Waves in Flows

Preface
Contents
1 A Priori Estimates from First Principles in Gas Dynamics
	1.1 Introduction
	1.2 Compensated Integrability
		1.2.1 Evolution Problems
			A Homogeneous Estimate
			Integrating in Time First
	1.3 Applications to Gas Dynamics (I): Euler Equations
		1.3.1 Euler Equations for a Compressible Inviscid Fluid
		1.3.2 Why Do We Care?
		1.3.3 Estimating the Velocity Field
		1.3.4 Flows in a Bounded Domain
		1.3.5 Relativistic Gas
		1.3.6 Flows with External Force
	1.4 Applications (II): Kinetic Models
		1.4.1 Boltzmann-Like Models
			The Cauchy Problem
			An Extra Estimate for Boltzmann-Like Equations
		1.4.2 What Should a Dissipative Model Be?
		1.4.3 Discrete Velocity Models
	1.5 Applications (III): Mean-Field Models
		1.5.1 Vlasov-Type Models
		1.5.2 The DPT of a Single Vlasov-Type Equation
		1.5.3 Genuine Plasmas
	1.6 Applications (IV): Molecular Dynamics
		1.6.1 Mass–Momentum Tensor of a Single Particle
		1.6.2 Long-Range Forces
		1.6.3 Hard Spheres
	References
2 Equatorial Wave–Current Interactions
	2.1 Introduction
	2.2 Preliminaries
	2.3 The Equatorial f-Plane Approximation
	2.4 The Equatorial β-Plane Approximation
	2.5 An Exact β-Plane Solution (Equatorially Trapped Wave)
		2.5.1 Analysis of the Equatorially Trapped Wave Motion
		2.5.2 Quantitative Aspects
	2.6 The Ocean Flow in the Equatorial Pacific
		2.6.1 The El Niño Phenomenon
		2.6.2 A Model for the Equatorial Currents
		2.6.3 Equatorial Wave–Current Interactions
		2.6.4 Linear Wave Theory
		2.6.5 Weakly Nonlinear Models
	References
3 Linear and Nonlinear Equatorial Waves in a Simple Modelof the Atmosphere
	3.1 Introduction
	3.2 Linear Equatorial Waves
	3.3 Weakly Nonlinear Long Equatorial Waves
		3.3.1 Long Linear Rossby and Kelvin Waves
		3.3.2 Nonlinear Slow Dynamics of Long Waves
	3.4 Equatorial Modons
	3.5 Equatorial Adjustment: Initial-Value Problem on the Equatorial Beta-Plane
	3.6 Brief Summary and Discussion
	References
4 The Water Wave Problem and Hamiltonian TransformationTheory
	4.1 Introduction
	4.2 Water Waves and Hamiltonian PDEs
		4.2.1 Physical Derivation of the Governing Equations
		4.2.2 General Notions on Hamiltonian Systems
		4.2.3 Examples of Hamiltonian PDEs
			Quasilinear Wave Equation
			Boussinesq System
			Korteweg–de Vries Equation
			Nonlinear Schrödinger Equation
		4.2.4 Zakharov's Hamiltonian for Water Waves
	4.3 Dirichlet–Neumann Operator and Its Analysis
		4.3.1 Legendre Transform
		4.3.2 Shape Derivative of H
		4.3.3 Invariants of Motion
		4.3.4 Taylor Expansion of G
	4.4 Birkhoff Normal Forms
		4.4.1 Significance of the Normal Form
		4.4.2 Complex Symplectic Coordinates and Poisson Brackets
		4.4.3 Resonances
		4.4.4 FormalTransformationTheoryandBirkhoffNormalForm
		4.4.5 Solving the Third-Order Cohomological Equation
		4.4.6 Normal Forms for Gravity Waves on Infinite Depth
			Third-Order Normal Form and Burgers' Equation
			Fourth-Order Normal Form
			Integrable Birkhoff Normal Form
	4.5 Model Equations for Water Waves
		4.5.1 Linearized Problem
		4.5.2 Non-dimensionalization
		4.5.3 Canonical Transformation Theory
		4.5.4 Calculus of Transformations
			Amplitude Scaling
			Spatial Scaling
			Surface Elevation-Velocity Coordinates
			Moving Reference Frame
			Characteristic Coordinates
		4.5.5 Boussinesq and KdV Scaling Limits
		4.5.6 Modulational Scaling Limit and the NLS Equation
			Normal Form Transformation
			Modulational Ansatz
			Expansion and Homogenization of Multiscale Functions
			NLS Equation
			Reconstruction of the Free Surface
	4.6 Initial Value Problems
		4.6.1 Local Well-Posedness
		4.6.2 Recent Results on Global Well-Posedness for Small Data
		4.6.3 Water Waves in a Periodic Geometry
	4.7 Numerical Simulation of Surface Gravity Waves
		4.7.1 Tanaka's Method for Solitary Waves
		4.7.2 High-Order Spectral Method
			Space Discretization
			Time Integration
		4.7.3 Collision of Solitary Waves
	References
5 Gravity Wave Propagation in Inhomogeneous Media
	5.1 Introduction
	5.2 Water Waves
		5.2.1 Propagation on Uneven Bottoms: First Order StokesWaves
		5.2.2 Second Order Stokes Waves
		5.2.3 Propagation in the Presence of Current or Through Porous Media
			Propagation in the Presence of Current
			Propagation Through Porous Media
	5.3 Wave Scattering: 2D Case
		5.3.1 Standing Wave in a Tank: Resonance and Sloshing
		5.3.2 Case of Smooth Bathymetries: Sinusoidal Beds
			Perturbation Method with Multiple-scale Expansion for Sinusoidal Beds of Finite Extend
			Mild-Slope and Modified Mild-Slope Equations
		5.3.3 Case of Abrupt Bathymetries
			General Expression of the Velocity Potentials
			Integral Matching Conditions Method
		5.3.4 Examples
			Sloping Beds
			Sinusoidal Beds
			Reflection Due to Structures
	5.4 Water Focusing: 3D Case
		5.4.1 Refraction—Snell—Descartes' Law
		5.4.2 Refraction-Diffraction
		5.4.3 Diffraction
			Analytic Solution: Semi-Infinite Dike
			Channels of Finite Width
		5.4.4 Examples
			Wave Scattering in the Presence of Underwater Mound
			Wave Scattering by Surface Piercing Structures
			Wave Scattering by Emerging Porous Media
	5.5 Application to Wave Energy Device
		5.5.1 Oscillating Water Column
		5.5.2 Pressure Oscillation
	References
6 Physical Models for Flow: Acoustic Interaction
	6.1 Introduction
	6.2 Fluid Dynamics
		6.2.1 Conservation Equations
			Conservation of Mass
			Conservation of Momentum
			Conservation of Energy
		6.2.2 Constitutive Equations
		6.2.3 Characterization of Flows by Dimensionless Numbers
		6.2.4 Vorticity
		6.2.5 Towards Acoustics
			Formulation for Scalar Potential
			Formulation for Vector Potential
	6.3 Acoustics
		6.3.1 Wave Equation
		6.3.2 Simple Solutions: d'Alembert
		6.3.3 Impulsive Sound Sources
		6.3.4 Free-Space Green's Functions
		6.3.5 Monopoles, Dipoles, and Quadrupoles
		6.3.6 Calculation of Acoustic Far Field
		6.3.7 Compactness
		6.3.8 Solution of Wave Equation Using Green's Function
	6.4 Aeroacoustics
		6.4.1 Lighthill's Acoustic Analogy
		6.4.2 Curle's Theory
		6.4.3 Vortex Sound
		6.4.4 Perturbation Equations
		6.4.5 Comparison of Different Formulations
		6.4.6 Acoustic Feedback Mechanisms
	6.5 Applications
		Coupling strategy of flow and acoustics
		Fluid dynamics
		Acoustics
		Conclusions of workflow
		6.5.1 Human Phonation
		6.5.2 Axial Fan
		6.5.3 Cavity at Low Mach Number schoder2020numerical
		6.5.4 Cavity at High Mach Number
	Appendix
	References