Real and imaginary parts of ω for a standing ion-acoustic wave derived from hybrid simulations (circles) as a function of the plasma-β for electrons. The continuous line corresponds to the numerical solution to the dispersion relation given by Eq. (19), while the dashed line corresponds to the analytical approximation given by Eqs. (20) and (21). In this plot β_p = 10⁻¹, L = 20 d_i (d_i is the ion-inertial length), and k = 2 × 2π/L.