Waves within a network of slowly time-modulated interfaces: time-dependent effective properties, reciprocity and high-order dispersion

Abstract

We consider wave propagation through a 1D periodic network of slowly time-modulated interfaces. Each interface is modelled by time-dependent spring-mass jump conditions, where mass and rigidity interface parameters are modulated in time. Low-frequency homogenisation yields a leading-order model described by an effective time-dependent wave equation, i.e.\ a wave equation with effective mass density and Young's modulus which are homogeneous in space but depend on time. This means that time-dependent bulk effective properties can be created by an array where only interfaces are modulated in time. The occurrence of k-gaps in case of a periodic modulation is also analysed. Second-order homogenisation is then performed and leads to an effective model which is reciprocal but encapsulates higher-order dispersive effects. These findings and the limitations of the models are illustrated through time-domain simulations.

Figures (7)

• Figure 1: Periodic network of $N=4$ imperfect interfaces located at $X_{n,\ell}$ and repeated periodically with a period $h$. A typical wavelength is represented in red.
• Figure 2: Case of a single modulated interface per unit cell, with $f_c=20$ Hz with $f_m=30$ Hz ($\eta_0=0.86$ and $\eta_1=2.16$). (a): Comparison of the field $U_h$ (red dots) obtained by full-field simulation in the microstructure and the leading-order field $U_0$ (blue plain line). (b): Zoom on the light-green zone of the left panel. (c) Space-time evolution of the displacement fields: microstructured medium in red and homogenised medium in blue, showing Floquet amplification. (d): Comparison of the dispersion diagrams. The background map is the logarithm of the norm of the double Fourier transform of the field obtained by a time-domain full-field simulation. The darkest parts therefore represent the branches of the exact dispersion relation and the location of the $k$-gaps where fields are amplified. The red dotted lines stand for the dispersion diagram of the leading-order homogenised model obtained by PWE. (e): Evolution of the energy over time for the exact field (blue) and its leading-order counterpart (orange). The background represents the time-modulation of the interface properties. (f): Time evolution of the energy when dissipative interface parameters \ref{['Q-T']} are considered, illustrating the competition between intrinsic losses and Floquet amplification.
• Figure 3: Case of a single modulated interface per unit cell with interface properties set to have impedance matching \ref{['eq:impedance_matching']}. (Left): comparison of the field $U_h$ (red dots) obtained by performing a full-field simulation in the microstructure and the leading-order field $U_0$ (blue plain line). (Right): Zoom on the light-green zone of the left panel. Simulations done with $f_c=10$ Hz and $f_m=20$ Hz (i.e. $\eta_0=0.44$ and $\eta_1=1.31$).
• Figure 4: Case of two modulated interfaces per unit cell. (Left): comparison of the field $U_h$ (red dots) obtained by performing a full-field simulation in the microstructure and the leading-order field $U_0$ (blue plain line). (Right): Zoom on the light-green zone of the left panel. Simulations done with $f_c=10$ Hz and $f_m=20$ Hz (i.e. $\eta_0=0.640$ and $\eta_1=1.92$).
• Figure 5: Comparaison betweeen $U_h$ (full-field simulation in the microstructure), and the homogenised fields $U_0$ and $U^{(2)}$ without modulation ($f_m=0$) for an increasing value of $\eta_0$. From top to bottom : $f_c=\{10,20,30\}$ Hz corresponding to $\eta_0=\{0.31,0.63,0.94\}$.

Theorems & Definitions (8)

• Lemma 1
• Remark 1
• Remark 2
• Proposition 1: Reciprocity
• Proposition 2
• Remark 3
• Remark 4
• Remark 5