Projection of the distribution of the frequency in the planes R = R₀, k_R = 0.2k₀, and k_z = 0.2k₀ for an idealized filament inspired by TMC-1, with R₀ = 0.043 pc and k₀ = 74 pc^-1. The color scale is normalized to its maximum for positive values of and to its minimum for negative values. The contours are equally spaced. Negative values of correspond to an unstable filament, whereas the filament should be stable in the region where is positive. The solid black curve separates the stable and unstable regimes, and corresponds to k = k_crit. This threshold corresponds to a length scale λ_crit = 2π/k_crit = 0.13 pc at R = R₀. The dashed line corresponds to the minimum value of , i.e., to the most unstable mode, and is obtained from equation 25 for the first panel. It intersects the line k_z = k_R at k_R,threshold = 0.31k₀, which corresponds to λ_threshold = 0.28 pc.