(a) Effective mass as a function of the angular frequency in the propagating A and O regions and in the BG region. Dotted black lines: harmonic regime in the propagating region. Solid red lines: hardening anharmonic regime in the propagating region. Solid green lines: softening anharmonic regime in the propagating region. Dotted red lines: harmonic regime in the BG region. Solid blue lines: hardening anharmonic regime in the BG region. Solid cyan lines: softening anharmonic regime in the “BG” region. (b) Representation of the complex wave vector (κ = π/L + i Im[κ]) in the complex plane for 1) M_eff ⁺ >0 Kg and k⁽⁴⁾ > 0 N/m³ and for 2) M_eff ⁻ < 0 Kg and k⁽⁴⁾ < 0 N/m³. (c) As in panel (b) but for 3) M_eff ⁺ >0 Kg and k⁽⁴⁾ < 0 N/m³ and for 4) M_eff ⁻ < 0 Kg and k⁽⁴⁾ > 0 N/m³. The double arrow marks the jump of the real part of the wave vector either from κ = 0 to κ = π/L (case 3)) or from κ = π/L to κ = 0 (case 4)).