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23. WAVES IN A UNIFORM MEDIUM: SPECIAL CASES
We consider the special case of an infinite, uniform medium with B0  B0 êz and
J 0  0 . In that case we can expand an arbitrary displacement in plane wave solutions as
 r,t     k ei kr  k t  ,
(23.1)
k
where k is the wave vector, and the addition of the complex conjugate is implied. When
we substitute this into the ideal MHD wave equation, we find that   ik and
 / t  i k , so that the problem is reduced to an algebra. The result of the substitution
is



 2   VA2 k   k    b̂   b̂  CS2 k    ,
(23.2)
where VA2  B02 / 2 0 0 is the square of the Alfvén speed and CS2  p0 / 0 is the square
of the sound speed, and b̂( ê z ) is a unit vector in the direction of the equilibrium
magnetic field. This represents three coupled homogenous equations of the form
A   2I    0 in the three unknowns x , y , and z . It has non-trivial solutions when




det A   2I  0 .
(23.3)
Equation (23.3) is the dispersion relation for waves in a uniform medium. The solutions
 2 (k) are the characteristic vibrational frequencies of the system
Expanding the vector identities, Equation (23.2) can be rewritten as
 


  
V k  b̂ k   b̂
 2  V 2 k  b̂ 2     C 2  V 2 k     V 2 k  b̂   b̂  k
A
A
A
 S



2
A
.
(23.4)
This can be put in another form by defining a unit vector in the direction of k as k  kk̂ ,
and defining  as the angle between the direction of propagation and the magnetic field,
k̂  b̂  cos . The result is
 2
2
2 
2
2
2
 k 2  VA cos      CS  VA k̂      b̂ VA cos  k̂



   
 
 k̂   VA2 cos b̂ .
(23.5)
Equation (23.5) contains three indepednent vectors: the displacement  , the
direction of propagation k̂ , and the direction of the magnetic field b̂ . The properties of
the waves depend on their relative orientation. In this Section we will consider several
special cases.
Case I. Transverse Waves.
1
Here the displacement is perpendicular to the direction of propagation, or k̂    0 ,
as shown in the figure.
Then Equation (23.5) becomes
 2
2
2 
2
 k 2  VA cos       b̂ VA cos k̂  0 .


 
(23.6)
Since  and k̂ are linearly independent, their coefficients must vanish individually.
Case IA. We first examine the vanishing of the coefficient of k̂ ,
  b̂V
2
A
cos  0 .
(23.7)
There are two possibilities.
1.   b̂  0 . If cos  k̂  b̂  0 , then b̂ must be perpendicular to  , so  is
perpendicular to both k̂ and b̂ , as shown in the figure.
2. cos  k̂  b̂  0 , or    / 2 . Then k̂ , b̂ , and  are mutually
perpendicular. This is just a special case of 1, above. Together, 1 and 2
determine the polarization of the wave.
Case IB. The vanishing if the coefficient of  in Equation (23.6) leads to

k
 VA cos  ,
(23.8)
or
  kPVA ,
(23.9)
where kP  k cos is the component of k parallel to the magnetic field. The
polarization of this wave is given by IA1 and IA2, above. It is a transverse wave that
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propagates along the magnetic field. It does not propagate across the magnetic field.
This is another manifestation of the anisotropy introduced by the presence of the
magnetic field. Its phase velocity is  / kP= VA . Note that if there is no magnetic field,
VA  0 and the wave does not exist. It only occurs in magnetized fluids. This is the
famous Alfvén wave for which Hannes Alfvén won the Nobel Prize in 1970. If only it
were so easy!
The behavior of the perturbed magnetic field in the Alfvén wave is found from
Faraday’s law and Ohm’s law. In this case, the former is
 B1  k  E1 ,
(23.10)
and the latter is

E1  i B0   b̂
.
(23.11)
These can be combined as
   
B1  ikB0   k̂  b̂  b̂ k̂    .
(23.12)
The last term on the right hand side vanishes because the wave is transverse in this case,
i.e., k̂    0 . Then B1  ikB0 k̂  b̂   ikB0 cos . The perturbed magnetic field B1 is
 
thus in the same direction as  , but is  / 2 out of phase (due to the i ).
Case II. Longitudinal Waves.
For a longitudinal wave the displacement is parallel to the direction of propagation,
or    k̂ , as shown in the figure.
After some algebra, Equation (23.5) becomes
 2
2
2 
2
 k 2  CS  VA  k̂  VA cos b̂  0 .




(23.13)
We consider the following special cases.
Case IIA. Here cos  k̂  b̂  0 , so that k̂ is perpendicular to b̂ and    / 2 .
Then Equation (23.13) reduces to
   CS2  VA2  k .
1/2
(23.14)
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This is a longitudinal wave that propagates across the magnetic field. The square of its
phase velocity is the sum of the squares of the sound speed and the Aflvén speed. It is
called the magneto-acoustic (or MA) wave.
The perturbed magnetic field is found from Equation (23.12) with k̂  b̂  0 and
k̂     :
B1  ikB0b̂ .
(23.15)
The perturbed field is parallel to the mean field and  / 2 out of phase with the
displacement. The pertubed field thus reinforces the mean field during part of the cycle,
and weakens it during another part. This causes a perturbed magnetic pressure that acts
in the same manner as the perturbed fluid pressure. It can support a longitudinal wave.
Case IIB. In this special case, b̂ , k̂ , and  are all parallel. The Equation (23.13)
becomes
  kCs .
(23.16)
The perturbed magnetic field vanishes:
   
B1  ikB0   k̂  b̂  b̂ k̂     ikB0 cos  cos   0 .
(23.17)
This is a sound wave propagating parallel to the magnetic field.
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