Page 45 - swp0000.dvi

2
(1)
 (1) = −     0  
 1 2 1
  
(1)
 (1) = −    
 1 1
 
2
    2
(1) 0 (1)
 =  
 1 2 2 1
3  0 −     0
(1)    0 (1)
 =   (2.7)
 1 2 2 1
3  0 −     0
From Poisson’s equation, the linear dispersion relation reads:

∙ µ ¶ ¸
2   2  0 ( − 1) 0
2
  0
2
4 2  2  +  0 − + −  =0
2 2 2  
       0 − 3  0
(2.8)
(1) (1) (1)
The first-order zeroth-harmonic  ,  ,and  would be set equal to
0 0 0
(1) (1)
zero, which results in  =  =0.
0 0
The second-order ( =2) perturbed quantities with  =1, are given by

∙ ¸
(1)

   0  (2) − 2  1 ( −   )
1
 (2) = −  (2.9)
 1 3
  
∙ ¸
(1)
(2) 
  − 1 +   1 ( −   )
 (2) =  (2.10)
 1 2
  

∙ µ ¶¸
(1)
3
2
   0 3  0  (2) +    0 − (2) +2  1 ( −   )
1
1

 (2) =  (2.11)
(    0 − 3  0 )
 1 2 2 2
33