# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 3 Numerical Examples and Discussions
### Wave Attenuation Based on \(\mathbf{k}(\omega)\) Formulations  
The attenuation characteristics of flexural waves of the heavy fluid-loaded periodic plate is evaluated by the \(\mathbf{k}(\omega)\) approach in this subsection. The simulation is performed using the finite element method combined with the wave field transformation. To validate the derived formulas, the imaginary parts of wavenumbers from the \(\Gamma\)X direction are also calculated by the general finite element method for comparison. The sound absorbing condition is replaced by a rigid wall condition when the general finite element method is  
Figure 4: The finite phononic plate composed of \(1\times 10\) unit cells. applied. The imaginary component of the wavenumber represents the spatial attenuation and, thus, the wave decaying rate \(\mu_{\theta}\) can be evaluated by:  
\[\mu_{\theta}=real(\max(\lambda l\cos\theta,\ \lambda l\sin\theta)) \tag{27}\]  
Focusing on the wave attenuation from the \(\Gamma\)X direction, the propagating angle \(\theta\) is specified as zero. Figure 6 presents the spatial attenuation of flexural waves obtained by the wave field transformation and general FEM, from which the directional band gap is easily distinguished from the frequency range with non-zero attenuation constants. In Figure 6a, the dispersion curve of the attenuation constant based on the proposed method is in good agreement with the result by the finite element method, which not only validates the proposed approach but also indicates that the boundary condition imposed on surface \(S_{\text{II}}\) has little influence on the band structure. The first band gap is shown as a local resonance gap and the second one is shown as a Bragg gap, whose locations are consistent with that displayed in Section 3.1. Figure 5: The vibration transmittance of finite plate composed of 1 \(\times\) 10 unit cells: (**a**) with fluid loading; (**b**) without fluid loading. Figure 6: Attenuation constants versus excitation frequencies for the phononic plate in the case of \(\theta=0\): (**a**) with fluid loading; (**b**) without fluid loading. It can be seen that the maximum attenuation constant of the local resonance gap is higher than that of the Bragg gap. This is why the transmission loss in \(g_{1}\) is larger than that in \(g_{2}\). The corresponding results of the bare periodic plate are given in Figure 6b. From Figure 6, one can notice that the value of the attenuation peak is decreased within the local resonance gap while increased in the Bragg gap due to the effects of heavy fluid loading. Such a phenomenon is able to explain the transmission drops of finite plates in Section 3.2, where the maximum absolute value in \(g_{1}\) of Figure 5a is smaller than that in Figure 5b. Likewise, the depth profiles of gaps in the transmittance function plots also demonstrate the viewpoint that the fluid loading is capable of weakening the maximum attenuation performance of the resonance gap and enhancing that of the Bragg gap. To further observe the effects of \(r_{e}\) on the attenuation performance, the parameter \(r_{e}\) is designed ranging from 0.001 to 0.002. Considering the fluid-structure interaction effects, Figure 7a shows the influence of \(r_{e}\) on the wave decaying inside the directional band gaps of the unit cell with fluid loading. From Figure 7a, one can see that the locations of band gaps are shifting to high frequencies as the elastic modulus ratio is further increased. In addition, a special phenomenon is observed in this attenuation variations diagram. When the ratio \(r_{e}\) is tuned to 0.00135, the two directional band gaps are nearly coupled together and form a super-wide gap with which only a very narrow pass band between them exists. Furthermore, as the ratio \(r_{e}\) is smaller than this specific value, the first directional band gap is generated by the local resonance mechanisms and the second one is induced by the Bragg scattering.