Introduction

This chapter is concerned with various applications of the full wave methods discussed in ch. 18. The object is to illustrate general principles but not to give details of the results. The number of possible applications is very large and only a selection can be given here. The topics can be divided into two groups: (a) problems where the solutions can be expressed in terms of known functions, and (b) problems where computer integration of the differential equations, or an equivalent method as in §§ 18.2–18.11, is used. In nearly all applications of group (a) it is necessary to make substantial approximations. The group (a) can be further subdivided into those cases where the fourth order governing differential equations are separated into two independent equations each of the second order, and those where the full fourth order system must be used. For the separated second order equations the theory is an extension of ch. 15 which applied for an isotropic medium, and many of its results can be used here.

In all the examples of this chapter the ionosphere is assumed to be horizontally stratified, with the z axis vertical. The incident wave is taken to be a plane wave with its wave normal in the x–z plane. It is assumed that the only effective charges in the plasma are the electrons.

Introduction

The first order coupled wave equations (16.22) were derived and studied in § 16.3 for the neighbourhood of a coupling point z = zp where two roots q1, q2 of the Booker quartic equation are equal. If this coupling point is sufficiently isolated, a uniform approximation solution can be used in its neighbourhood and this was given by (16.29), (16.30). It was used in § 16.7 to study the phase integral formula for coupling. The approximations may fail, however, if there is another coupling point near to zp, and then a more elaborate treatment is needed. When two coupling points coincide this is called ‘coalescence’. This chapter is concerned with coupled wave equations when conditions are at or near coalescence.

A coupling point that does not coincide with any other will be called a ‘single coupling point’. It is isolated only if it is far enough away from other coupling points and singularities for the uniform approximation solution (16.30) to apply with small error for values of |ζ| up to about unity. Thus an isolated coupling point is single, but the reverse is not necessarily true.

Various types of coalescence are possible. The two coupling points that coalesce may be associated with different pairs q1, q2 and q3, q4 of roots of the Booker quartic. This is not a true coalescence because the two pairs of waves are propagated independently of each other.

Introduction

Although the earth's magnetic field has a very important influence on the propagation of radio waves in the ionospheric plasma, it is nevertheless of interest to study propagation when its effect is neglected. This was done in the early days of research in the probing of the ionosphere by radio waves, and it led to an understanding of some of the underlying physical principles; see, for example, Appleton (1928, 1930). In this chapter, therefore, the earth's magnetic field is ignored. For most of the chapter the effect of electron collisions is also ignored, but they are discussed in §§ 12.3, 12.11. The effect both of collisions and of the earth's magnetic field is small at sufficiently high frequencies, so that some of the results are then useful, for example with frequencies of order 40 MHz or more, as used in radio astronomy. For long distance radio communication the frequencies used are often comparable with the maximum usable frequency, §§ 12.8–12.10, which may be three to six times the penetration frequency of the F-layer. They are therefore in the range 10 to 40 MHz. This is large compared with the electron gyro-frequency which is of order 1 MHz, so that here again results for an isotropic ionosphere are useful, although effects of the earth's magnetic field have to be considered for some purposes.

This chapter is largely concerned with the use of pulses of radio waves and the propagation of wave packets, and uses results from ch. 10, especially §§ 10.2–10.6.

Introduction

In ch. 7 it was shown that at most levels in a slowly varying stratified ionosphere, and for radio waves of frequency greater than about 100 kHz, the propagation can be described by approximate solutions of the differential equations, known as W.K.B. solutions. The approximations fail, however, near levels where two roots q of the Booker quartic equation approach equality. In this chapter we begin the study of how to solve the differential equations when this failure occurs. For an isotropic ionosphere there are only two values of q and they are equal and opposite, and given by (7.1), (7.2). The present chapter examines this case. It leads on to a detailed study of the Airy integral function, which is needed also for the solution of other problems. Its use for studying propagation in an anisotropic ionosphere where two qs approach equality is described in § 16.3.

For an isotropic ionosphere, a level where q = 0 is a level of reflection. In § 7.19 it was implied that the W.K.B. solution for an upgoing wave is somehow converted, at the reflection level, into the W.K.B. solution for a downgoing wave with the same amplitude factor, and this led to the expression (7.151) for the reflection coefficient R. The justification for this assertion is examined in this chapter and it is shown in § 8.20 to require only a small modification, as in (7.152).

Introduction

This chapter continues the discussion of radio waves in a stratified ionosphere and uses a coordinate system x, y, z as defined in §6.1. Thus the electric permittivity ε(z) of the plasma is a function only of the height z. It is assumed that the incident wave below the ionosphere is a plane wave (6.1) with S2 = 0, S1 = S so that wave normals, at all heights z, are parallel to the plane y = constant, and for all field components (6.48) is satisfied. It was shown in § 6.10 that if the four components Ex, Ey, ℋx, ℋy of the total field are known at any height z, they can be expressed as the sum of the fields of the four characteristic waves, with factors f1, f2, f3, f4. In a homogeneous medium the four waves would be progressive waves, and these factors would be exp (− ikqiz), i = 1, 2, 3, 4. We now enquire how they depend on z in a variable medium. This question is equivalent to asking whether there is, for a variable medium, any analogue of the progressive characteristic waves in a homogeneous medium. The answer is that there is no exact analogue. There are, however, approximate solutions, the W.K.B. solutions, which have many of the properties of progressive waves.

This problem has proved to be of the greatest importance in all those branches of physics concerned with wave propagation.

Introduction

The earlier chapters have discussed the propagation of a plane progressive radio wave in a homogeneous plasma. It is now necessary to study propagation in a medium that varies in space, and this is the main subject of the rest of this book. The most important case is a medium that is plane stratified, and the later chapters are largely concerned with the earth's ionosphere that is assumed to be horizontally stratified. Because of the earth's curvature the stratification is not exactly plane, but for many purposes this curvature can be neglected. In cases of curved stratification it is often possible to neglect the curvature and treat the medium as locally plane stratified. Some cases where the earth's curvature is allowed for are discussed in §§ 10.4, 18.8.

The theory of this chapter is given in terms of the earth's ionosphere. A Cartesian coordinate system x, y, z is used with x, y horizontal and z vertically upwards. These coordinates are not in general the same as the x, y, z of chs. 2–4. The composition of the plasma, that is the electron and ion concentrations Ne, Ni are assumed to be functions of z only. For the study of radio propagation the frequency is usually great enough for the effect of the ions to be ignored. When radio waves are reflected in the ionosphere the important effects occur in a range of height that is small enough for the spatial variation of the earth's magnetic induction B to be ignored.

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.