Broadband attenuation of Lamb waves through a periodic array of thin rectangular junctions

We study theoretically subwavelength physical phenomena, such as resonant transmission and broadband sound shielding for Lamb waves propagating in an acoustic metamaterial made of a thin plate drilled with one or two row(s) of rectangular holes. The resonances and antiresonances of periodically arranged rectangular junctions separated by holes are investigated as a function of the geometrical parameters of the junctions. With one and two row(s) of holes, high frequency specific features in the transmission coefficient are explained in terms of a coupling of incident waves with both Fabry-Perot oscillations inside the junctions and induced surface acoustic waves between the homogeneous part of the plate and the row of holes. With two rows of holes, low frequency peaks and dips appear in the transmission spectrum. The choice of the distance between the two rows of holes allows the realization of a broadband low frequency acoustic shielding with attenuation over 99% for symmetric waves in a wide low frequency range and over 90% for antisymmetric ones. The origin of the transmission gap is discussed in terms of localized modes of the “H” element made by the junctions, connecting the two homogeneous parts of the plate.

Figure 1

Schematic view of the unit cell of the acoustic metamaterial silicon slab of thickness $e$ drilled with one or two rows of rectangular holes of width $d$ and length $l$. The lattice parameter is $a$ and the distance between the two holes is $w$. Periodic boundary conditions are applied to each side of the unit cell, in the direction $x$ while perfect matching layers (dark gray areas) are applied in the direction of propagation $z$. The input symmetric Lamb wave $S_0$ is launched from the left side of the structure.

Figure 2

(a) Transmission coefficients calculation as a function of the reduced frequency for the three components $u_x$ (green), $u_y$ (red), and $u_z$ (black) of the symmetric Lamb wave $S_0$ through one row of rectangular holes for the geometrical parameters: $d$ = $0.9a$, $l$ = $0.4a$. (b) Cartography of the displacement fields for the three components at the frequencies of the two peaks $P_1$ (left) and $P_2$ (right) and the dip $Z_1$ (center). The red (blue color) corresponds to a positive (negative) displacement with a zero represented with the white color. For the components $u_x$ and $u_y$ the color bar has been magnified for a better observation of the field.

Figure 3

Evolution of the transmission curve for the symmetric Lamb wave $S_0$ with respect to the main component $u_z$ as a function of the length $l$ of the hole, with $d$ = $0.9a$. $P_i$, where $i$ is an integer, represents the positions of the Fabry-Perot maxima inside the vertical junction.

Figure 4

Evolution of the transmission coefficient for the symmetric Lamb wave $S_0$ with respect to the main component $u_z$ as a function of (a) the width $d$ when $l$ = $0.4a$ and (b) the length $l$ when $d$ = $0.9a$. The yellow (resp. black) color corresponds to the high (resp. low) transmission coefficient represented in scalar scale.

Figure 5

Evolution of the transmission curve for the three components $u_x$ (green), $u_y$ (red), and $u_z$ (black) of the antisymmetric Lamb wave $A_0$ with $d$ = $0.9a$ and for three values of the length $l$ of the holes. Bottom right panel: Evolution of the transmission coefficient $T_y$ of the $A_0$ mode as a function of the length $l$ when $d$ = $0.9a$. The yellow (resp. black) color corresponds to the high (resp. low) transmission coefficient represented in scalar scale.

Figure 6

(a) Transmission coefficients for each component calculated as a function of the reduced frequency for the three components $u_x$ (green), $u_y$ (red), and $u_z$ (black) of the symmetric Lamb wave $S_0$ through two rows of rectangular holes for the geometrical parameters: $d$ = $0.9a$, $l$ = $0.4a$, and $w$ = $0.4a$ (b) Transversal, perpendicular, and longitudinal components $u_x$, $u_y$, and $u_z$, respectively, of the displacement fields at the frequencies of the three peaks $P_1^{\prime }$, $P_2^{\prime }$, and $P_3^{\prime }$ and the two dips $Z_1^{\prime }$ and $Z_2^{\prime }$. The red (blue color) corresponds to a positive (negative) displacement while the white color represents the zero. For the components $u_x$ and $u_y$ the color bars have been chosen differently to magnify the corresponding field.

Figure 7

(a) Definition of an H element extracted from the periodic unit cell. (b) Useful symmetric eigenmodes of the H element at 1580 ($S_1$), 2840 ($S_2$), and 3530 m/s ($S_3$) and antisymmetric at 1820 ($A_1$) and 3840 m/s ($A_2$) with fixed boundaries condition applied on each vertical junction [hatched area in (a)]. The symmetric mode $S_2$ is also an eigenmode of the H element but cannot be excited by the incident wave since it is antisymmetric with respect to the $z$ axis. The red (blue color) corresponds to a positive (negative) displacement with a zero represented with the white color.

Figure 8

Evolution of the transmission coefficient of the symmetric Lamb wave $S_0$ with respect to the main component $u_z$ at low frequency ($fa<4500$ m/s) as a function of the width $w$ between the two rows of rectangular holes for $l$ = $0.4a$. Upper right panel: Displacement field distributions at the frequency of the peak (left) and the dip (right) for $w=0.1a$; the red (blue color) corresponds to a positive (negative) displacement with a zero represented with the white color.

Figure 9

(a) Evolution of the transmission coefficient of the symmetric Lamb wave $S_0$ with respect to the main component $u_z$ as a function of the distance $w$ between the two rows of rectangular holes: the yellow (resp. dark blue) color corresponds to the high (resp. low) coefficient. (b) Highlight of the widest low frequency band gap for the optimized set of parameters: $d$ = $0.9a$, $l$ = $0.4a$, and $w$ = $0.8a$: the attenuation reaches 99% of the input signal.

Figure 10

Evolution of the transmission coefficient for the $S_0$ mode with respect to the main component $u_z$ as a function of the length of the holes when $w$ = $0.8a$ and $d$ = $0.9a$.

Figure 11

(a) Transmission coefficient of the $A_0$ mode for $w=0.4a$. (b) Evolution of the transmission coefficient for the $A_0$ mode with respect to the main component $u_y$ as a function of the distance $w$ between the two rows of rectangular holes: the yellow (resp. dark blue) color corresponds to the high (resp. low) coefficient. (c) $u_x$, $u_y$, and $u_z$ displacement fields at $A_{P_1}$ and $A_{P_2}$ for $w=0.4a$.

Figure 12

(a) Unit cell for the band structure calculation with periodic boundaries conditions applied in the $x$ and $y$ directions. (b) Band structure calculated in the direction parallel to the holes lines for $w=0.2a$. Band sorting calculation for $P$ = $9a$, $d$ = $0.9a$, $l$ = $0.4a$, and $w$ = $0.2a$, selecting the modes (c) symmetric and (d) antisymmetric with respect to the middle plane of the plate. The grayscale color bar defines the amount of displacement of the eigenmodes concentrated in the H element with respect to the displacement field in the whole unit cell.

Figure 13

Band sorting calculation for $P$ = $9a$, $d$ = $0.9a$, $l$ = $0.4a$, and $w$ = $0.4a$, selecting the modes (a) symmetric and (b) antisymmetric with respect to the middle plane of the plate. The grayscale color bar defines the amount of displacement of the eigenmodes concentrated in the H element with respect to the displacement field in the whole unit cell. (c) Transversal, perpendicular, and longitudinal components $u_x$, $u_y$, and $u_z$ of the displacement fields at the frequencies of the two modes $S_1$, $S_2$, $S_3$ and $A_1$, $A_2$ indicated with red font in (a) and (b). The red (blue) color corresponds to a positive (negative) displacement while the white color represents the zero. For the components $u_x$ and $u_y$ the color bars have been chosen differently from time to time to magnify the corresponding field.

Figure 14

(a) Elementary unit cell of the phononic crystal with periodic conditions applied in the $x$ and $y$ directions. Band structure of the phononic crystal in $\Gamma X$ direction for the modes symmetric (red), antisymmetric (black), and shear horizontal (blue) for the geometrical parameters $w$ = $0.35a$ and $l$ = $0.4a$. (b) Overview of the low frequency gap formation ($fa$ $<$ 3000 m/s) from the transmission curves for an increasing number of unit cells for the symmetric Lamb modes.