# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 4 Conclusions  
As it is necessary to control wave propagation and attenuation in a strongly coupled structural acoustic wave guide, the complex band diagram of flexural waves in the fluid-loaded phononic plates is investigated in the present work. The numerical models based on the \(\omega(\mathbf{k})\) approach and the \(\mathbf{k}(\omega)\) approach are developed, respectively, to study the propagating and evanescent flexural waves. It is noted that the \(\omega(\mathbf{k})\) formulation is in accordance with the general finite element form but the \(\mathbf{k}(\omega)\) approach capable of determining the attenuation constants of flexural waves from all directions provides a new avenue to deal with the fluid-structure interaction problem in periodic wave guides. The numerical model in the \(\mathbf{k}(\omega)\) form is established by applying the wave field expansion to the out-of-plane  
Figure 8: Directional attenuation constants of the phononic plate with fluid loading at: (**a**) 360 Hz and (**b**) 490 Hz; the phononic plate without fluid loading at: (**c**) 560 Hz and (**d**) 970 Hz. displacement and the acoustic pressure, while using the Galerkin method with the finite element discretization technique to acquire the discrete formulations. The comparison results of the real band structure of the square phononic plate with and without external heavy fluid show that the locations of the band gaps are modulated into lower frequencies and the bandwidths are expanded to larger frequency ranges due to the external heavy fluid. This trend can be found from both complete and directional band gaps generated by the mechanisms of the local resonance and the Bragg scattering. Further, from the results of the attenuation diagrams, one can notice that the fluid loading is able to reduce the maximum attenuation constant inside the local resonance gap but enhance the decaying level of the Bragg gap. The location and width of the stop bands are also consistent with those observed from the real band structure. Moreover, these influences caused by the heavy fluid loading are demonstrated by the calculation results of the vibration transmittance of finite phononic plate composed of numerous unit cells. It is seen that the transmission loss regions match well with the stop bands. The directionality of wave attenuation in the fluid-loaded plate is also investigated. Regarding the complete stop band induced by the local resonance, the propagation of flexural waves from all directions can be prohibited and, thus, the attenuation level varies slightly with respect to the wave direction. In contrast, the Bragg gap is an angular gap and shows a strong attenuation anisotropy. The results indicate that the heavy fluid loading is able to reduce the level of the anisotropy degree of the Bragg gap. Additionally, a special phenomenon of forming a super-wide directional band gap occurs by varying the elastic modulus ratio, in which case the local resonance frequency is expected to coincide with the Bragg frequency and only a narrow pass band exists inside the stop band. However, due to the heavy fluid loading, the specific values of the elastic modulus ratio corresponding to such a phenomenon are different. As a result, the mechanisms to induce the first and second directional band gaps are switched for the considered cases if the ratio is tuned to range from 0.00135 to 0.0028; in other words, the first stop band is a Bragg scattering gap and the second one is a local resonance gap for the fluid-loaded phononic plate, while that for the bare phononic plate is just the opposite. The presented study provides a new numerical framework for characterizing wave propagation and attenuation in the phononic plate with heavy fluid loading. Potentially, it could be extended to more complex structures such as composite plates, shells, and membranes. In the case of heavy fluid loading, the anisotropic dissipation of metamaterial plates with viscoelastic damping will be investigated in further works. Conceptualization, J.L. and S.L. ; data curation, S.L.