Title:

Propagation of waves in inhomogeneous media

Waves propagating in an inhomogeneous medium differ from waves in uniform surroundings through the dependence of their properties on the variation of the physical parameters of the medium. In this thesis, we will investigate the effects of inhomogeneity on wave propagation in two different cases  introducing the subject via propagation of waves in nonuniform atmospheres and culminating in a full analytic solution of the cold plasma equations describing wave propagation in a plasma with a spatially rotating magnetic field. Plasmas can support a wide variety of waves. In nonuniform plasmas, it is of great interest to consider the possibility of one type of wave undergoing mode conversion to a completely different wave  a phenomenon used to heat plasmas in fusion reactor experiments. In Chapter 1, we present an overview of plasmas in general and consider their widespread natural occurrence and the vast range of their characteristic parameters. This chapter also contains the definition of certain basic plasma quantities, such as plasma frequency, which will be used extensively in later chapters and a discussion of plasma confinement systems, illustrating the magnetic field configurations of interest in the context of fusion reactors. Mode conversion is introduced formally in Chapter 2 where various approaches are discussed. The philosophy behind the powerful WKBJ theory, which applies to slowlyvarying media, is presented as a natural extension of the description of waves in uniform media. The "local dispersion relation" method favoured by many authors is examined critically and an alternative description is outlined which is derived from the full differential equation including the gradient terms. The equations of fluid theory are derived from kinetic theory in Chapter 3 using the method of moments. We discuss how a cold (pressureless) plasma may be described satisfactorily using these equations. Thus, we conclude that the same basic set of equations may be applied to propagation of waves in a cold plasma and a neutral atmosphere (with the removal of electromagnetic forces and addition of a gravitational field). As a simple introduction to the study of wave propagation in inhomogeneous media, we consider the propagation of waves in an atmosphere where the temperature (and hence sound speed) varies with height. Oscillations in an isothermal atmosphere are shown to possess two branches corresponding to acoustic and gravity waves. In considering nonuniform atmospheres, the work of authors in the fields of atmospheric and solar physics is combined and reviewed. The isothermal dispersion relation is demonstrated to be inadequate in describing waves in temperature stratified atmospheres. It is found that the ordinary differential equation constructed only has solutions for a limited number of special cases of temperature variation and, in general, requires numerical solution. Chapter 5 contains the analysis of the propagation of cold plasma waves in a constant magnetic field. In particular, we examine propagation perpendicular to the magnetic field direction in order to provide the background for the equivalent inhomogeneous case of Chapter 6. In Chapter 6 we solve the problem of wave propagation in a spatially rotating magnetic field. It is shown that, in order to balance the gradient of the equilibrium magnetic field, a current is required which we partition between the ions and electrons. The number of nonvanishing equilibrium quantities is therefore considerably extended from the uniform situation of Chapter 5, leading to a significant alteration of the form of the Ohm's Law which must now contain the electric field plus its first two derivatives. By transforming reference frame, it becomes possible to eliminate the positiondependent coefficients of the differential equations and thus derive a dispersion relation describing waves in such a field structure. It is shown that the waves consist of a propagating part modulated by a periodic envelope induced by the periodicity of the field. Finally, in Chapter 7, we make suggestions for extensions of the work of Chapter 6.
