Introduction

This chapter deals with problems where the approximations of ray theory cannot be used and where it is necessary to take account of the anisotropy of the medium. It is therefore mainly concerned with low and very low frequencies, so that the change of the medium within one wavelength is large. The problem is discussed here for the ionosphere, which is assumed to be a plane stratified plasma in the earth's magnetic field. A radio wave of specified polarisation is incident obliquely and it is required to find the reflection coefficient as defined in §§ 11.2–11.6. In some problems it is also required to find the transmission coefficients. It is therefore necessary to find solutions of the governing differential equations, and the form used here is (7.80), (7.81). This type of problem has been very widely studied and there are many methods of tackling it and many different forms of the differential equations. These methods are too numerous to be discussed in detail, but the main features of some of them are given in §§ 18.3–18.5. The object in this chapter is to clarify the physical principles on which the various operations are based, and for this the equations (7.80), (7.81) are a useful starting point. In a few cases solutions can be expressed in terms of known functions, and some examples are given in §§ 19.2–19.6.

Introduction

For wave propagation in a stratified medium, the idea of writing the governing differential equations as a set of coupled equations has already been used in §§ 7.8, 7.15, for the study of W.K.B. solutions. The term ‘coupled equations’ is usually given to a set of simultaneous ordinary linear differential equations with the following properties:

1. (1) There is one independent variable which in this book is the height z or a linear function of it.

2. (2) The number of equations is the same as the number of dependent variables.

3. (3) In each equation one dependent variable appears in derivatives up to a higher order than any other. The terms in this variable are called ‘principal’ terms and the remaining terms are called ‘coupling’ terms.

4. (4) The principal terms contain a different dependent variable in each equation, so that each dependent variable appears in the principal terms of one and only one equation.

It is often possible to choose the dependent variables so that the coupling terms are small over some range of z. Then the equations may be solved by successive approximations. As a first approximation the coupling terms are neglected, and the resulting equations can then be solved. The values thus obtained for the dependent variables are substituted in the coupling terms and the resulting equations are solved to give a better approximation. Some examples of this process are given in § 16.13.

Introduction

The term ‘ray’ was used in § 5.3 when discussing the field of a radio wave that travels out from a source of small dimensions at the origin of coordinates ξ, η, ζ in a homogeneous medium. The field was expressed as an integral (5.29) representing an angular spectrum of plane waves. The main contribution to the integral was from ‘predominant’ values of the components nξ, nη of the refractive index vector n such that the phase of the integrand was stationary for small variations δnζ, δnη For any point ξ, η, ζ there were one or more predominant values of the refractive index vector n, such that the line from the origin to ξ, η, ζ was normal to the refractive index surface. Each predominant n defines a progressive plane wave that travels through the whole of the homogeneous medium.

Instead of selecting a point ξ, η, ζ and finding the predominant ns, let ns choose a fixed n and find the locus of points ξ, η, ζ for which this n is predominant. For a homogeneous medium this locus is called the ‘ray’. The arguments leading to (5.31) still apply and show that it is the straight line through the origin, in a direction normal to the refractive index surface at the point given by the chosen n.

FINDING A CLOCK THAT WOULDN'T GET SEASICK

Navigation has provided one of the most persistent motives for measuring time accurately. All navigators depend on continuous time information to find out where they are and to chart their course. But until about two centuries ago, no one was able to make a clock that could keep time accurately at sea.

COOLING OFF

How do you make something colder? Making something hotter is easy. For example, if you need to warm yourself on a chilly night, you can build a fire with little or no technology. But to cool yourself on a hot day is quite another matter.

THE GENESIS OF AN IDEA

The year was 1665, the month was August, and Cambridge, England, was besieged by bubonic plague. Isaac Newton, then a 23-year-old university student, retired to the solitude of his family's farm in Lincolnshire until the plague subsided and the university reopened. Not taking kindly to inactivity, Newton composed 22 questions for himself to tackle, ranging from geometric constructions to Galileo's new mechanics to Kepler's planetary laws. During the next 18 months, he immersed himself in the search for answers and along the way discovered calculus, the laws of motion, and the universal law of gravity.

THE UNIVERSE AS A MACHINE

In the seventeenth century science assumed its modern form and the scientific spirit infected Europe. It was then that Aristotle's view of nature was rejected and Galileo's great book of the universe was adopted. The new science was nourished by an optimism that mankind could discover the laws of nature.

One of the most significant and influential figures in seventeenth-century natural philosophy was René Descartes. Early in his life, Descartes rebelled against the traditions in which he had been thoroughly educated. He sought new foundations for knowledge, foundations which could underpin confidence in our understanding of nature. Convinced of the indubitable logic of mathematics, Descartes chose to identify mathematics with physics.

Descartes is credited with having been the first person to state the law of inertia correctly.

FREEWAYS IN THE SKY

Not many years ago, the only conceivable use of the beautiful celestial mechanics developed over hundreds of years was to compute the positions of bodies in the heavens. Today that situation has changed radically.

FORCED OSCILLATIONS

Galileo was not the only famous member of the Galilei family; his father Vincenzo was an accomplished and articulate musician.

WINDING UP THE MECHANICAL UNIVERSE

We've now arrived at the final chapter in our study of the mechanical universe. In our story we've introduced revolutionary ideas and heroes from Copernicus to Newton, and just as they did before us, we've linked the physics of the heavens to the physics of the earth.

THE AGE OF STEAM

The age of steam is past. The steam engine is a curiosity, an object of nostalgia that has been replaced by diesel engines, electric motors, turbine engines, and gasoline engines to drive the wheels of civilization. Nonetheless, steam did have its day. The steam engine not only caused the industrial revolution, which changed our lives, it also led to discoveries in physics so profound that they changed the way we think. How did investigations into the nature of steam engines lead to a deeper understanding of the universe?

First, we need to understand how a steam engine operates. In essence, a steam engine is a device which heats water in a dosed container, a boiler, thereby converting it to steam.

ANTIDIFFERENTIATION, THE REVERSE OF DIFFERENTIATION

We saw by examples in earlier chapters that laws of physics are often expressed as equations about the rate at which things change – that is, equations about derivatives.

In discussing falling bodies in Chapter 2, we started with a knowledge of the distance function (how far a body falls in a given time), then took its derivative to find its speed (how fast it is falling), and then took the derivative of the speed to find its acceleration (how fast it was getting faster).

FINDING THE CONNECTION BETWEEN ELECTRICITY AND MAGNETISM

The flood of forces identified and classified in the eighteenth century was reduced to a trickle after it was realized that electric forces were responsible for many phenomena. Nature, it seemed, acknowledged only a handful of forces. Each force had its own “universal constant.” For electric forces, it was Ke ; for gravity, G; for magnetism, there was an additional constant Km ; and for light, there was the speed of light, known since 1630 to be 3 × 108 m/s. Surely many physicists wondered whether these constants and the forces they represent are somehow related.