1. Eikonal Method for the Ideal MHD
[2] In the geometric acoustics short-wave
perturbations are described by functions
of the form
f( r, t)=A( r, t) exp(iS( r, t)),
where
A is called the wave amplitude and
S is called the eikonal.
For the monochromatic waves the eikonal has been taken as
S( r, t)=w (t ( r)-t).
The eikonal method may be used when the wavelengths are
small as compared with equilibrium state scale
l0.
Let
v A0 and
cs0 be the scales of Alfvén and sound velocities,
then the frequency should satisfy the following conditions
1/w 
l0/v A0 and
1/w l0/cs0, i.e.
the frequency should be high. The function
t ( r) is also called the eikonal.
For the MHD-waves it is derived from the equations
 | (1) |
where
P0=P0( r),
r 0=r 0( r),
B0= B0( r) are the equilibrium state.
The first equation corresponds to the Alfvén wave,
the second gives the fast and slow magnetosonic waves.
In the small plasma beta approximation the eikonal equation
for the fast waves is
 | (2) |
The zero order approximation for the Alfvén waves is
 | (3) |
where
A is some coefficient that is found from the solvability
condition of the equations for the first order approximation
The zero order approximation for the magnetosonic waves is
The equation for the coefficient
A in case of the fast waves is
Here plus and minus are taken for the fast and slow waves respectively.
This results allow us to draw the conclusion that the short-wave
perturbations in nonuniform mediums have the main properties of the MHD-waves
in uniform mediums. The Alfvén and the magnetosonic waves are
not coupled. The Alfvén and the slow waves do not propagate
across the equilibrium magnetic field.
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