Example of the dispersion relation of Langmuir waves driven by the bump-on-tail instability using Eq. (2). (a) The distribution function used to calculate the dispersion relation (black line). The vertical dashed black line shows the v_ph of the wave correspond to maximum growth. (b) Dispersion relation of Langmuir waves (black line) and associated growth rate (red line) shifted up in frequency by ω_pe. The upper and lower dashed lines show the Doppler-shifted dispersion dispersion relation for k aligned and anti-aligned with s solar wind flow of 350 km s⁻¹. The blue shaded region shows the expected range of k and frequencies for Langmuir waves.