The discovery of solitary waves

This book is an introduction to the theory of solitons and to the applications of the theory. Solitons are a special kind of localized wave, an essentially nonlinear kind. We shall define them at the end of this chapter, describing their discovery by Zabusky & Kruskal (1965). A solitary wave is the first and most celebrated example of a soliton to have been discovered, although more than 150 years elapsed after the discovery before a solitary wave was recognized as an example of a soliton. To lead to the definition of a soliton, it is helpful to study solitary waves on shallow water. We shall describe briefly in this section the properties of these waves, and then revise the elements of the theory of linear and nonlinear waves in order to build a foundation of the theory of solitons. Let us begin at the beginning, and relate a little history.

The solitary wave, or great wave of translation, was first observed on the Edinburgh to Glasgow canal in 1834 by J. Scott Russell. Russell reported his discovery to the British Association in 1844 as follows:

Epilogue

The essence of this book is the description of the method of inverse scattering. The book is too short to do more than outline the chief properties of solitons and indicate some lesser properties by a few remarks and problems. For that reason the chief properties stand out more clearly.

Research into the physical, earth and life sciences has led to the study of hundreds of nonlinear equations. We have mentioned only a little of this. Indeed, it is too wide to be described in any single volume. Each reader of this introduction to solitons may, however, go on to study the derivation of the equations used in his own specialist field. On having obtained an appropriate nonlinear system, it is natural to seek the waves of permanent form and to test whether they are stable and whether they are solitons. The existence of soliton interactions is the exception rather than the rule. Yet scores of nonlinear systems are already known to have soliton solutions. Even now it is not known how to ascertain definitely whether any given nonlinear system has solitons which may preserve their identities after interacting with one another. The preceding chapters, nonetheless, outline many ideas which may determine, or at least suggest, that a given nonlinear system has, or has not, soliton solutions and which indicate how the properties of the solitons may be found. The ideas are summarized in the following points.

Introduction

In chapter 3 we studied the way in which stretching and bending effects combine to carry axially symmetric loads applied to a uniform elastic cylindrical shell. In chapter 4 we investigated the way in which applied loads of a more general kind are carried by in-plane stress resultants alone, according to the membrane hypothesis. It became clear when we examined the resulting deflections of the shells in chapter 7 that this hypothesis is untenable in certain circumstances, and that indeed, in such cases the bending effects, explicitly neglected in the membrane hypothesis, might well sustain a major portion of the applied loading.

We have reached the point, therefore, where we must consider the more general problem of a shell which is capable of carrying the loads applied to it by a combination of bending and stretching effects. This is our task in the present chapter.

An important idea which we shall develop is that it is advantageous to regard the shell as consisting of two distinct surfaces which are so arranged to sustain the ‘stretching’ and ‘bending’ stress resultants, respectively; and indeed, the chapter as a whole explores various consequences which flow directly and indirectly from this idea.

In section 8.2 we introduce the ‘two-surface’ idea to the equilibrium equations, and show that we may treat the two surfaces separately provided we introduce appropriate force-interactions between them. The ‘stretching surface’ is identical to a shell analysed according to the membrane hypothesis, and we may therefore use directly the work of chapters 4–7 for this part.

Introduction

The main task of this chapter is to investigate some purely geometrical aspects of the distortion of curved surfaces. In general, if a given surface is distorted from its original configuration, every point on the surface will under-go a displacement; and at every point the surface will experience strain (‘stretching’) and change of curvature (‘bending’). Clearly the components of strain and change of curvature are, in general, functions not only of the components of displacement but also of the geometry of the surface in its original configuration.

In the present chapter (and indeed throughout the book) we shall consider only the limited class of distortions in which displacements, strains and changes of curvature are regarded as small, just as they are in the classical theory of simpler structures. In consequence of this simplification, the functional relationships between strain, change of curvature, and displacement will be relatively simple, and indeed linear.

In chapter 3 we have already established ad hoc expressions for change of curvature and hoop strain in terms of radial displacement for symmetrical deformations of a cylindrical shell surface. Our present task includes not only the investigation of more general types of distortion of cylindrical shells but also the consideration of distortion of other kinds of surface.

The arrangement of the chapter is as follows. First we investigate some aspects of the distortion of initially plane and cylindrical surfaces: these have the advantage that they may be described in terms of simple Cartesian coordinates.

Introduction

Piping systems are an indispensable feature of many industrial installations. In such systems straight tubes predominate; but problems of plant layout, etc., obviously make it necessary for pipes to turn corners. There are, broadly, four ways of getting the line of a pipe to turn a corner. First, fig. 13.1a shows a so-called long-radius bend in which the radius b of the centre-line of the curved portion is much larger than the radius a of the tube itself. A rightangle bend is illustrated, but it is obvious that the angle through which the line of the pipe turns is arbitrary, in general. On the domestic scale, bends of this sort may be made, ad hoc, in ductile metal pipes by the use of a pipe-bending machine; but the resulting cross-section of the curved portion is usually not circular: see later. Second, fig. 13.1b shows a so-called shortradius bend, in which the ratio b/a has a value of less than 4, say. The curved section is specially fabricated by casting, or welding together suitably curved panels; and the curved unit is connected to the straight pieces by bolted or welded joints. The types shown in fig. 13.1a and b are known as smooth bends. Third, fig. 13.1c shows a single-mitre bend, which is made by joining a pipe which has been ‘mitred’ by a plane oblique cut. A mitre joint may either be unreinforced (as shown) or reinforced by an elliptical ring or flange.

Introduction

Most of this book is concerned with the performance of shells under static loading. In contrast, the present chapter is concerned with an aspect of the response of shells to dynamic loading. The response of structures to dynamic loads is an important part of design in many branches of engineering: examples are the impact loading of vehicles, the aeroelastic flutter of aircraft, and wave-loading on large marine structures.

In this chapter we shall be concerned with the vibration of cylindrical shells, and in particular with the calculation of undamped natural frequencies. Calculations of this kind sometimes give the designer a clear indication that trouble lies ahead for a proposed structure; but if the design can be altered so that the natural frequencies of vibration of the structure are sufficiently different from the frequencies of the exciting agency, the occurrence of vibration can often be avoided.

For reasons of brevity, this chapter is restricted to cylindrical shells. The methods of the chapter may be adapted to the study of other sorts of shell, e.g. hyperboloidal shells used for large natural-draught water-cooling towers: see Calladine (1982).

Two very early papers on the subject of shell structures, by Rayleigh and Love, respectively, were on the subject of vibration, and the present chapters represent in fact only a relatively small advance on their work. Rayleigh (1881) was concerned with the estimation of the natural frequencies and modes of vibration of bells.

Introduction

Reinforced concrete shells have been used in the construction of roofs for many large buildings such as airport terminals, exhibition halls and factories. From a structural point of view a shell is attractive for this purpose, since the continuity of surface which is required to keep out the weather is provided by the structural member itself. From an economic point of view, however, reinforced-concrete shell roofs cast in situ are less attractive, largely on account of the labour-intensive effort which is needed in the construction of the formwork.

According to chapter 5, it is easy to construct a surface having zero Gaussian curvature from rectangular plywood sheets, whereas the construction of other kinds of surface makes it necessary to cut the sheets individually into non-rectangular shapes. It is not surprising therefore that cylindrical shells have been popular for the roofing of relatively simple rectangular buildings according to the scheme shown in fig. 10.1 and extensions of it. Shells of this kind, simply supported at their ends, form the subject of the present chapter.

Several authors have written on the structural analysis of cylindrical shell roofs of this sort, and at least one conference has been devoted to this subject: see Timoshenko & Woinowsky-Krieger (1959, § 126), Flügge (1973, §5.4.4.2), Gibson & Cooper (1954) and Witt (1954).

Almost all of the work which has been reported, however, is devoted to the analysis of particular examples having specific dimensions, and it cannot be said that any clear design principles have yet emerged from these studies.