Broadband cloaking of flexural waves

The governing equation for elastic waves in flexural plates is not form invariant, and hence designing a cloak for such waves faces a major challenge. Here, we present the design of a perfect broadband cloak for flexural waves through the use of a nonlinear transformation in the region of the cloak and by matching term by term the original and transformed equations and also assuming a prestressed material with body forces. For a readily achievable flexural cloak in a physical setting, we further present an approximate adoption of our perfect cloak under more restrictive physical constraints. Through direct simulation of the governing equations, we show that this cloak, as well, maintains a consistently high cloaking efficiency over a broad range of frequencies. The methodology developed here may be used for steering waves and designing cloaks in other physical systems with non-form-invariant governing equations.

Figure 1

Profile of the rigidity as a function of $r$ required to achieve a cloak for flexural waves. Each layer is made up of a homogeneous and isotropic material, but the averaged properties provide an inhomogeneous and orthotropic apparent rigidity according to (5). In the design presented here, the number of layers is 15, each layer is divided into two sublayers, and $b/a=4$.

Figure 2

Nondimensionalized displacement field $\eta /a_0$ for a nonlinear cloak (left column), without any cloak (middle column), and for a linear cloak (right column). Each row corresponds to a different frequency: $200\mathrm{Hz}$ (first row), $300\mathrm{Hz}$ (second row), $400\mathrm{Hz}$ (third row), and $500\mathrm{Hz}$ (the last row). Coordinates are nondimensionalized with radius of the inner cylinder $a$ and the cloak size for both linear and nonlinear cloak is $b/a=4$. The cloaks are approximated with $N=15$ layers of homogeneous anisotropic materials with each layer composed of two sub-layers made up of different homogeneous and isotropic materials. For a direct comparison with Stenger et al. [20], frequencies are obtained using the values of $h=1$ mm, $a=1.5$ cm, $\rho =2000\mathrm{kg}/\mathrm{m}{}^3$, and $D_0=0.1037\mathrm{Nm}{}^2$. A nonlinear cloak shows a consistent cloaking efficiency for different frequencies, while the performance of linear cloak drops significantly as the frequency increases. The scattered field is also shown in Fig. 3. For a quantitative comparison of performance, see Figs. 5 and 6.

Figure 3

Nondimensionalized scattered field ${\eta }^{\mathrm{sc}}/a_0=(\eta -{\eta }^{\mathrm{inc}})/a_0$, where ${\eta }^{\mathrm{inc}}$ is the incident wave for a nonlinear cloak (left column), without any cloak (middle column), and for a linear cloak (right column). Each row corresponds to a different frequency as follows: $200\mathrm{Hz}$ (first row), $300\mathrm{Hz}$ (second row), $400\mathrm{Hz}$ (third row), and $500\mathrm{Hz}$ (the last row). Coordinates are nondimensionalized with radius of the inner cylinder $a$ and the cloak size for both linear and nonlinear cloak is $b/a=4$.

Figure 4

Nondimensionalized scattered amplitude ${\eta }^{\mathrm{sc}}/a_0$ along the $x$ axis at the frequency $f=300\mathrm{Hz}$ (i) without a cloak ($\mathbf{——}$), (ii) with a linear cloak ($\mathbf{–}\mathbf{–}\mathbf{–}$), and (iii) with our nonlinear cloak ($-\mathit{\text{−}}-$). We also show the nondimensionalized incident wave amplitude ${\eta }^{\mathrm{inc}}$ ($···$).

Figure 5

Polar plot of absolute value of scattered amplitude $\left|f\right(\theta \left)\right|$ for the linear cloak ($\mathbf{–}\mathbf{–}\mathbf{–}$), without cloak ($\mathbf{——}$), and with nonlinear cloak ($-\mathit{\text{−}}-$) for different frequencies of (a) $200\mathrm{Hz}$, (b) $300\mathrm{Hz}$, (c) $400\mathrm{Hz}$, and (d) $500\mathrm{Hz}$.

Figure 6

Nondimensionalized total scattering cross section with wavelength, i.e., $k{\sigma }^{\mathrm{sc}}$ for the a range of frequencies for three different cases of without cloak, with a linear cloak, and with a nonlinear cloak.