Domains of existence of sub-harmonic resonances (13), as a function of Ω₀ = Ω_orb/Ω_s and N₀/Ω_s. In white regions, no waves can satisfy sub-harmonic resonance condition (13). Stars (yellow area): hyperbolic waves ℋ₁. Right slash (purple area): hyperbolic waves ℋ₂. Dots (green area): elliptic waves ℰ₁. Back slash (blue area): elliptic waves ℰ₂. The classical allowable region of tidal instability (for neutral fluids) is −1 ≤ Ω₀ <  3. wave-like domains [ℋ₁, ℋ₂] are illustrated in Fig. 3. Similarly, wave-like domains [ℰ₁, ℰ₂] are illustrated in Fig. 4.