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.

The reflection and transmission processes that occur when a travelling wave meets a discontinuity in a string, and the standing waves set up when incident and reflected waves are superposed, were analyzed in chapter 9. We were able to adapt the results to other situations, such as the reflection of electromagnetic plane waves meeting a dielectric surface at right angles, or the formation of acoustic standing waves in a pipe.

Some of the most interesting ways in which the presence of a boundary between two media can affect the behaviour of waves in those media become apparent only when the waves are travelling in directions other than normal to the boundary. Likewise, standing waves in a region of three-dimensional space have characteristics which we cannot learn about by studying standing waves on strings. In this chapter, therefore, we think about plane waves meeting plane boundaries, not in general at normal incidence.

As a preliminary step, we learn how to handle plane waves in three-dimensional problems.

Reflection and refraction

Although we began by discussing transverse waves on strings, we have seen already that the disturbance in a one-dimensional wave need not be concentrated along a line. In a sinusoidal travelling wave ψ(z, t) for which ψ is a quantity like acoustic pressure or the strength of an electric field, which can have a value at any point in space, the phase angle ωt – kz has the same value at all points which have the same value of z; these points lie on a surface perpendicular to the direction of propagation.

Before going on to discuss other wave equations, we pause to examine a few physical systems which obey (exactly or approximately) the nondispersive form of wave equation (9.3), or its damped version (9.37). We start by outlining our general approach, which involves three steps:

(1) We first identify the return force acting at a point z. In our examples we shall always find that the size of this force is proportional to some gradient ∂ψ/∂z where ψ is the variable which measures the particular disturbance being propagated. (For a transverse disturbance on a string, ∂ψ/∂z is simply the slope of the string.)

(2) From the return force at the point z we deduce the return force acting on a segment of length Δz. That there is such a force follows from the fact that the return force will usually vary with its position z, so that the forces at the two ends of a segment do not balance each other. If the return force at point z is F, and the return force at point z + Δz is F + ΔF, then the segment will be both strained (by the balanced forces F) and accelerated (by the unbalanced force ΔF). Since F is proportional to ∂ψ/∂z then ΔF will be proportional to ∂2ψ/∂z2. (For the model system this quantity simply measures the curvature of the string.)

(3) We apply Newton's second law to the segment. For a vanishingly small segment the result is the wave equation.

We now take a quick look at a few representative physical systems, so that we can appreciate the great variety of circumstances in which the mathematics of harmonic motion can be applied. The model system has taught us what properties we must look for if we wish to know whether vibration is possible in any given real system. The mathematics of the prototype can be used in a production-line spirit once we have isolated the ingredients (inertia and stiffness) necessary for vibration.

As well as appreciating the essential similarity of different kinds of vibration, we shall in this chapter be vitally interested in the numerical sizes of the physical quantities involved, and particularly in the values of the free vibration frequencies of the various systems. One of the most striking things we shall discover is the very wide spread of frequencies to be found.

Angular vibrations

The mass in the model system vibrates to and fro along a straight line. Some of the most familiar vibrations involve massive objects rotating back and forth under the influence of a return torque. We shall consider two kinds of return torque: first an elastic torque provided by a stiff suspension, and then one due to gravity.

Torsional vibrations. The system shown in fig. 2.1 is the exact rotational analogue of the model system. The coordinate ψ now measures the angular displacement of the mass from its equilibrium position.

Systems which can be described in terms of a single coordinate are exceptional, and we must expect the vibrational possibilities to become more numerous and more complex as the required number of coordinates increases.

Most of the new features can be revealed by thinking about a system with only two coordinates. Even a two-coordinate system can exhibit very complicated vibrational behaviour, but we shall find that there are two very simple basic motions, known as modes, and that any other possible vibration can be treated as a superposition of these.

In the first two sections of this chapter we shall discover, without undue rigour, the physical nature of the modes and the way in which the mathematics can be simplified by the choice of a special coordinate system. The third section outlines the technique for finding these ‘mode coordinates’ for a given system, and the technique is applied to a physical example in section 8.4. Only sections 8.1 and 8.2 are strictly necessary to enable you to understand the rest of the book.

Modes and mode coordinates

We start with the simple system shown in fig. 8.1. It consists of two identical masses, connected to each other and to two fixed points by a symmetric arrangement of three springs, of which the outer ones each have stiffness s, and the central one has stiffness S. We ignore all forces other than the spring forces, and we consider one-dimensional motion of each mass along the line of the springs.

It would be generally agreed that an undergraduate course on vibrations and waves should lead the student to a thorough understanding of the basic concepts, demonstrate how these concepts unify a wide variety of familiar physics, and open doors to some more advanced topics on which they shed light. The fundamental ideas can be introduced with reference to mechanical systems which are easy to visualize and to illustrate; less tangible phenomena can then be treated with the same mathematics, leaving one free to concentrate on more interesting problems of formulation and interpretation.

In such a course there is always a risk that springs and strings will take over completely, submerging the real physics. The theory must be developed with some care and thoroughness if the student is to comprehend it in sufficient depth to be able to apply it readily, and all too often the physics has to be left half baked. This textbook is an attempt to provide, in a volume of moderate size, an account which is systematic and coherent, but which also treats the physical examples in some depth.

The diagram on p. viii shows the plan of the book, and indicates the relative weights attached to fundamental and illustrative material. The theoretical development (indicated by the downward flow through the toned areas on the left) is continuous, but is punctuated by regular excursions into physics.

The boundaries between different wave-carrying media discussed in the previous chapter were assumed to be extensive. Reflected and refracted waves generated at such boundaries have a family resemblance to the incident wave giving rise to them; plane waves arriving at plane boundaries, for example, result in plane reflected and refracted waves.

If the second medium occupies only a small pocket of space embedded within the first medium, we think of it less as a medium and more as an object obstructing the progress of the wave through the medium proper. A new type of behaviour, known as diffraction, is apparent in such cases. Particularly striking effects are obtained when large numbers of identical diffracting objects are arranged in some regular way. A familiar example is provided by a street lamp as seen through a fog of water particles: the halo round the lamp would not be there if the lamp were viewed through a continuous mass of water.

The most characteristic feature of diffraction is that incident waves appear to ‘bend round’ objects made of materials which, in the form of extended media, are quite opaque. In a similar way, waves passing through a transparent hole in an opaque object are ‘spread out’. We shall refer to an obstacle in a transparent medium, or an aperture in an opaque object, as a diffraction centre.

We shall find that these effects are appreciable when the diffraction centres have dimensions which are comparable with the wavelength of the waves involved.