Parallel streaming instabilities and the dispersion relations of ion-acoustic waves. (a) The electron and ion distribution functions of the electron-ion streaming instability. For the electron distribution we use n_e = 30 cm⁻³, T_e = 15 eV, and V_e = 0.25v_e and for the ion distribution we use n_i = 30 cm⁻³, T_i = 3 eV. (c) The distribution for the electron-electron-ion streaming instability, consisting of core electrons and ions, and an electron beam. For core electrons we use n_e = 28 cm⁻³ and T_e = 15 eV, for core ions n_i = 30 cm⁻³ and T_i = 3 eV, and beam electrons n_b = 2 cm⁻³, T_b = 2 eV, and V_b = v_b, where V_b and v_b are the bulk and thermal speeds of the electron beam. (e) The distribution for the ion beam-driven instability, consisting of core electrons and ions and an ion beam. For core electrons we use n_e = 30 cm⁻³, T_e = 15 eV, for core ions n_i = 25 cm⁻³ and T_i = 2 eV, and beam ions n_b = 5 cm⁻³, T_b = 2 eV, and V_b = 4v_b. Panels a, c, and e: the vertical dashed line indicates the phase speed of the unstable mode where the growth rate peaks. Panels b, d, and f: dispersion relations calculated from the distributions in panels a, c, and e, respectively. The solid black lines show the dispersion relations in the plasma rest frame or θ_Bx = 90°, while the dashed black lines show the dispersion relations Doppler shifted due to the solar wind flow past the spacecraft for θ_Bx = 0° and V_sw = 350 km s⁻¹. The red curves show the growth rate γ. In panel f the higher-frequency dispersion relation is stable γ < 0 for all k, while the lower-frequency mode is unstable.