Spacetime modulation in floating thin elastic plates

We consider the propagation of flexural-gravity waves in thin elastic plates floating atop nonviscous fluids, e.g., seawater, which are governed by a partial differential equation with Laplacian and tri-Laplacian terms. We investigate the effect of time modulation as well as spacetime modulation on thin floating elastic plates and show the peculiarity of the phenomena of the $k$-band gap and the rotated $k$-band gap in the context of flexural-gravity waves. This makes possible floating plates with nonreciprocal features and behaving as elastodynamic analogs of luminal electromagnetic metamaterials, with exotic applications in enhanced control of ocean waves, such as filtering devices, unidirectional acoustic propagation, and isolation effects and energy harvesting in maritime engineering.

Figure 1

Schematic of a floating thin plate of thickness $\delta$ atop shallow seawater of depth $h$, in the $x\text{−}z$ plane. The thin plate is supposed to be infinitely extended along the $y$ direction. The parameters of the plate are as follows: Young's modulus $E=50$ GPa, thickness $\delta =10$ m, Poisson's ratio $\nu =0.34$, and density of $900\mathrm{kg}/{\mathrm{m}}^3$, to make it float atop water.

Figure 2

(a) Dispersion curves of free plane waves (i.e., ${\delta }_m=0$) in floating thin plates for $N=5$ (i.e., considering 11 bands) and for ${\mathrm{\Omega }}_m=10$ rad/s. The inset shows an enlarged view at low frequency using two different PDEs (see Appendix pp1). (b) Dispersion curves for ${\delta }_m=0.4$, where the inset shows the $k$-band gap around the angular frequency $\omega ={\mathrm{\Omega }}_m/2$. The number of bands and modulation frequency are the same as in (a). The inset shows a magnified view around the second $k$-band gap. (c) Two-band dispersion curves for the case of ${\delta }_m=1.75$, i.e., considering only the bands $n=0$ and $n=-1$. The width of the $k$-band gap ($[{\beta }_{-},{\beta }_{+}]$) can be calculated analytically in this case. (d) Broader domain for the dispersion similar to (b), i.e., with ${\mathrm{\Omega }}_m=10$ rad/s, but for ${\delta }_m=1.9$ and considering $N=10$, i.e., 21 bands.

Figure 3

Two-band dispersion curves for the case of ${\delta }_m=1.75$ and considering only the bands $n=0$ and $n=-1$, for (a) ${\mathrm{\Omega }}_m=0$ rad/s and ${\kappa }_m=0.06$ rad/m, i.e., classical space modulation with frequency band gap, (b) ${\mathrm{\Omega }}_m=20$ rad/s and ${\kappa }_m=0.06$ rad/m, i.e., spacetime modulation with an oblique (indirect) band gap, and (c) ${\mathrm{\Omega }}_m=20$ rad/s and ${\kappa }_m=0$ rad/m, i.e., classical time modulation with $k$-band gap.

Figure 4

Top: Dispersion curves for $N=5$, i.e., considering 11 bands for ${\delta }_m=0.8, {\mathrm{\Omega }}_m=30$ rad/s, and ${\kappa }_m=0.0065$ rad/m, where the inset (red curves) is a magnified view of the oblique (rotated) $k$-band gap. Bottom: Imaginary part of the first eigenfrequency $\mathrm{Im}\left(\omega \right)$ in the same range of flexural-gravity wave numbers. The blue (green) dots correspond to negative (positive) ${\beta }_1=\mp 0.034$ rad/m.

Figure 5

(a) Comparison of the strength of the terms $T_2$ and $T_3$, using dimensional analysis, normalized by the first term $T_1$, vs the water wavelength ${\lambda }_0=2\pi /k_0$. (b) Dispersion relation (flexural wavelength vs water wavelength) of flexural-gravity waves, using the PDE of Eq. (A7) and the other PDE [same as in Eq. (A7) without the term $T_2$]. (c) Two-band dispersion curves in free space, i.e., no coupling, ${\delta }_m=0$. (d) Two-band dispersion considering only the bands $n=0$ and $n=-1$, for ${\mathrm{\Omega }}_m=20$ rad/s, ${\kappa }_m=0.06$ rad/m [spacetime modulation with an oblique (indirect) band gap], and ${\delta }_m=1.75$.

Figure 6

Variation of the eigenfrequencies $\omega$ vs the normalized perturbation parameter $\mathrm{\Delta }$ computed numerically for $\mathrm{\Lambda }={\mathrm{\Omega }}_m, \mathrm{\Lambda }={\gamma }_0, \mathrm{\Lambda }={\delta }_m$, and $\mathrm{\Lambda }={\gamma }_1$ reading from top to bottom, respectively. The parameters of the spacetime modulation are ${\mathrm{\Omega }}_m=10$ rad/s and ${\delta }_m=1.75$, while the other parameters are the same as in the main text.

Figure 7

Analytical (blue dashed line) and numerical (black squares) calculation of the variation of the eigenvalues vs the normalized perturbation parameter $\mathrm{\Delta }$, with the remaining properties being the same as in Fig. 6.

Figure 8

Two-band dispersion curves for the case of ${\delta }_m=1$ and considering only the bands $n=0$ and $n=-1$, for ${\mathrm{\Omega }}_m=20$ rad/s and for varying ${\kappa }_m$ from 0 to 0.05 rad/m (spacetime modulation with an oblique, i.e., indirect, band gap). The black curves show the real part of the eigenfrequencies, while the red curves show their imaginary part. The remaining physical parameters of the elastic plate and liquid are the same as those defined in Sec. 2 of the main text.

Figure 9

Two-band dispersion curves for the case of ${\delta }_m=1$ and considering only the bands $n=0$ and $n=-1$, for ${\mathrm{\Omega }}_m=20$ rad/s and for varying ${\kappa }_m$ from 0.06 to 0.25 rad/m [spacetime modulation with an oblique (indirect) band gap]. The black curves show the real parts of the eigenfrequencies, while the red curves show their imaginary parts. The remaining physical parameters of the elastic plate and liquid are the same as those defined in Sec. 2 of the main text.