial parameters of the wave: f = 0.32Hz; λ = 15m; c = 4.8m/s; H = 5m; a = 1m; kH = 2.1; ε = 0.2; time (from top to bottom) t = 5.1s; t = 7.6s; t = 8.3s.
Figure 5.4.10 Dynamics of changes in the profile of a surface gravity wave that runs into the coastal slopes with different steepness, initial wave parameters: f = 0.39Hz; λ = 10m; c = 4m/s; H = 5m; a = 0.5m; kH = 3.14; ε = 0.1; (from top to bottom) 1) steep rise t = 6.2s; 2) linear rise t = 7.3s; 3) loping rise t = 6.8s.
Figure 5.4.11 The dynamics of the change in the profile of the incident surface gravity wave and the complete flooding of the long shallow shore, the initial parameters of the wave: f = 0.28Hz; λ = 15m; c = 4.3m/s; H = 2.5m; a = 0.25m; kH = 1; ε = 0.2; time (from top to bottom) t = 2.1s; t = 5.6s; t = 18.6s.
Figure 5.4.12 Run-up of the surface gravity wave on the coastal slope with initial parameters: λ = 10m; c = 4m/s; H = 5m; a = 0.5m; kH = 3.14; ε = 0.1; t = 6.5s (grid with a larger step size h_x = h_z = 0.1m).
Figure 5.4.13 Run-up of surface gravity wave on the coastal slope with initial parameters: λ = 10m; c = 4m/s; H = 6.5m; a = 0.5m; kH = 4; ε = 0.07; t = 5.6s (grid step h_x = h_z = 0.047m, size of area 30x12.7m).
Figure 5.5.1 The comparison of run-up surface gravity wave on a flat coastal slope on the basis of different models: (a) λ = 5m; H = 0.5m; T = 2.0s; a = 0.15m; t = 12s [Kawasaki et al., 2010]; (b) λ = 3m; H = 0.2m; T = 5.0s; a = 0.128m [Zhao et al., 2004]; (c) λ = 10m; H = 2m; a = 1m; c = 3.6m/s; kH = 1.3; ε = 0.5; t = 5.4s (investigated model).
Figure 5.5.2 The experimental setup (a) and stage-by-stage distortions of the profile of the breaking surface wave propagating along the sloping bottom (b), the initial wave parameters: λ = 2.5m; H = 0.735m; T = 1.3s; a = 0.07m; distance of breaking 12.375m [Kimmoun, Branger, 2007].

Chapter 6

Figure 6.1.1 Geometry of the three-dimensional run-up of a surface gravity wave.
Figure 6.3.1 Geometry of a three-dimensional cell
Figure 6.7.2 Schematic view of a revetment.
Figure 6.7.3 Three-dimensional grid of a still shallow water basin.
Figure 6.7.4 Consecutive stages of the run-up of a surface gravity wave with initial parameters: f= 0.13Hz; λ = 50m; c = 6.6m/s; H = 5m; a = 4.2m; kH = 0.6; ε = 0.8; at various times (a) t = 3.7s; (b) t = 4.3s; (c) t = 6.3s.
Figure 6.7.5 Consecutive stages of the run-up of a surface gravity wave with initial parameters: f = 0.13Hz; λ = 50m; c = 6.6m/s; H = 5m; a = 4.2m; kH = 0.6; ε = 0.8; at various times (a) t = 7.7s; (b) t = 9.9s; (c) t = 10.8s.
Figure 6.7.6 Consecutive stages of the run-up of a surface gravity wave with initial of shallow water parameters: (a) f = 0.25Hz; λ = 25m; c = 6.2m/s; H = 8.4m; a = 2.5m; kH = 2.1; ε = 0.3; t = 8.8s; (b) f = 0.12Hz; λ = 75m; c = 8.9m/s; H = 10m; a = 2.5m; kH = 0.8; ε = 0.2; t = 5.7s.
Figure 6.7.7 Consecutive stages of the run-up of a surface gravity wave with initial parameters: f = 0.12Hz; λ = 60m; c = 6.7m/s; H = 5m; a = 5.5m; kH = 0.5; ε = 1.1; at various times t: (a) 3.1s; (b) 4.8s; (c) 6.9s.
Figure 6.7.8 Consecutive stages of the run-up of a surface gravity wave with initial parameters: f = 0.12Hz; λ = 60m; c = 6.7m/s; H = 5m; a = 5.5m; kH = 0.5; ε = 1.1; at various times t: (d) 9.7s; (e) 12.7c; (f) 20.9s.
Figure 6.7.9 Indication (color scale) of the water mass pressure inside the basin for the run-up of a surface gravity wave with f = 0.12Hz; λ = 60m; c = 6.7m/s; H = 5m; a = 5.5m; kH = 0.5; ε = 1.1; t = 16.4s.
Figure 6.8.1 Comparison of the process of run-up of a surface gravity wave on the basis of different models: (a) λ = 2.5m; H = 0.7m; T = 1.3s; a = 0.07m [Lubin et al., 2006]; (b) A = 25m; H = 6.7m; c = 7.5m/s; a = 4.2m; t = 4.7c (developed model).

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Preface

How mesmerizing is the beauty of the waves approaching the seashore against a background of the sunset: they try to catch up with each other in a continuous cycle of water flow, then they subside, then intensify, rolling up on the shore, crashing into a sparkling foam, creating an endless symphony of surf. You can endlessly admire this landscape, which has existed for billions of years, from the time when there were no living beings on the planet Earth. Also, primeval ocean waves wash ashore, as is happening now in the presence of a person watching this picture. These waves have attracted the attention of artists and researchers for more than a century. Despite their beauty and simplicity, however, they are not always easy to describe. Moreover, to verify the plausibility of the created model, special knowledge is not necessarily required. It’s enough to go to the beach, and everything will become clear.

At the same time, neglecting the power of this beauty can lead to devastating consequences in storm surges and earthquakes. Therefore, the study of waves on the sea surface is not an easy task, and attempts are made in this work to describe and simulate some wave events on the surface of the aquatic environment. By their nature, these waves are inherently nonlinear, although some approximations may be considered linear. Consequently, the most appropriate theory of surface wave description is nonlinear theory.

This book presents the work done by the author for the research and modeling of nonlinear wave activities on the shallow water surface. An attempt was made to describe the run-up of surface waves to various coastal formations in shallow waters. Photographic illustrations of wave activities on the shallow water surface, made by the author, are also provided to illustrate the work.

I want to express my appreciation to my teachers, and promote a love for mathematics, art, and beauty.

Iftikhar B. Abbasov

Introduction

In the context of the study of the ecosystems of the shallow coastal areas of the world’s oceans, physical phenomena occurring on the surface of the aquatic environment play an important role. These phenomena, like all natural phenomena, are complex and nonlinear. Therefore, this leads to the nonlinear mathematical models of the actual processes.

The theory of wave motion fluids is a classical section of hydrodynamics and has a three-hundred-year history. The interest in wave activities on the surface of the fluid could be explained by the prevalence and accessibility of this physical phenomenon. Despite a great deal of research, the theory of wave fluid movements is still incomplete.

Of great importance is the matter of researching and modeling the wave activities at shallow water and the impact of surface gravity waves to coast formations and hydrotechnical structures. Therefore, the question of 3D modeling of the distribution, run-up and refraction of nonlinear surface waves can play an important role in monitoring and forecasting the sustainable development of the ecosystems of these areas.

The results of the research and numerical modeling of the dynamic of nonlinear surface gravity waves at shallow water are introduced in this work. Corresponding equations of mathematical physics and methods of mathematical modeling are used for describing and modeling.

Analytical descriptions of these nonlinear wave activities often use different modifications of the shallow water equations. For the numerical modeling, shallow water equations are also used in a 1D case. 2D and 3D numerical modeling of nonlinear surface gravity waves to beach approaches are based on Navier-Stokes equations. Navier-Stokes equations allow for both nonlinear effects and turbulent processes to be considered in the incompressible fluid.

Therefore, appropriate nonlinear waves of hydrodynamic equations will be used to adequately model nonlinear wave activities in shallow water conditions.