Having seen how Landau damping scuttles the notion of ATTIP military -sighted UFOs as holograms (designed to confuse pilots), e.g.
It is instructive to see how the Landau damping rate is derived. We recall here that Landau damping is a result of a solution to the dispersion relation involving w and k, i.e. the phase velocity resulting is v f = w / k. We proceed then by considering the effect of a small oscillating (Langmuir) wave associated with a perturbed plasma distribution f1(x,v,t) = 0.
Consider then a linear wave of form:
E1 (x’,t) = E o sin (kx’ - w r t)
Where x’ denotes the lab frame of reference. E o is a small constant for amplitude. In the frame of reference x moving with phase speed v f = w / k > 0 with respect to the laboratory frame the wave field is independent of time t and is given by: E1 (x’,t) = E o sin (kx’) This is represented in the graphic below. (E o amplitude position marked by bold horizontal line)
Note that all the particles in the background distribution plasma (f o (v)) are affected by this non-time dependent E-field. Some are speeded up, others slowed down.
For simplicity, we focus on only those that have speed v o in the lab frame at t = 0. Hence, they have speed v o - v f in the frame moving with the wave phase speed v f = w / k. (See top graphic.)
Change in energy (d E ) of a given particle:
d E = ½ (m v o + d v) 2 - ½ m v o2
= m v o d v + O (d v 2)
where we ignore the term in d v 2
Averaging the energy change over one wavelength:
< d E >xo = m v o < d v > xo
Since d v can be calculated in any frame, we work in the wave frame. Then the change in velocity is:
d v = v(t) - v o = (-e) E o / m òt o dt’ sin kx(t) dt’
x(t) must be along particle orbit so:
x(t’) = x o + òt’ o v(t”) dt”
Examine again the equation of motion:
mx” = - e E o sin kx
For which we may write:
mx’ x” = - e E o sin kx (dx/ dt)
d/dt (½ m x’2) = -e E o/ k [ d/dt (cos kx)]
x’ = Ö 2 e E o/ km [cos kx - cos kx o ]
Now integrate both sides:
dx/[cos kx - cos kx o ] ½ = Ö (2 e E o/ km) dt
We subst. x = x o + v o t"' into integral:
òt’’ o sin (kx o + k v o t"') dt’" =
- 1/k v o [cos (kx o + k v o t"') - cos kx o ]
v(t") = v o + e E o /ne kv [cos (kx o + k v o t"') - cos kx o ]
The preceding parameter to be subst. into:
x(t') = x o + òt’ o v (t") dt"
(To be continued)