Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 85

3. Slow magnetosonic waves: ion sound which at β<< 1 almost does not
disturb the magnetic field. The plasma is seemingly 1D-density perturba-
tions caused by longitudinal plasma displacements are transmitted along
the magnetic field at sound velocity c s .

If the parameter β is finite, magnetosonic waves cannot be precisely sepa-
rated into the FMS and ion sound. Coupling equations (4.33) and (4.34) must
be investigated. Let us apply Fourier transform decomposing the signal into
its harmonic components

ξ ⊥ exp( − iωt + i kr ) d b exp( − iωt + i kr ) .

Equations (4.27), (4.28) for Alfven waves and (4.33), (4.34) for FMS-waves
reduce to:
Alfven waves

k 2 ξ ⊥ =0 ,

ω 2
c 2 A

−

( k · ξ ⊥ )=0 . (4.37)

Magnetosonic waves

βξ + ω 2

b B 0

ik k 2 k 2 βk 2

⊥

c 2 A −

− =0 , (4.38a)

⊥

ω 2
c s −

ξ −

b B 0

k 2 ik

=0 ,

(4.38b)

where k ⊥ and k are the values of the perpendicular and parallel to the mag-
netic field B 0 components of the wave vector k .
A uniform linear system of algebraic equations has a non-trivial solution
only in the case of the zero determinant ∆ of this system. Since the deter-
minant ∆

∆ ( ω, k ) is a function of ω and k , this condition results in a
functional relation between frequency ω and wave vector k , designated as the
dispersion equations which are given by
Alfven waves

≡

ω 2 k 2 c 2 A =0 . − (4.39)

Magnetosonic waves

− c 2 A + c s k 2 ω 4 ω 2 + c 2 A c s k 2 k 2 =0 .

(4.40)

A presentation of magnetohydrodynamic waves in homogeneous plasma
can be given in the form of an angle dependency of phase velocity V ph = ω/k
on angle θ between k and B 0 . Then substitution k = k cos θ into (4.39) and
(4.40) leads to

V ph = V A = c 2 A cos 2 θ

for the Alfven waves and

± − c 2 A + c s V 2

V 4 + c 2 A c s cos 2 θ =0

±