1. paraxial approximation
microwave horn

2. z Eq Hj  r
I DL y j x

3. far field

4. x y z  ’
x’, y’ a b
uniform illumination

6. Be careful with the procedure of calculating the limits on the integration – this is an anomaly and it is not a real effect!

7. b a
microwave horn

8. b a
p. 624 ?

9. b a
limits??

16. plasma

17. Waves in plasmas– high frequency
Only electrons can move! Mi > > me

18. Waves in plasmas– high frequency

19. Waves in plasmas– high frequency
Derive a wave equation
 > pe

20. Microwave experiment x z y a b
I  ne E E ne r
resonances

21. Waves in plasmas– low frequency – normalized variables
Equation of continuity for the ions
Equation of motion for the ions
Electrons can respond quickly to electric fields and we can assume that their density is given by a Maxwell-Boltzmann relation.
Poisson’s equation
Electrostatic approximation

22. Waves in plasmas– low frequencyb

23. Linear experiments Vacuum chamber to reduce collisions plasma creation
Ion acoustic wave excitation, propagation, and detection

24. Waves in plasmas– low frequencyz t
???

25. Nonlinear fluid equations for ion motion
Mel Widner Ph.D. thesis

26. What about the following experiment?
Reflection ???

27. w

28. Nonlinear ion acoustic solitons
Scott-Russell experiment 1834
KdV equation 1895
Recurrence calculation 1960
Nonlinear plasma equations  KdV equation
Plasma experiments

29. Plasma experiment

30. Experimental results

32. In order to minimize this energy, V must be a solution of Laplace’s equation.
Let there be another solution U that satisfies the boundary conditions. Linear media implies superposition!

33. Either V is specified (U= 0) or the normal derivative of V = 0.