Thursday 28 June 2018

Integrating Factor Solution for Plasma Wave Equation


As part of a derivation in Bernstein '58 [1] a linear first-order (eqn. (9) in the image) appears:


Bernstein '58



But the general solution I would usually take (as appears in Gradshteyn and Ryzhik and checked in Mathematica) here would be


$$F=c_1G+\frac1G\int^\phi_1\frac{G(\phi^\prime)\Phi(\phi^\prime)}\Omega$$ where $\Phi$ is the RHS of eqn. (9). I suppose it depends on the requirement of periodicity and that $s$ has positive complex part (it comes from a Laplace transform) but I can't see how Bernstein's solution is obtained or equivalent, i.e. where the factor of $1/G$ goes. I'm sure I'm missing something silly, but I've wasted a bit of time on this already and can't get it to work out. The same derivation appears in slightly neater form in Montgomery and Tidman[2] but without further insight. Any help justifying this result would be greatly appreciated.


[1] Bernstein, I. (1958). Waves in a plasma in a magnetic field. Physical Review. Retrieved from http://journals.aps.org/pr/abstract/10.1103/PhysRev.109.10


[2] Montgomery D.C., Tidman, D.A. (1964). Plasma Kinetic Theory, McGraw-Hill advanced physics monograph series




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...