Real and imaginary parts of ω for a standing ion-acoustic wave derived from hybrid simulations as a function of the amplitude of the Alfvénic pump. For the top curve β_e = 6 × 10⁻² and β_p = 10⁻², while for the bottom curve β_e = 10⁻¹ and β_p = 1.25 × 10⁻². In this simulation k₀ = 2π/L. The attenuation is calculated from the Fourier analysis in space and the power of the mode m′ = 2 (or m = 4). The horizontal dashed line corresponds to the expected real and imaginary frequency of the corresponding non-driven ion-acoustic mode attenuated by LD and calculated from the numerical solution of Eq. (19).