# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 1 Introduction
An efficient reduced formulation of the Bloch Operator Finite Element method was conducted to calculate the complex band structure by Palermo et al. [31] based on the wave field transformation, with accurate predictions of propagative, evanescent, and complex wave solutions with a reduction in computational time of more than one order of magnitude with respect to the full model calculations. The authors also extended this technique to observe the complex band structure of elastic waves in a three-dimensional phononic slab with viscoelastic effects [32]. One can notice that the studies applying the \(\mathbf{k}(\omega)\) approach were mainly focused on the wave band gaps in the purely elastic medium and the present work aims to present a general numerical framework that is able to predict the location, the bandwidth, and the attenuation characteristics of flexural wave band gaps when multi-field coupling effects are taken into consideration. In order to observe not only the real band diagram but also the degree of wave attenuation, the numerical formulations from \(\omega(\mathbf{k})\) and \(\mathbf{k}(\omega)\) approaches are derived, respectively, and the results can support the correctness of each other. It is acknowledged that the coupled fluid-structure mechanical systems have been widely applied in various engineering applications such as submarines, anechoic coatings, and offshore platforms, and, thus, it is necessary to tailor the propagation and attenuation of waves in continuous periodic structures with heavy fluid loading. Especially, plate-like periodic structures generally serve as the fundamental components in the underwater applications and, thus, the investigation of the flexural wave band gaps of the PnC plate has significant research value. However, to the best knowledge of the authors, the analysis of the band structure for the fluid-structure coupled system is concentrated on the periodic pipe structures with internal fluid, lacking accurate and efficient models to calculate the complex band structure of flexural waves with external heavy fluid. By proposing a periodic waveguide made of an array of expansionary mufflers, Liu et al. showed that the introduction of a muffler array to form a periodic structure in the piping system can suppress the propagation of acoustic waves with longer wavelengths [33]. The band structure of deploying pipes conveying fluid with periodic varying parameters was investigated by Liang et al. based on the Bragg scattering mechanisms [34], the results of which demonstrated that the fluid-structure interaction, deploying motion, and component geometry have significant effects on the location and width of the Bragg band gaps of the pipe. With a periodically varying cross-section of fluid-conveying pipes, Yu et al. attainedthe band gap distribution by the spectral element method and showed the effects of the substructure length and fluid-structure interaction on the band gaps [35]. For the numerical simulations on the periodic structures with external fluid, Sorokin [36] gave an analytical algorithm to compute Bloch parameters for the one-dimensional case incorporating fluid loading, and Zhang et al. proposed a tunable fluid-solid metamaterial consisting of several liquid inclusions in a solid matrix where the tunability of the band gaps is achieved by controlling the liquid distribution while the theoretical analysis is only available at the sub-wavelength scale [37]. In order to accurately predict the propagation and attenuation characteristics of flexural waves, the present work is concerned with the complex band structure of the phononic plate with external infinite heavy fluid, which is more widely used, and aims to construct a general numerical model that is available at full wavelength to discover the effects of fluid loading on the real and imaginary parts of dispersion relations.