Estimating the damping coefficient of a wave assuming a very small ratio


by loto
Tags: approximations, plasma, waves
loto
loto is offline
#1
Nov14-11, 07:38 PM
P: 17
1. The problem statement, all variables and given/known data
The original problem is determining the dispersion relation of an ordinary wave in plasma damped by collisions. That part was easy enough but the next part is to find the damping rate (-Im(ω)) of the wave, assuming k is real and [itex]\nu << \omega[/itex] where [itex]\nu[/itex] is the collision frequency.

2. Relevant equations

I've found the dispersion relation to be:

[itex]\omega^2 - k^2c^2 = \frac{\omega_{pe}^2(1-i\frac{\nu}{\omega})}{(1 + \frac{\nu^2}{\omega^2})}[/itex]

And we are told the damping rate is:
[itex]\gamma = \frac{\omega_{pe}^2\nu}{\omega^2 2}[/itex]

3. The attempt at a solution
Since we want only the negative of the imaginary part:

[itex]\gamma = \frac{\nu\omega_{pe}^2}{\omega^2 (1+\frac{\nu^2}{\omega^2 })}[/itex]

However, I can't think of an approximation that would give me that factor of 1/2. Series expansion or small number approximations don't seem to do it. If anyone has ideas, I'd just like a push in the right direction?
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
AlephZero
AlephZero is offline
#2
Nov17-11, 05:16 PM
Engineering
Sci Advisor
HW Helper
Thanks
P: 6,341
Apologies if this is irrelevant, but some people write the equation for damped oscillations as
[itex] \ddot x + b\omega \dot x + \omega^2 x = 0[/itex]
and other people write it as
[itex] \ddot x + 2 b\omega \dot x + \omega^2 x = 0[/itex]
I deliberately wrote [itex]b[/itex] in those equations rather than the usual greek letters, because I don't know what notation convention is used in plasma dynamics!

Assuming you got the first part of the question right, is that where your factor of 2 has come from?


Register to reply

Related Discussions
Neper freq, Damping factor, ratio, coefficient confusion. Electrical Engineering 0
Damping ratio from transfer function Classical Physics 2
Damping Ratio Classical Physics 1
Gilbert Damping Ratio in Ferromagnets Atomic, Solid State, Comp. Physics 0
Damping & Damping Coefficient Atomic, Solid State, Comp. Physics 0