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.

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.