Fig. 2: Structure of the isofrequency curves.

Three isofrequency curves representing elliptic-like (green) and hyperbolic-like (black and blue) dispersion. Bloch waves have elliptic-like dispersion in the first passing band, and hyperbolic-like dispersion in the second band. Transition occurs through the band gap. Dash-dotted blue curve is the result of the effective medium theory (EMT) obtained from Eqs. (2) and (3). The frequencies corresponding to hyperbolic-like dispersion lie slightly above and slightly below the saddle point at f₀ = 104.8 kHz. The saddle point is the point of crossing of two parabolas marked by red lines in the inset. The arrows show the directions of the phase and group velocities for the Bloch waves with the same projection k_y, which is conserved at the water–steel boundaries. The crossing of the vertical line k_y = 89 (k_y = 71) with solid (dash-dotted) blue curve occurs at k_x = −51 m⁻¹ (k_x = −71 m⁻¹) that corresponds to the angle γ = 60° (γ = 45°).