# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 1 Introduction  
During the past two decades, the phononic crystals [1; 2; 3; 4; 5] have attracted the great interests of researchers for their novel physical characteristics. The most striking feature of such periodic structures is the ability of exhibiting band gaps in certain frequency ranges where no acoustic or elastic wave propagation is allowed. This physical phenomenon of the phononic crystal makes it particularly useful in designing applications for vibration suppression and sound attenuation. By employing the Floquet-Bloch theorem, a representative cell with less computation costs is allowed to be modeled to investigate the wave motion of the infinite periodic medium. Several numerical approaches such as the transfer matrix method [6; 7], the multi-scattering theory [8; 9; 10], the plane wave expansion method [11; 12], and the finite element method [13; 14] have been extensively used to analyze the band gap properties. In phononic crystals, band gaps due to Bragg scattering were required to satisfy the constraint that the wavelength should be on the order of the lattice spacing. Thus, to generate low-frequency Bragg gaps, low-wave speeds or large-sized structures are needed [15]. To overcome this shortcoming, Liu [16] proposed a new type of composite material based on the localized resonant mechanisms. Local resonance gaps have been widely used to improve the vibration and sound reduction performance from low to medium frequency ranges [17; 18; 19]. In addition, the mechanisms of inertial amplification have attracted increasing attention as the induced band gaps can be created as a result of amplifying the motion of the small mass [20; 21; 22]. Regardless of the Bragg gaps, the local resonance gaps, or the inertial-amplification-induced gaps, the dispersion curves of phononic crystals and metamaterials are commonlypresented in the form of the function of frequencies for real wavenumbers, or conversely, complex wavenumbers as a function of real frequencies. The former condition corresponds to the \(\omega(\mathbf{k})\) form, where the wavenumbers are given in Bloch conditions and the underlying eigenvalue problem is solved for frequencies to detect wave modes within pass bands. The latter condition, on the other hand, follows the form of \(\mathbf{k}(\omega)\), which can be used to calculate complex wave vectors for a given frequency so that evanescent wave modes inside gaps can be obtained. It can be seen that the location, width, and attenuation characteristics of stop bands can be obtained simultaneously via the \(\mathbf{k}(\omega)\) form. The formulation based on the \(\omega(\mathbf{k})\) path is easily implemented as the resulting characteristic determinant is usually set-up from a linear equation and, thus, a large number of studies have been conducted to examine the location and width of elastic wave band gaps [13; 23; 24; 25; 26; 27]. In contrast, the \(\mathbf{k}(\omega)\) path is more computationally complicated as the calculations of complex Bloch wavenumbers are enforced to find solutions from transcendent equations [28], especially for the case concerned with detecting the decaying level of elastic waves from an arbitrary direction. In order to reduce the transcendental complexity of the \(\mathbf{k}(\omega)\) form, a wave field transformation capable of eliminating the dependence of phase constants is employed by researchers. Lewinska et al. [29] measured the effects of viscoelastic behavior on the longitudinal and shear wave band gaps by the formulation in the \(\mathbf{k}(\omega)\) form where the Bloch displacement function with special periodicity was used to reformulate the eigenvalue problem. Combing the Dirichlet-to-Neumann method with cylindrical wave expansions, Li et al. analyzed the effects of viscosity on the anti-plane transverse wave band gaps of a two-dimensional phononic crystal composed of square or triangular lattices of elastic circular solid cylinders in a viscoelastic solid matrix [30].