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

• loto
In summary, the conversation is about determining the dispersion relation and damping rate of an ordinary wave in plasma damped by collisions. The dispersion relation has been found to be \omega^2 - k^2c^2 = \frac{\omega_{pe}^2(1-i\frac{\nu}{\omega})}{(1 + \frac{\nu^2}{\omega^2})} and the damping rate is given by \gamma = \frac{\omega_{pe}^2\nu}{\omega^2 2}. The question is about finding an approximation that would give a factor of 1/2 in the damping rate equation. There is also a discussion about the different notation conventions used in plasma dynamics.
loto

## Homework Statement

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 $\nu << \omega$ where $\nu$ is the collision frequency.

## Homework Equations

I've found the dispersion relation to be:

$\omega^2 - k^2c^2 = \frac{\omega_{pe}^2(1-i\frac{\nu}{\omega})}{(1 + \frac{\nu^2}{\omega^2})}$

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

## The Attempt at a Solution

Since we want only the negative of the imaginary part:

$\gamma = \frac{\nu\omega_{pe}^2}{\omega^2 (1+\frac{\nu^2}{\omega^2 })}$

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?

Apologies if this is irrelevant, but some people write the equation for damped oscillations as
$\ddot x + b\omega \dot x + \omega^2 x = 0$
and other people write it as
$\ddot x + 2 b\omega \dot x + \omega^2 x = 0$
I deliberately wrote $b$ 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?

## What is the damping coefficient of a wave?

The damping coefficient of a wave is a measure of how quickly the amplitude of the wave decreases over time. It is typically denoted as "b" and is often expressed in units of kg/s.

## How do you estimate the damping coefficient of a wave assuming a very small ratio?

To estimate the damping coefficient of a wave assuming a very small ratio (less than 1), you can use the formula b = (2πf)/Q, where f is the frequency of the wave and Q is the quality factor. This assumes that the damping ratio is significantly smaller than 1, which is often the case for many real-world systems.

## What is the significance of the damping coefficient in wave behavior?

The damping coefficient plays a crucial role in determining the behavior of a wave. A higher damping coefficient means that the wave will lose energy more quickly and have a shorter duration. On the other hand, a lower damping coefficient allows the wave to persist for a longer time and travel a greater distance.

## How does the damping coefficient affect the amplitude of a wave?

The damping coefficient has a direct impact on the amplitude of a wave. As the damping coefficient increases, the amplitude of the wave decreases more rapidly. This means that a wave with a higher damping coefficient will have a smaller maximum amplitude compared to a wave with a lower damping coefficient.

## Can the damping coefficient of a wave be negative?

No, the damping coefficient of a wave cannot be negative. This is because the damping coefficient is a measure of how quickly the amplitude of a wave decreases over time, and a negative value would imply that the amplitude is increasing. In reality, the amplitude of a wave can only decrease due to factors such as friction and resistance.

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