Introduction

The methods in this chapter are concerned with the analysis of a sample of independent observations θ1,…, θn from some common population of vectors or axes. A number of data sets were introduced in Chapter 2, in the context of methods for displaying data. Here, we begin with some other examples to illustrate the range of problems which occur with a single sample of data.

For this edition I have added one new topic, made a number of miscellaneous changes intended to improve the presentation and clarify some of the arguments, and adjusted a few emphases to attune with the changing times. (Microwave ovens are now commonplace, while ballistic galvanometers have been consigned to the laboratory basement.) To keep the length reasonable I have relegated to the problems a few peripheral topics, such as Lorentzians.

The teaching of vibrational physics at undergraduate level cannot ignore for ever the revolutionary ideas grouped under the title ‘chaos’. I believe that a straightforward description of the physical phenomena will provide the student with a platform from which to approach the more detailed, but also more abstract, consideration of these matters in terms of trajectories and attractors in phase space. I have therefore expanded chapter 7 to include a simple discussion of large-amplitude forced vibrations, introducing chaotic vibrations and related non-linear effects.

Our examination of various waves in the previous three chapters has highlighted a fact of great practical significance: most wave systems are dispersive. The chief characteristic of a dispersive system is that it distorts any travelling waves that are not sinusoidal (section 12.1). Sinusoidal waves are not very useful, since they cannot carry any information other than their frequency and amplitude: one is usually interested in sending signals as some kind of modulated wave, and in particular as a train of pulses. Such an enterprise might appear doomed to failure if the chosen wave system is a dispersive one.

The remarkable fact is, however, that dispersive systems can after all support the undegraded propagation of stable pulses of a special kind, known as solitary waves. The effective cancellation of the dispersion has a surprising source in another property of most real wave systems, namely non-linearity.

To see how this comes about, we must first learn how non-linearity on its own affects a wave system. To simplify the discussion, we shall consider in this chapter only travelling plane waves moving in the positive z-direction, and we shall also neglect all non-conservative processes like friction and viscosity, which dissipate energy.

Non-linear wave systems

In chapter 7 we examined free and forced vibrations of non-linear systems.

The wave equation we discussed in chapter 9 has the special property that it allows a disturbance of arbitrary form to be propagated indefinitely as a travelling wave, without having its shape changed. We met several examples of such non-dispersive waves in chapter 10.

Non-dispersive waves are exceptional. In this chapter we examine possible sources of dispersion in a stretched string.

Stiff strings

The stretched string of chapter 9 was assumed to be perfectly flexible, so that there were no transverse return forces other than those due to the tension. Real strings, such as violin and piano strings, are ‘stiff’ and tend to straighten out even when unstretched. The extra return forces due to this lateral stiffness make the string dispersive.

These return forces come from the stresses within curved parts of the string. The stress forces at any cross-section of the string will have components acting along the string direction, and components acting in the plane perpendicular to the string. Each set of components can be replaced by a single force and a single torque: for the components parallel to the string we have the force of tension, and a bending moment, while the perpendicular components give a shear force tending to break the string across, and a twisting moment. Since we already know the return force due to the tension, and since the string is presumably not twisted, we need consider only the other two.

The next three chapters are about waves which are, except in special circumstances, dispersive. For each type of wave we shall find the dispersion relation. When we have found it we shall quarry in it for physics, exploring particularly those extreme forms which can be so instructive.

To most people the word ‘wave’ suggests a stormy sea or ripples on the surface of a pond. Our discussion of these, the most familiar of all waves, falls into three main parts. In the first section we consider the purely geometrical problem of how the water must move in order to make a sinusoidal travelling wave on the surface. Then we find the dispersion relation, which tells us what sinusoidal waves are actually possible: this is a physical problem. Finally we conduct our discussion of the dispersion relation under various interesting extreme conditions.

Most of the physical results are to be found in the third section. If you wish to skip the preliminaries, you should first study fig. 13.5 and its caption, and the summary at the end of section 13.1, and then proceed directly to the dispersion relation (13.19).

The nature of the wave motion

We study the water in a long, rectangular canal of depth h (fig. 13.1). In equilibrium the water surface is flat and horizontal. With luck, a very special kind of breeze might blow along the length of the canal, exciting a sinusoidal travelling wave.

At this point it is worth bringing together several things we have discovered about vibrating systems in earlier chapters.

(1) If a number of harmonic driving forces act simultaneously on a linear system, the resulting steady-state vibration ψ(t) is a superposition of harmonic vibrations whose frequencies are those of the driving forces: each harmonic force makes its own independent contribution to ψ(t). This is an example of the principle of superposition (section 5.2).

(2) The free vibration of a non-linear system is not harmonic, but something more complicated; we found it possible, however, to express an anharmonic vibration ψ(t) as a series of terms consisting of the fundamental vibration and a series of harmonics (sections 7.1 and 7.2).

(3) At small amplitudes, the application of a harmonic driving force to a non-linear system leads to a steady-state ψ(t) which contains the driving frequency and harmonics of that frequency (section 7.3).

(4) The standing waves that are possible on a non-dispersive string of finite length have frequencies in a sequence like v1, 2v1, 3v1,…; when a number of standing waves are excited simultaneously, the vibration ψ(t) of any given point on the string must therefore consist of a series of superposed harmonics.

It is clear from these examples alone that the harmonic type of vibration on which we have spent so much time has a fundamental significance as the building block for more complicated motions.

Encouraged by the friendly reception given to the first edition, I have preserved its basic form and most of the details. The new and revised material occurs mainly in the latter half, on waves. There is one completely new chapter, intended to provide an elementary introduction to the so-called solitary wave, or soliton, which has become such a pervasive feature of physical science. It seemed to me that it should be possible to base an explanation of the solitary wave on the simplest notions of non-linear waves, and chapter 16 is my modest attempt. Consequential changes were necessary in chapter 12, but I believe the outcome is a more helpful treatment of dispersion, even for those who have no immediate need of the solitary wave material. Apart from these and other, smaller, changes, I have added some 30 new problems, the majority of which introduce new physical examples of vibrations and waves.

Most of the work was done during a further period of Study Leave from the University of Liverpool. Of the many people who made valuable comments on the first edition, I am particularly grateful to my Liverpool colleagues Professor J. R. Holt and Dr A. N. James, and their students.

A number of new diagrams were drawn for this edition by Roderick Main.