Introduction

The linear theory calculations performed so far have the advantage that analytic solutions can be found for a broad range of circumstances. However, the solutions are strictly valid only in the limit of infinitesimally small amplitude waves. Waves are said to be weakly nonlinear when their amplitude is sufficiently large that nonlinear effects arising from the advection terms in the material derivative begin to play an important role. Using perturbation theory, simplified equations can be derived that capture the leading-order nonlinear effects, referred to as ‘weak nonlinearity’. Analysis of such equations not only reveals how nonlinear effects change the evolution of the waves, but they serve to establish bounds on the wave amplitude for which linear theory is valid.

As well as modifying the structure of small-amplitude waves, nonlinear effects can give rise to new classes of steady waves, such as solitary waves, and they can result in the breakdown of steady waves through mechanisms such as modulational instability and parametric subharmonic instability.

In this chapter we focus primarily upon the weakly nonlinear dynamics of interfacial and internal waves in otherwise stationary fluid; there is no ambient mean flow. For those not familiar with perturbation theory for differential eigenvalue problems, we begin with a brief review of the mathematics necessary for the treatment of weakly nonlinear waves. This analysis shows that frequency is a function of amplitude as well as wavenumber. We then examine how weakly nonlinear effects modify the evolution of interfacial and internal waves.

Introduction

In this chapter we consider waves at the interface between two fluids of different density and waves in multi-layer fluids. We derive the fully nonlinear equations for waves of arbitrary amplitude and then determine approximate formulae appropriate for small-amplitude waves. These are expressed through coupled linear partial differential equations for which analytic solutions may be found through standard methods. Approximate solutions for nonlinear waves are considered in Chapter 4.

A ‘one-layer fluid’ has uniform density and is bounded above by a free surface that may oscillate up and down, for example, due to gravitational forces. Waves in a one-layer fluid are specifically referred to here as ‘surface waves’. Waves on the ocean surface can be treated as existing in a one-layer fluid if the density variation with depth in the ocean negligibly affects their dynamics.

A ‘two-layer fluid’ describes a fluid of one density underlying a second fluid of smaller density. ‘Interfacial waves’ propagate at the interface between the two fluids. In one sense, surface waves on the ocean are interfacial waves in that they propagate at the interface between water and air. However, in this book we distinguish surface waves from interfacial waves by requiring for the latter that the density difference between the upper and lower layer fluids is a small fraction of the density of either layer. Thus undular displacements of an interface between fresh and salt water are considered for waves in a two-layer fluid.

With the advent of faster computers, numerical simulation of physical phenomena is becoming more practical and more common. Computational prototyping is becoming a significant part of the design process for engineering systems. With ever-increasing computer performance the outlook is even brighter, and computer simulations are expected to replace expensive physical testing of design prototypes.

This book is an outgrowth of my lecture notes for a course in computational mathematics taught to first-year engineering graduate students at Stanford. The course is the third in a sequence of three quarter-courses in computational mathematics. The students are expected to have completed the first two courses in the sequence: numerical linear algebra and elementary partial differential equations. Although familiarity with linear algebra in some depth is essential, mastery of the analytical tools for the solution of partial differential equations (PDEs) is not; only familiarity with PDEs as governing equations for physical systems is desirable. There is a long tradition at Stanford of emphasizing that engineering students learn numerical analysis (as opposed to learning to run canned computer codes). I believe it is important for students to be educated about the fundamentals of numerical methods. My first lesson in numerics includes a warning to the students not to believe, at first glance, the numerical output spewed out from a computer.

In the next two chapters we develop a set of tools for discrete calculus. This chapter deals with the technique of finite differences for numerical differentiation of discrete data. We develop and discuss formulas for calculating the derivative of a smooth function, but only as defined on a discrete set of grid points x0, x1, …, xN. The data may already be tabulated or a table may have been generated from a complicated function or a process. We will focus on finite difference techniques for obtaining numerical values of the derivatives at the grid points. In Chapter 6 another more elaborate technique for numerical differentiation is introduced. Since we have learned from calculus how to differentiate any function, no matter how complicated, finite differences are seldom used for approximating the derivatives of explicit functions. This is in contrast to integration, where we frequently have to look up integrals in tables, and often solutions are not known. As will be seen in Chapters 4 and 5, the main application of finite differences is for obtaining numerical solution of differential equations.

Construction of Difference Formulas Using Taylor Series

Finite difference formulas can be easily derived from Taylor series expansions.

Since the original publication of this book ten years ago, the available computer power has increased by more than 2 orders of magnitude due to massive parallelism of computer processors and heterogeneous computer clusters. Today, scientific computing is playing an ever more prominent role as a tool in scientific discovery and engineering analysis.

In the second edition an introduction to the finite element method has been added. The finite element method is a widely used technique for solving partial differential equations (PDEs) in complex domains. As in the first edition, numerical solution of PDEs is treated in Chapter 5, and the development there is based on finite differences for spatial derivatives. This development is followed in Chapter 6 by an introduction to more advanced transform methods for solving PDEs: spectral methods and, now, the finite element method. These methods are compared to the finite difference methods in several places throughout Chapter 6.

Hopefully, most of the errors that remained in the 2007 reprint of the book have now been corrected. Several exercises have also been added to all the chapters. In addition, complete MATLAB programs used for all the worked examples are available at www.cambridge.org/Moin. Students should find this new feature helpful in attempting the exercises, as similar computer programs are used in many of them. Working out the exercises is critical to learning numerical analysis, especially using this book.

Most physical phenomena and processes encountered in engineering problems are governed by partial differential equations, PDEs. Disciplines that use partial differential equations to describe the phenomena of interest include fluid mechanics, where one is interested in predicting the flow of gases and liquids around objects such as cars and airplanes, flow in long distance pipelines, blood flow, ocean currents, atmospheric dynamics, air pollution, underground dispersion of contaminants, plasma reactors for semiconductor equipments, and flow in gas turbine and internal combustion engines. In solid mechanics, problems encountered in vibrations, elasticity, plasticity, fracture mechanics, and structure loadings are governed by partial differential equations. The propagation of acoustic and electromagnetic waves, and problems in heat and mass transfer are also governed by partial differential equations.

Numerical simulation of partial differential equations is far more demanding than that of ordinary differential equations. Also the diversity of types of partial differential equations precludes the availability of general purpose “canned” computer programs for their solutions. Although commercial codes are available in different disciplines, the user must be aware of the workings of these codes and/or perform some complementary computer programming and have a basic understanding of the numerical issues involved. However, with the advent of faster computers, numerical simulation of physical phenomena is becoming more practical and more common. Computational prototyping is becoming a significant part of the design process for engineering systems. With ever increasing computer performance the outlook is even brighter, and computer simulations are expected to replace expensive physical testing of design prototypes.