Wave Dynamics

Contents
Preface
About the Editors
1. Numerical Aspects of Nonlinear Wave–Wave Interactions in Operational-Wave Models
	1. Introduction: Discrete Spectral-Wave Models
	2. Mathematical Formulation: Nonlinear Wave–Wave Interactions Using Webb–Resio–Tracy Method
	3. Multi-Resolution Analysis and Haar Wavelets
		3.1. Multi-resolution Analysis or Multi-level Representation of Function
		3.2. Numerical Integration Using Haar Wavelets and Their Application to Transfer Integral
		3.3. Procedure for Approximating the Locus Curve Using Haar Wavelets
		3.4. Results of Haar-Wavelets Approximation to WRT (HWRT) Approximation on Locus Curves
	4. Computational Results Using HWRT to Transfer Integral
	References
2. Impact of Tropical Cyclones on the Ocean Surface Waves Over the Bay of Bengal
	1. Introduction
	2. Data and Methods
	3. Results and Discussion
		3.1. Climatology
		3.2. TC-Induced Significant Wave Height Variations
	4. Conclusions
	References
3. Theoretical and Numerical Studies of Boussinesq Equations for Onshore Shallow-Water Wave Propagation
	1. Introduction
	2. Mathematical Formulation
		2.1. Derivation of Nonlinear Boussinesq Equations (NBEs)
	3. Dispersion Characteristics
		3.1. Linear Analysis
		3.2. Nonlinear Analysis
	4. Analytical Solutions
		4.1. Analytical Solution of Nonlinear 1-D BEs
		4.2. Analytical Solution of Nonlinear 2-D BEs
	5. Numerical Solutions of 1-D and 2-D BEs
		5.1. Discretization Form of Nonlinear 1-D BEs
		5.2. Discretization Form of Nonlinear 2-D BEs
			5.3. 1-D Boundary Conditions
			5.4. 2-D Boundary Conditions
	6. Results
		6.1. Validation Results
		6.2. Simulation Results
	7. Conclusion
	References
4. Differential Quadrature-based Advanced Numerical Scheme for Simulation of Solitary Motion of Shallow Water Waves
	1. Introduction
	2. Mathematical Model of Solitary Wave
	3. Numerical Framework
		3.1. Differential Quadrature Method
		3.2. DQM-based Numerical Structure of the KdV Equation
	4. Numerical Examples
		4.1. Single Soliton
		4.2. Interaction of Two Solitons
		4.3. Interaction of Three Solitons
		4.4. Single Soliton with Imprecise Parameters
	5. Conclusions
	References
5. Seismic Waves and Their Effect on Structures
	1. Introduction
	2. Theory of Elasticity in 3D
	3. Derivation of Seismic Wave Equation Using Basic Principles of Physics
		3.1. Equation of Equilibrium
		3.2. Wave Equation in a Source-free Region
		3.3. Wave Equation in Vector Form
	4. Solution for Wave Equation
		4.1. D’ Alembert’s Solution for Plane Wave
		4.2. Wentzel, Kramer, Brioullion, and Jeffrey’s (WKBJ) Solution of Wave Equation for Heterogeneous Medium
	5. Derivation of Raypath Geometry from Eikonal Equation
	6. Derivation of Relation Between Travel Time and Epicentral Distance for Source at Surface When Velocity Increases with Depth
	7. Interpretation of Snell’s Law Using Simple Ray Geometry
	8. Travel Time Equations
		8.1. Travel Time Equations in a Layered Earth
		8.2. Travel Time Equations in a Sphere
		8.3. Ray Parameter for a Spherical Earth
		8.4. Travel Time Equation for a Spherical Body where Velocity Changes Continuously with Depth
	9. Amplitude of Seismic Wave
		9.1. Amplitude of Seismic Wave Recorded at Surface for a Medium with Continuous Velocity Variation
		9.2. Energy Partitioning Caused by Presence of an Interface Across where Velocity Changes Abruptly
		9.3. Reflection and Transmission Coefficients for Solid–Solid Boundary for SH Wave
		9.4. Reflection and Transmission Coefficients for Solid–Solid Boundary for Incident P Wave
	References
6. Mathematical Study of Reflection and Transmission Phenomenon of Plane Waves at the Interface of Two Dissimilar Initially Stressed Rotating Micro-Mechanically Modeled Piezoelectric Fiber-Reinforced Composite Half-spaces
	1. Introduction
	2. The Micro-Mechanics Modeling of PFRC
	3. Basic Constitutive Equations
	4. Formulation and Geometry
	5. Relevant Boundary Conditions
		5.1. Mechanical Conditions
		5.2. Electrical Conditions
	6. Mathematical Solution
	7. Energy Ratios
	8. Special Cases of Interest
		8.1. Normal Incidence of qP Wave
		8.2. Grazing Incidence of qP Wave
	9. Numerical Computations and Discussion
		9.1. Energy Ratio of Reflected qP Wave (|ER1|)
		9.2. Energy Ratio of Reflected qSV Wave (|ER2|)
		9.3. Energy Ratio of Reflected EA Wave (|ER3|)
		9.4. Energy Ratio of Transmitted qP Wave (|ET1|)
		9.5. Energy Ratio of Transmitted qSV Wave (|ET2|)
		9.6. Energy Ratio of Transmitted EA Wave (|ET3|)
		9.7. Interaction Energy Ratio (|EINT|)
		9.8. Net Energy Ratio (|E|)
	10. Conclusions
	References
	Appendix A. Mathematical Expressions
7. Analyzing Shocks and Traveling Waves in Non-Newtonian Viscoelastic Fluids
	1. Introduction
	2. Shock Structure in Non-Newtonian Viscoelastic Fluids
	3. Shock Waves in Viscoelastic Burgers’ Equation
	4. Shock-Wave Solutions of Viscoelastic Fluids of Differential Type
	References
8. Wave Propagation through Resonators, Resonators in Series and Multi-Resonator
	1. Introduction
	2. Design of Resonators
	3. Performance of Resonators
		3.1. Transmission Loss
		3.2. Effective Metamaterial Properties
	4. Conclusion
	References
9. Solution of Interval-Modified Kawahara Differential Equations Using Homotopy Perturbation Transform Method
	1. Introduction
	2. Homotopy Perturbation Transform Method
	3. Solution of Interval-Modified Kawahara Equation
	4. Solution of Interval-Modified Kawahara Equation
	5. Numerical Results
	6. Conclusion
	References
10. Natural Convection of Sodium Alginate and Copper Nanofluid Inside Parallel Plates with Uncertain Parametric Behavior
	1. Introduction
	2. Natural Convection of Nanofluid Inside Parallel Plates
	3. Interval Homotopy Perturbation Method (IHPM)
	4. Solution of Natural Convection of Sodium Alginate and Copper Nanofluid Inside Parallel Plates using Percentage-Parametric HPM
	5. Concluding Remarks
	Nomenclature
	References
11. Mathematical Modeling of Radon-Transport Mechanism with Imprecise Parameters
	1. Introduction
	2. Interval-Midpoint Approach
	3. Interval Bounds to Radon Transport in Soil Chamber
		3.1. Radon Buildup in a Soil Chamber
		3.2. Mathematical Modeling by Interval-Midpoint Approach
		3.3. Experimental Results and Discussion
	4. Radon-Transport Mechanism with Imprecise Parameters
		4.1. Modeling Radon Transport with Imprecise Parameters
	5. Numerical Results and Discussion
	6. Conclusion
	Acknowledgment
	References
12. An Efficient Numerical Scheme for Time-Fractional Coupled Shallow-Water Equations with a Non-Singular Fractional Derivative
	1. Introduction
	2. Preliminaries
	3. Implementation of the MHATM to Time-Fractional SWEs
	4. Convergence Analysis of MHATM Solution
	5. Numerical Results and Discussions
	6. Conclusion
	References
13. Solution of Fractional Wave Equation by Homotopy Perturbation Method
	1. Introduction
	2. Preliminaries
	3. Homotopy Perturbation Method for One-Dimensional TFWE
	4. Numerical Implementation
	5. Homotopy Perturbation Method for Two-Dimensional TFWE
	6. Numerical Implementation
	7. Conclusion
	References
Index