## Monday, September 13, 2021

### Deriving the Landau Damping Rate Of an E-Field (2)

We left off with:

x(t')   =   x o   +  ò t’ o v (t") dt"

Now define:

d v(t’)   =    v(t)  -   v o  =  (-e)E o/ m e  ò t’’  o    dt’ sin kx (t’)

è

d v(t’)   =    v o  -  (-e)E o/ m e  ò t’’  o    dt”’ sin kx (t’”)

Where:  x   =   x o   +  v o  (t"’)

v   (t") =

v o  +  (e)E o/ m e  k v o  [ cos(kx o) + k v o t”) - cos(kx o)]

And the next to last integration yields:

x (t’)  = x o   +  v o  t’  -  t’ eE o/ m e  k v o  cos (kx o)

+  (e)E o/ m e  k 2 v o 2 [sin (kx o) + k v o t’) – sin (kx o)]

Note the first integrand is of the form:

sin (kx o) + k v o t’)   +    D^ ]

Where:

D^   =    t’ e E o/ m e v o  cos (kx o)

+ eE o/ m e  k v o 2   [sin kx o) + k v o t’)  -   sin kx o]

Given we seek the lowest order correction to E o  we can use Taylor expansion  of the form:

sin (a +  D^)  =  sin a  +   D^ cos a

to lowest order in D^.    Then:

d v(t)   =

- e E o/ m e   ò t’  o dt [D^cos (kx o) + k v o t’)

+   sin (kx o + k v o t’)]

We then average d v   over one wavelength upon which the sine term disappears.  The other terms are evaluated using the identities:

<sin (u – a) cos (u – b)>u  =   -  ½  sin (a- b)

<sin (u – a) sin (u – b)>u  =  <cos (u – a) cos (u – b)>u  =

½  cos (a - b)

Where < > means an average over one period of the variable u.  From this we find:

<d v(t) )>u   =

(e E o/ m e ) 2  k/2 ò t’o    dt’ {- 1/ k2  v o 2   sin (k v o t’)  +

t’/ m e  k v o   + cos (k v o t’)

Integration yields:

<d v(t) )>u   =   (e E o/ m e ) 2  ½ k2  v o 2   | 2 | cos (k v o t) –

1] +   k v o t  sin (k v o t) ]

(To be continued)