1. Waves in Plasma
Very short course

2. Outline Introduction – some definitions
Self-consistency, linear theory
Transverse waves in unmagnetized plasma
Longitudinal waves, Landau damping
Waves in magnetized plasma
Beam-plasma instabilities, plasma masers

3. Introduction – some definitions
Self-consistency, linear theory
Transverse waves in unmagnetized plasma
Longitudinal waves, Landau damping
Waves in magnetized plasma
Beam-plasma instabilities, plasma masers

4. What is a wave? A wave is a perturbation propagating in space
a lot of physical phenomena
Gerald B. Whitham, ``Linear and Nonlinear Waves''

5. Nonlinear waves We restrict ourselves by considering only linear waves
Solitary wave
Shock wave
Out of scope of the course
Solitary wave in a lab
We restrict ourselves by considering only linear waves

6. Linear waves Mathematically, a process described by linear equations
Wave equation
is a linear equation
Partial solution in Cartesian coordinates – monochromatic plain wave
Particularly, Maxwell equations in vacuum are linear equations and can be reduced to the wave equation
What about plasma?

7. Plasma is a ``fourth state of a matter'‘ as defined by Russian physicist Frank-Kamenetsky)
Temperature
Solid state
Liquid
Gas
Plasma

8. Most general definition: plasma is an ensemble of charged particles
(less general one includes the condition of quasi-neutrality)
to be described by Maxwell equations
In the ensemble of charged particles, the long-range forces appear.
The motion of each charged particle is determined by electric and magnetic fields, which, in their turn, are created by a large number of other charged particles, i.e., depend on their velocities and locations.
The motion of ions and electrons occurs in self-consistent fields.
The self-consistent long-range e/m fields allow passing information about particles motion from one region to another without collisions. This is the main difference between plasma waves and, for instance, sound waves and waves at a water surface.

9. Self-consistency, linear theory
Introduction – some definitions
Self-consistency, linear theory
Transverse waves in unmagnetized plasma
Longitudinal waves, Landau damping
Waves in magnetized plasma
Beam-plasma instabilities, plasma masers

10. Maxwell equations in vacuum
external current and charge densities do not depend on fields
In plasma, assuming
Only currents and charges remain that depend on fields
How to find this dependence?
 means a sort of particles
Equation of motion
Continuity equation
Simplest model for the self-consistent set of equations

11. Kinetic equation in the self-consistent fields
Distribution function
thermal motion is accounted
natural normalizing
Number density
No particles are born
No particles are died
No collisions (free motion in e/m fields)
Liuville equation:
Vlasov equation:
Vlasov + Maxwell = Vlasov-Maxwell set of equations
One more example is the magnetohydrodynamic model

12. Plasma as a medium Maxwell equations in medium
Dielectric permittivity
For linear waves, only linear terms should be kept for
and 
Accordingly, the permittivity tensor
does not depend on fields
Permittivity tensor contains all information about linear properties of a medium and depends on parameters of the medium steady-state: density, temperature, external fields, etc.

13. Linear theory
Dispersion relation

14. Transverse waves and longitudinal waves
For isotropic medium
Dispersion relation
Dispersion relation
In vacuum
no way!