134 Chapter 4. Elementary plasma waves
4.3.3 Fast mode (mode whereBz 1 =0)
In this case only thezcomponent of Faraday’s law is used and after crossing the top equa-
tion withz,ˆEqs.(4.51) become
∂
∂tE⊥ 1 ׈z = vA^2(
ˆz×∂B⊥ 1
∂z+∇⊥Bz 1 ׈z)
×zˆ=v^2 A(
∂B⊥ 1
∂z−∇⊥Bz 1)
∂B 1 z
∂t= −zˆ·∇⊥×E⊥ 1 =−∇·(E⊥ 1 ×zˆ). (4.59)Taking a time derivative of the bottom equation and then substituting forE⊥ 1 ×zˆgives
∂^2 B 1 z
∂t^2=−vA^2 ∇·(
∂B⊥ 1
∂z−∇⊥Bz 1)
. (4.60)
However, using∇·B 1 =0it is seen that
∇·B⊥ 1 =−
∂Bz 1
∂z(4.61)
and so the fast wave equation becomes
∂^2 B 1 z
∂t^2= −v^2 A(
∂∇·B⊥ 1
∂z−∇^2 ⊥Bz 1)
= −v^2 A(
−
∂^2 Bz 1
∂z^2−∇^2 ⊥Bz 1)
= vA^2 ∇^2 Bz 1. (4.62)4.3.4 Comparison of the two modes
The slow mode Eq.(4.58) involveszonly derivatives and so has a dispersion relation
ω^2 =k^2 zvA^2 (4.63)whereas the fast mode involves the∇^2 operator and so has the dispersion relation
ω^2 =k^2 v^2 A. (4.64)The slow mode hasBz 1 =0and so its perturbed magnetic field is entirely orthogonal to
the equilibrium field. Thus the slow mode magnetic perturbation is entirely perpendicular
to the equilibrium field and corresponds to a twisting or plucking of the equilibrium field.
The fast mode hasBz 1 =0which corresponds to a compression of the equilibrium field as
shown in Fig.4.2.