Fundamentals of Plasma Physics

4.2 Two-fluid theory of unmagnetized plasma waves 125

Eq. (4.6) is then

mσnσ 0

∂uσ 1

∂t

=qσnσ 0 E 1 −∇Pσ 1. (4.7)

The electric field can be expressed as

E=−∇φ−

∂A

∂t

, (4.8)

a form that automatically satisfies Faraday’s law. The vector potentialAis undefined with
respect to a gauge sinceB=∇×(A+∇ψ) =∇×A.It is convenient to chooseψso as to
have∇·A= 0.This is called Coulomb gauge and causes the divergence of Eq.(4.8) to give
Poisson’s equation so that charge density provides the only source term for the electrostatic
potentialφ.Since Eq. (4.8) is linear to begin with, its linearized form is just

E 1 =−∇φ 1 −

∂A 1

∂t

. (4.9)

4.2.1 Electrostatic (compressional caves)

These waves are characterized by having finite∇·u 1 and are variously called compres-
sional, electrostatic, or longitudinal waves. The first step in the analysis is to take the
divergence of Eq.(4.7) to obtain

mσnσ 0

∂∇·uσ 1

∂t

=−qσnσ 0 ∇^2 φ 1 −∇^2 Pσ 1. (4.10)

Because Eq.(4.10) involves three variables (i.e.,uσ 1 ,φ 1 ,Pσ 1 )two more equations are re-
quired to provide a complete description. One of these additional equations is the linearized
continuity equation
∂nσ 1
∂t

+n 0 ∇·uσ 1 = 0 (4.11)

which, after substitution into Eq.(4.10), gives

mσ

∂^2 nσ 1

∂t^2

=qσnσ 0 ∇^2 φ 1 +∇^2 Pσ 1. (4.12)

For adiabatic processes the pressure and density are related by

Pσ

nγσ

=const. (4.13)

whereγ= (N+ 2)/NandNis the dimensionality of the system, whereas for isothermal
processes
Pσ
nσ

=const. (4.14)

The same formalism can therefore be used for both isothermal and adiabaticprocesses
by using Eq. (4.13) for both and then simply settingγ= 1if the process is isothermal.
Linearization of Eq.(4.13) gives
Pσ 1
Pσ 0

=γ

nσ 1

nσ 0

(4.15)