Fundamentals of Plasma Physics

4

Elementary plasma waves

4.1 General method for analyzing small amplitude waves

All plasma phenomena can be described by combining Maxwell’s equations with the Lorentz
equation where the Lorentz equation is represented by the Vlasov, two-fluid or MHD ap-
proximations. The subject of linear plasma waves provides a good introduction to the study
of plasma phenomena because linear waves are relatively simple to analyze and yet demon-
strate many of the essential features of plasma behavior.
Linear analysis, a straightforward method applicable to any set of partial differential
equations describing a physical system, reveals the physical system’s simplest non-trivial,
self-consistent dynamical behavior. In the context of plasma dynamics, themethod is as
follows:

1. By making appropriate physical assumptions, the general Maxwell-Lorentz system of
   equations is reduced to the simplest set of equations characterizing the phenomena
   under consideration.


2. An equilibrium solution is determined for this set of equations. The equilibrium might
   be trivial such that densities are uniform, the plasma is neutral, andall velocities are
   zero. However, less trivial equilibria could also be invoked wherethere are density
   gradients orflow velocities. Equilibrium quantities are designated by the subscript 0 ,
   indicating ‘zero-order’ in smallness.


3. Iff,g,h,etc. represent the dependent variables and it is assumed that a specificpertur-
   bation is prescribed for one of these variables, then solving the system ofdifferential
   equations will give the responses of all the other dependent variables to this prescribed
   perturbation. For example, suppose that a perturbationǫf 1 is prescribed for the depen-
   dent variablefso thatfbecomes

f=f 0 +ǫf 1. (4.1)

The system of differential equations gives the functional dependence of the other

variables onf, and for example, would giveg = g(f) =g(f 0 +ǫf 1 ). Since

the functional dependence ofgonfis in general nonlinear, Taylor expansion gives

g=g 0 +ǫg 1 +ǫ^2 g 2 +ǫ^3 g 3 +....Theǫ’s are, from now on, considered implicit and

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