Influence of the beam density on the wave growth and fractional bandwidths, investigated by varying the beam-to-background density ratio r_n from 10⁻⁴ to 1: Panel a: maximum growth rates ω_i for different beam-to-background density ratios r_n obtained by solving the linear dispersion relation (dots) and from simulations (crosses) with fit function (solid line). Panel b: integrated growth rates calculated from numerical solutions of the dispersion relation (dots) and from simulation results (crosses) with fit function (solid line). Panel c: fractional bandwidth Δω_r at the point where ω_i is at its maximum, normalised to the corresponding real part of the frequency ω_r, obtained by linear theory (dots) and simulation (crosses). All three figures are for ρ₀ = ρ₁ = 1 and γ_b = 26. Details regarding the fit functions can be found in Table 1.