# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 3 Numerical Examples and Discussions
### Directionality of Flexural Wave Decaying  
The above discussions are mainly focused on the attenuation constant in the horizontal direction. In order to observe the directionality of wave attenuation of the coupled plate, the spatial attenuation constants from all directions are calculated at the frequencies of the attenuation peaks, that is, 360 Hz and 490 Hz. As the considered lattice possesses the highest-level symmetry characteristics, the complete attenuation profile can be obtained by calculating the attenuation constants only within the range of \(\theta\) from 0 to \(\pi/4\). The  
Figure 7: Effects of the ratio \(r_{e}\) on the attenuation performance of flexural waves: (**a**) with fluid loading; (**b**) without fluid loading.  
evaluations of attenuation constants with respect to the propagating direction \(\theta\) at 360 Hz and 490 Hz are shown in Figure 8a,b, respectively. It is worth noting that the complete band gap is generated by the local resonance around 360 Hz and, thus, a comparable attenuation degree of flexural waves can be observed from all directions in Figure 8a. Notably, the magnitude varies slightly with respect to changing \(\theta\). In contrast, the Bragg band gap is an angular gap, which shows a great dependence on \(\theta\) as observed from Figure 8b. The maximum attenuation in the Bragg gap is not found in horizontal or vertical directions--it is attained around the directional angle \(\theta\) of \(\pi/12\).  
The results for the plate without fluid loading at 560 Hz and 970 Hz, i.e., the attenuation peaks, are given in Figure 8c,d. The shape of the wave attenuation profile of the local resonance gap is similar but with larger amplitudes, while the Bragg gap deviates a lot. By comparing Figure 8b-d, one can also notice that the angular dependence of wave attenuation in the fluid-loaded plate is not as strong as that in the plate without fluid loading; in other words, the external fluid has the ability to reduce the attenuation anisotropy of the Bragg stop band in the phononic plate. To quantify the degree of anisotropy in the wave attenuation profile, a new quantity \(s\) defined as the standard deviation in the angle-dependent attenuation constant \(\mu_{\theta}\) for a given frequency \(\omega\) is introduced. This quantity can be obtained by:  
\[s=\frac{1}{\overline{\mu}(\omega)}\sqrt{\frac{1}{n_{\theta}-1}\sum_{i=1}^{n_{ \theta}}(\mu_{i}(\omega,\theta_{i})-\overline{\mu}(\omega))^{2}},\quad \overline{\mu}(\omega)=\sum_{i=1}^{n_{\theta}}\mu(\theta,\omega) \tag{28}\]  
where \(\theta_{i}\) indicates the direction of wave propagation and \(\mu_{i}\) is the attenuation constant at \(\theta_{i}\); \(i\) is the sampling index and \(n_{\theta}\) is the total number of sampling points in the range of 0 to \(\pi/4\). For the fluid-loaded phononic plate, the value of \(s\) inside the Bragg gap (at 490 Hz) is 1.109, and for the bare phononic plate, the corresponding value (at 970 Hz) is 2.33. A lower value indicates that the fluid loading tends to smooth the results of the attenuation directionality and make the phononic plate behave less anisotropically.