Figure 3

(a) Sound absorption coefficients of metamaterials with and without rubber coating and a homogeneous rubber layer of the same thickness. Geometric and material parameters: \(d\) = 4 mm, \(t_{1}\) = 1 mm, \(t_{2}\) = 4.7 mm, \(t_{3}\) = 1 mm, \(L\) = 30.8 mm, \(w\) = 5 mm, \(h\) = 50 mm, \(L_{eff}\) = 203 mm, \(\eta_{r}\) = 0.3, \(\rho_{r} = 1100\, {\text{kg/m}}^{{3}}\), \(E_{r}\) = 10 MPa, \(\mu_{r}\) = 0.49, \(\rho_{0} = 1000\,{\text{kg/m}}^{{3}}\) and \(c_{0} = {1500}\) m/s. Solid lines and circles represent theoretical and numerical results, respectively. (b) The corresponding effective sound speed \(c_{z}\) on the water channel, the real part represented by the blue line and the imaginary part represented by the green line. (c) Sound pressure distribution at the first resonance frequency of 181 Hz. (d) Vibration velocity distribution of the rubber coating on the cross-section of the metamaterial at 181 Hz. The red arrows indicate the vibration velocity distribution on the water channel boundary, the length represents the magnitude of the velocity and the directions represents the direction of the vibration velocity. (e) Slow-sound propagation with inherent dissipation on the water channel. The thickness of the red line represents the intensity of the sound wave. (f) Energy dissipation in different parts of the metamaterial.