- #1

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- Homework Statement
- Given the relation (treat ##k, \omega## as variables and ##v_0, \omega_p## as constants).

$$y^2=1+2\alpha^2 \pm(1+8\alpha^2)^{1/2}, \ \text{where} \ \alpha:= kv_0/\omega_p, \ y:= \sqrt{2}\omega/\omega_p$$

a) Find the range of wave numbers ##k## for which the oscillation frequency ##\omega## is imaginary

b) Find ##k_{max}## and ##\text{Im}(\omega_{max})##. Hint: sketch the solution for ##\text{Im}(\omega)##

- Relevant Equations
- N/A

Out of the given information I see no way of solving neither of the two sections.

So I did some reading and it turns out that the given relation is a solution of the following function (I won't prove it here)

$$\epsilon (k, \omega) = 1 - \frac 1 2 \left[ \frac{\omega_p^2}{(\omega-kv_0)^2} + \frac{\omega_p^2}{(\omega+kv_0)^2}\right] = 0$$

So I think that finding the range of wave numbers ##k## for which the oscillation frequency ##\omega## is imaginary on the ##\epsilon## function should be equivalent. I did some manipulations

$$\frac{\omega_p^2}{(\omega-kv_0)^2} + \frac{\omega_p^2}{(\omega+kv_0)^2} = 2 \Rightarrow \omega_p^2\left( \frac{\omega^2+2kv_0+k^2v_0^2+\omega^2-2kv_0+k^2v_0^2}{(\omega^2-k^2v_0^2)^2} \right) = 2$$

$$\frac{(\omega^2+k^2v_0^2)}{(\omega^2-k^2v_0^2)^2} = \frac{1}{\omega_p^2}$$

But got stuck in the above step...

I strongly think we better avoid using the function and focus on the given relation to answer the questions.

Might you please guide me through?

Thanks.

So I did some reading and it turns out that the given relation is a solution of the following function (I won't prove it here)

$$\epsilon (k, \omega) = 1 - \frac 1 2 \left[ \frac{\omega_p^2}{(\omega-kv_0)^2} + \frac{\omega_p^2}{(\omega+kv_0)^2}\right] = 0$$

So I think that finding the range of wave numbers ##k## for which the oscillation frequency ##\omega## is imaginary on the ##\epsilon## function should be equivalent. I did some manipulations

$$\frac{\omega_p^2}{(\omega-kv_0)^2} + \frac{\omega_p^2}{(\omega+kv_0)^2} = 2 \Rightarrow \omega_p^2\left( \frac{\omega^2+2kv_0+k^2v_0^2+\omega^2-2kv_0+k^2v_0^2}{(\omega^2-k^2v_0^2)^2} \right) = 2$$

$$\frac{(\omega^2+k^2v_0^2)}{(\omega^2-k^2v_0^2)^2} = \frac{1}{\omega_p^2}$$

But got stuck in the above step...

I strongly think we better avoid using the function and focus on the given relation to answer the questions.

Might you please guide me through?

Thanks.