In Chapter 2 we saw how the motion of a guitar string could be established by considering the physical mechanisms that governed it and using them to determine the wave equation for the system. In this chapter we shall see that the same approach can be applied to a wide range of physical systems. We begin with detailed derivations of the wave equations for electromagnetic waves along a coaxial cable and in free space, and then examine ocean waves and ripples on a fluid surface, showing how they may be extended to describe a variety of atmospheric and oceanic phenomena.

In each case, we begin by determining how the disturbance or displacement at any point is affected by that at adjacent points, and how the physical properties of the system determine how quickly it can respond. This allows us to derive a partial differential equation that describes the wave propagation and embodies all the relevant physics. What remains, as before, is the purely mathematical solution of this wave equation.

Although this chapter provides many useful illustrations of the principles outlined in Chapter 2, the reader may safely skip these examples without missing any fundamental principles upon which we shall build later.

Waves along a coaxial cable

Coaxial cables are used for the distribution of electrical signals, and include microphone cables, television aerial leads and, indeed, most of the connections between audio and video apparatus.

Introduction

The physics of waves is too often presented only in a few rather straightforward and sometimes uninspiring contexts: the motion of strings, sound, light and so on. Students may be led to regard the topic with disdain; and they may be left with some crucial misconceptions, such as that all waves are sinusoidal. Wave physics may hence be considered an old-fashioned field with little relevance to the more modern, exciting areas of quantum physics, nanotechnology and cosmology. Yet there is plenty to find interesting just in classical and modern optics and the physics of musical instruments; and wave phenomena prove to be central to most of the fascinating and newly emerging branches of both fundamental and applied physics.

Most aspects of physics may be viewed from two perspectives: one, named after Lagrange, addresses particles, while the other, due to Euler, considers fields. We may, for example, establish the electromagnetic properties of matter by considering the Coulomb forces among all the constituent charged particles; or we may describe the material's bulk response to a field and tackle the problem that way. This duality pervades most areas of physics and, while one of the alternatives often proves vastly more convenient than the other, the two are ultimately quite consistent, equivalent viewpoints.

When we extend our analysis to dynamic systems, the particle approach becomes a form of ‘kinetics’ or ballistics, and changes in the field description are manifest as waves.