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B
0
B
0
β<1
β>1
c
s
V
F
c
A
c
s
c
A
V
F
V
A
V
A
k
V
S
k
V
S
π/2−θ
π/2−θ
(c
A
+c
s
)
1/2
(c
A
+c
s
)
1/2
Fig. 4.2.
Polar diagrams of phase velocities for the Alfven and magnetosonic waves
for the magnetosonic waves. Solutions for these equations are
V
ph
=
V
A
=
c
A
|
cos
θ
|
,
(4.41a)
⎡
⎤
1
1
/
2
2
1+
β
2
2
±
β
2
−
⎣
⎦
+
β
2
sin
2
θ
V
±
=
c
A
.
(4.41b)
2
The root
V
F
=
V
+
given by the upper sign corresponds to the FMS-wave, and
by the lower sign
V
S
=
V
−
to the Slow MagnetoSonic (SMS) wave.
Dependencies of phase velocities of an Alfven wave
V
A
, an FMS-wave
V
F
and an SMS wave
V
S
on the angle between the wave vector and the
external magnetic field are given by the polar diagrams in Fig. 4.2 (left panel
at
c
A
>c
s
, right panel at
c
A
<c
s
). The polar for an FMS-wave is an oval
compressed along the magnetic field direction, and for an SMS-wave it has the
shape of two osculating ovals compressed transversely. At
β
→
0 the FMS-
polar is transformed into a circle of radius
c
A
, while the SMS-polar turns
into two osculating circles of radius
c
s
. In the limiting case of incompressible
liquid, when
β
, the polar diagram for the SMS is transformed into two
osculating circles of diameter
c
s
. A similar diagram corresponds to an Alfven
wave at any
β
.
→∞
4.3 Inhomogeneous Plasma
Basic Equations Cold Plasma
If a wavelength is significantly less than the characteristic scale of spatial in-
homogeneity in plasma, the wave propagation can be described within the
ray approximation, which is suciently universal and applicable to many in-
teresting wave phenomena. However, a number of important effects observed
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