The Dynamical Projectors Method

Hydro and Electrodynamics

Sergey Leble, Anna Perelomova

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9780367849573
• eBook ISBN: 9781351107976
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

The dynamical projectors method proves to reduce a multicomponent problem to the simplest one-component problem with its solution determined by specific initial or boundary conditions. Its universality and application in many different physical problems make it particularly useful in hydrodynamics, electrodynamics, plasma physics, and boundary layer problems. A great variety of underlying mechanisms are included making this book useful for those working in wave theory, hydrodynamics, electromagnetism, and applications.

"The authors developed a universal and elegant tool – dynamical projector method. Using this method for very complicated hydro-thermodynamic and electrodynamics problem settings, they were able to get a lot of interesting analytical results in areas where before often just numerical methods were applicable."

—L. A. Bordag, University of Applied Sciences Zittau/Görlitz, Zittau, Germany

"The book is intended for professionals working in various fields of linear and nonlinear mathematical physics, partial differential equations and theoretical physics. The book is written clearly, and in my opinion, its material will be useful and easy to understand for professionals and for students familiar with ordinary and partial differential equations."

—Sergey Dobrokhotov, Russian Academy of Sciences, Moscow, Russia

1 Introduction

This book discusses the results published in papers devoted to the technique of derivation of model evolution equations in terms of fundamental systems such as hydrodynamics and electrodynamics. It also naturally includes initial or boundary conditions that fix a type of a medium perturbation. Such universal approach is absent in monograph or textbook literature. The contents of the book provide readers with a practical tool that allows them to divide a space of solutions into subspaces with specific behavior marked by such terms as direction of wave propagation, frequency range, and, that is important to stress, characteristics of a medium. The characteristics are expressed in terms of standard parameters of a matter, such as permeability in case of electrodynamics or dissipation terms (equilibrium state and attenuation) in hydrodynamics.

One such way is the introduction of a combinations of field variables, named as modes, which correspond to their specific behavior. The simplest example that appears already in one-dimensional (1D), or planar, dynamics may be described as unidirectional waves; that is, waves with a fixed direction of propagation. In the case of celebrated wave equation (the string equation in 1D) we have the d’Alembert formula that gives the general solution of the initial (Cauchy) problem, with the natural inclusion of the directed waves. A choice of initial condition that corresponds to a wave, propagating, say, rightwards, is readily realized very easily with the help of a technique discussed in this book, namely the dynamic projecting. This technique splits the initial or boundary conditions space to the modal subspaces of the directed waves. Readings of dispersion and/or dissipation (terms with higher derivatives) are automatically taken into account within the algorithmic formalism, described already in the example of introduction and with general detalization in Chapter 2. Account of nonlinearity leads to interaction of the modes that are also automatically specified. The simple case of nonlinearly directed wave leads to the celebrated Korteweg-de Vries and nonlinear Schrödinger equations.

The general technique is universal, the steps are quite similar in both huge fields of possible applications, in hydro- and electrodynamics. A transition to two- and three-dimensional (3D) problems complicates calculations, but lies in the same road, and may be either realized mechanically or one can effectively include the symbolic computation programs because all the steps are written analytically. A good chance to demonstrate a power of projecting the method relates to a case of atmosphere-ocean waves physics, whose exponential equilibrium stratification admits, however, a transition to equations with constant coefficients and separate directed acoustic and internal waves.

An account of combined dissipation and nonlinearity in 3D hydrodynamics leads to effective description of such important phenomena as heating and streaming. It is a direct excitation of so-called entropy and rotation modes (non-wave or near-zero frequency ranges). The application of the dynamic projecting technique allows to use both periodic and nonperiodic acoustic sources for control of such processes. The impulse excitation is also included! Cumulatively, the combined dispersion and nonlinearity allows to effectively describe electromagnetic pulses up to very short ones (video) on the basis of generalization of Shafer-Wayne equations that ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Epigraphs
• Table of Contents
• List of Figures
• Preface
• Authors
• Chapter 1 Introduction
• Chapter 2 General Technique
• Chapter 3 One-Dimensional Problem in Hydrodynamics
• Chapter 4 Coupling of Sound with Vorticity: Acoustic Streaming
• Chapter 5 Projecting in Flows with Relaxation: Effects of Sound in Acoustically Active Fluids
• Chapter 6 Boundary Layer Problem: Acoustic and TollmiennSchlichting Waves
• Chapter 7 1D Electrodynamics
• Chapter 8 Metamaterials
• Chapter 9 Waves in Waveguides
• Chapter 10 Waves in 3D Space
• Index