Exercises

1. Derive the dispersion relation (7.28) from Equations (7.23)–(7.27).

2. Show that the dispersion relation (7.28) can be written
   $$1-\frac{1}{x} - \frac{3\,y^{2}}{x^2} + {\rm i}\,\epsilon(x,y) = 0,$$
   where $x=(\omega/{\mit\Pi}_e)^2$, $y=k\,\lambda_D$, ${\mit\Pi}_e=(n\,e^2/\epsilon_0\,m_e)^{1/2}$, $\lambda_D =(T_e/m_e\,{\mit\Pi}_e^{2})^{1/2}$, and
   $$\epsilon(x,y) = \left(\frac{\pi}{2}\right)^{1/2}\frac{x^{1/2}}{y^{3}}\,\exp\left(-\frac{x}{2\,y^{2}}\right).$$
   Demonstrate that, in the limit $y$, $\epsilon \ll 1$, the approximate solution is
   $$x\simeq 1 + 3\,y^{2}-{\rm i}\,\epsilon(1,y).$$

3. Show that, when combined with the Maxwellian velocity distribution (7.24), the dispersion relation (7.23) reduces to
   $$1- \frac{Z'(\zeta) }{2\,(k\,\lambda_D)^2}= 0,$$
   where $\zeta = (\omega/{\mit\Pi}_e)/(k\,\lambda_D)/\!\sqrt{2}$, ${\mit\Pi}_e=(n\,e^2/\epsilon_0\,m_e)^{1/2}$, $\lambda_D =(T_e/m_e\,{\mit\Pi}_e^{2})^{1/2}$, and $Z(\zeta)$ is the plasma dispersion function. Hence, deduce from the large argument asymptotic form of the plasma dispersion function that
   $$-2\,{\rm i}\,\pi^{1/2}\,\zeta\,{\rm e}^{-\zeta^{2}} + \frac{1}{\z... ...{2\,\zeta^{4}} + {\cal O}\left(\frac{1}{\zeta^{6}}\right)= 2\,(k\,\lambda_D)^2$$
   in the limit $k\,\lambda_D\ll 1$. Show that the approximate solution of the previous equation is
   $$\frac{\omega}{{\mit\Pi}_e}= \sqrt{2}\,(k\,\lambda_D)\,\zeta \sime... ...1/2}\frac{1}{(k\,\lambda_D)^3}\exp\left[-\frac{1}{2\,(k\,\lambda_D)^2}\right].$$

4. Show that, when combined with the Maxwellian velocity distribution (7.24), the dispersion relation (7.49) reduces to
   $$1-\frac{Z'(\zeta_e)}{2\,(k\,\lambda_{D\,e})^2} - \frac{Z'(\zeta_i)}{2\,(k\,\lambda_{D\,i})^2} = 0,$$
   where ${\mit\Pi}_s=(n\,e^2/\epsilon_0\,m_s)^{1/2}$, $\lambda_{D\,s}=(T_s/m_s\,{\mit\Pi}_s^{2})^{1/2}$, $\zeta_s= (m_s/2\,T_s)^{1/2}\,\omega/k$, and $Z(\zeta)$ is the plasma dispersion function. Use the large-argument expansion of the plasma dispersion function for the ions,
   $$Z'(\zeta_i)\simeq - {\rm i}\,2\!\sqrt{\pi}\,\zeta_i\,{\rm e}^{-\zeta_i^{2}} + \frac{1}{\zeta_i^{2}},$$
   and the small-argument expansion for the electrons,
   $$Z'(\zeta_e) \simeq - {\rm i}\,2\!\sqrt{\pi}\,\zeta_e\,{\rm e}^{-\zeta_e^{2}} - 2.$$
   Substituting these expansions into the dispersion relation, writing $\omega=\omega_r + {\rm i}\,\gamma$, where $\omega_r$ and $\gamma$ are both real, and $\vert\gamma\vert\ll \omega_r$, demonstrate that
   $$\frac{\omega_r}{k} \simeq \left(\frac{T_e}{m_i}\right)^{1/2}\frac{1}{\left[1+(k\,\lambda_{D\,e})^{2}\right]^{1/2}},$$
   and
   $$\frac{\gamma}{\omega_r} \simeq -\frac{(\pi/8)^{1/2}}{\left[1+(k\,... ...ac{T_e}{2\,T_i}\,\frac{1}{\left[1+(k\,\lambda_{D\,e})^2\right]}\right)\right].$$

5. Derive Equation (7.74) from Equations (7.69) and (7.73).

6. Derive Equation (7.82) from Equations (7.74) and (7.79).

7. Derive Equations (7.89)–(7.91) from Equation (7.82).

8. Derive Equations (7.94)–(7.96) from Equation (7.82).

9. Derive Equations (7.102)–(7.105) from Equation (7.82).

10. Derive Equation (7.119) from Equation (7.118).

11. Derive Equation (7.120) from Equations (7.79) and (7.119).

12. Derive Equation (7.124) from Equation (7.123).

13. Demonstrate that the distribution function (7.131) possesses a minimum at $u=0$ when $v_e < \sqrt{3}\,V$, but not otherwise.

14. Verify formula (7.133).

15. Consider an unmagnetized quasi-neutral plasma with stationary ions in which the electron velocity distribution function takes the form
    $$F_0(u) = n_e\,\frac{v_e}{2\pi}\left[\frac{1}{v_e^{2}+(u-V)^{2}} + \frac{1}{v_e^{2}+(u+V)^{2}}\right].$$
    Demonstrate that the dispersion relation for electrostatic plasma waves can be written

    $$k^{2} = {\mit\Pi}_e^{2}\,\frac{v_e}{2\pi}\left[\int_{-\infty}^\infty \f... ...\int_{-\infty}^\infty \frac{du}{(u-\omega/k)^2\,[v_e^{2} + (u+V)^{2}]} \right],$$

    where ${\mit\Pi}_e =(n_e\,e^2/\epsilon_0\,m_e)^{1/2}$. Assuming that $k$ is real and positive, and that $\omega/k$ lies in the upper half of the complex plane, show that when the integrals are evaluated as contour integrals in the complex $u$-plane (closed in the lower half of the plane), making use of the residue theorem (Riley 1974), the previous dispersion relation reduces to
    $$2 = {\mit\Pi}_e^{2}\left[\frac{1}{(k\,V - \zeta)^{2}}+ \frac{1}{(k\,V+\zeta)^{2}}\right],$$
    where $\zeta= \omega+{\rm i}\,k\,v_e$. Finally, in the small-$k$ limit, $k\ll {\mit\Pi}_e/V$, demonstrate that the growth-rate of the most unstable mode is
    $$\gamma\equiv -{\rm i}\,\omega \simeq k\,(V-v_e).$$

16. Derive Equation (7.137) from Equations (7.134)–(7.136).

17. Derive Equations (7.160) and (7.161) from Equations (7.154), (7.155), and (7.159).