Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- Video resources
- 1 Introduction
- 2 Instabilities of fluids at rest
- 3 Stability of open flows: basic ideas
- 4 Inviscid instability of parallel flows
- 5 Viscous instability of parallel flows
- 6 Instabilities at low Reynolds number
- 7 Avalanches, ripples, and dunes
- 8 Nonlinear dynamics of systems with few degrees of freedom
- 9 Nonlinear dispersive waves
- 10 Nonlinear dynamics of dissipative systems
- 11 Dynamical systems and bifurcations
- Appendix A The Saint-Venant equations
- References
- Index
9 - Nonlinear dispersive waves
Published online by Cambridge University Press: 05 August 2011
- Frontmatter
- Contents
- Foreword
- Preface
- Video resources
- 1 Introduction
- 2 Instabilities of fluids at rest
- 3 Stability of open flows: basic ideas
- 4 Inviscid instability of parallel flows
- 5 Viscous instability of parallel flows
- 6 Instabilities at low Reynolds number
- 7 Avalanches, ripples, and dunes
- 8 Nonlinear dynamics of systems with few degrees of freedom
- 9 Nonlinear dispersive waves
- 10 Nonlinear dynamics of dissipative systems
- 11 Dynamical systems and bifurcations
- Appendix A The Saint-Venant equations
- References
- Index
Summary
Introduction
When a fluid is locally perturbed by an impulse (for example, by an impact) or a periodic excitation (the vibration of a membrane, string, or mechanical blade), the perturbation may propagate from the source in the form of a wave. Examples include acoustic waves, surface waves, and internal waves in a stratified fluid (Lighthill, 1978). The solution of the linearized equations for small amplitude perturbations leads to a major result: the wave number k and frequency ω (or the wave speed c = ω/k) are not independent, they are related by a dispersion relation. Since this relation is obtained from linearized equations, another major result is that dispersion does not depend on the amplitude of the perturbation. However, if the amplitude exceeds some level, new effects arise that the dispersion relation of linear theory obviously does not describe. To explain these new effects it is necessary to include nonlinear terms neglected in the linear study, i.e., to develop a theory of nonlinear waves. Such waves are also referred to as finite-amplitude waves, in contrast to the waves of infinitesimal amplitude considered in a linear analysis.
The objective of the present chapter is to give an elementary account of the theory of nonlinear waves. We will show (i) how nonlinear waves can be constructed by a perturbation method (essentially the multiple-scale method presented in the preceding chapter), and (ii) how the linear stability of these waves can be studied.
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- Hydrodynamic Instabilities , pp. 274 - 298Publisher: Cambridge University PressPrint publication year: 2011