124 Chapter 4. Elementary plasma waves
so the variables are written as
f = f 0 +f 1
g = g 0 +g 1 +g 2 +....
h = h 0 +h 1 +h 2 +.... (4.2)and it is assumed that the order of magnitude off 1 is smaller than the magnitude off 0
by a factorǫ,etc. The smallness of the perturbation is an assumption which obviously
must be satisfied in the real situation being modeled. Note that there is nof 2 or higher
fterms because the perturbation tofwas prescribed as beingf 1.- Each partial differential equation is re-written with all dependent quantities expanded
to first order as in Eq.(4.2). For example, the twofluid continuity equation becomes
∂(n 0 +n 1 )
∂t
+∇·[(n 0 +n 1 )(u 0 +u 1 )] = 0. (4.3)
By assumption, equilibrium quantities satisfy
∂n 0
∂t+∇·(n 0 u 0 ) = 0. (4.4)
The essence of linearization consists of subtracting the equilibrium equation (e.g.,
Eq.(4.4)) from the expanded equation (e.g., Eq.(4.3)). For this example such a sub-
traction yields
∂n 1
∂t+∇·[n 1 u 0 +n 0 u 1 +n 1 u 1 ] = 0. (4.5)
The nonlinear termn 1 u 1 which is a product oftwofirst order quantities is discarded
because it is of orderǫ^2 whereas all the other terms are of orderǫ. What remains
is called the linearized equation, i.e., the equation which consists ofonly first-order
terms. For the example here, the linearized equation would be
∂n 1
∂t+∇·[n 1 u 0 +n 0 u 1 ] = 0.
The linearized equation is in a sense the differential of the original equation.
Before engaging in a methodical study of the large variety of waves that canpropagate
in a plasma, a few special cases of fundamental importance will first beexamined.
4.2 Two-fluid theory of unmagnetized plasma waves
The simplest plasma waves are those described by two-fluid theory in an unmagnetized
plasma, i.e., a plasma which has no equilibrium magnetic field. The theory for these waves
also applies to magnetized plasmas in the special situation where allfluid motions are
strictly parallel to the equilibrium magnetic field becausefluidflowing along a magnetic
field experience nou×Bforce and so behaves as if there were no magnetic field.
The two-fluid equation of motion corresponding to an unmagnetized plasma is
mσnσ
duσ
dt=qσnσE−∇Pσ (4.6)and these simple plasma waves are found by linearizing about an equilibriumwhereuσ 0 =
0 ,E 0 = 0, andPσ 0 are all constant in time and uniform in space. The linearized form of