Asymmetric lateral sound beaming in a Willis medium

Open Access

Asymmetric lateral sound beaming in a Willis medium

Xiaoshi Su and Debasish Banerjee

Phys. Rev. Research 3, 033080 – Published 22 July 2021

Abstract

Willis coupling, also known as pressure-velocity cross coupling, in acoustic materials has received much attention in the past years. This effect has been found useful in acoustic metasurface designs for wave redirection. We find that Willis coupling in phononic crystals also provides rich physics to manipulate waves. Here, we report the extreme asymmetric lateral sound beaming effect in a two-dimensional phononic crystal composed of Willis scatterers. By matching the second-order Bragg scattering with two leaky guided modes (a quadrupole resonance and a cross-coupling-induced dipole resonance), a normally incident wave is redirected towards the positive and negative directions orthogonal to the incident wave with different amounts of energy. Simulation and experimental results demonstrate the extraordinary asymmetric lateral beaming effect in the Willis medium. The results presented here may find applications in the design of tunable beam splitters and waveguides.

Received 18 April 2021
Revised 6 July 2021
Accepted 8 July 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.033080

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic wave phenomena Acoustics

General Physics

Authors & Affiliations

Xiaoshi Su ^* and Debasish Banerjee

Toyota Research Institute of North America, 1555 Woodridge Avenue, Ann Arbor, Michigan 48105, USA

^*xiaoshi.su@toyota.com

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 3, Iss. 3 — July - September 2021

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Description of the Willis medium. The material is composed of Willis scatterers arranged in a square lattice pattern as shown in (a). A plane wave is normally incident along the $+ y$ direction from port 1; the transmitted waves can exit from ports 2, 3, and 4. (b) shows the details of the unit cell. The lattice constant is denoted by $d$ ; $r_{i}, r_{o}$ , and $w$ are the inner radius, outer radius, and neck width, respectively.
Reuse & Permissions
Figure 2
Normalized dipole scattering coefficients for a normally incident plane wave along the $+ y$ direction. (a) corresponds to the acoustic pressure excited dipole orthogonal to the direction of the incident wave. (b) corresponds to the acoustic velocity excited dipole along the incident direction. The blue and black lines represent the dipole scattering coefficients for the C-shaped scatterer and a rigid scatterer of the same outer radius, respectively.
Reuse & Permissions
Figure 3
Lateral beaming in periodic structures. (a) is a square lattice of rigid cylinders; the radius of each cylinder is $r = 1.05$ cm, and the lattice constant is $d = 6.466$ cm. (b) illustrates the first antisymmetric (4924.9 Hz) and symmetric ( $5283.9 + 173.39 i$ Hz) modes and the corresponding dipole and quadrupole modes in the unit cell. (c) is the simulated sound pressure field at 5305 Hz showing the symmetric lateral beaming effect in the periodic structure; a Gaussian beam is normally incident along the $+ y$ direction. (d) is a Willis medium composed of C-shaped scatterers; the outer radius of the scatterer is $r_{o} = 1.05$ cm, and the lattice constant is $d = 6.466$ cm. (e) shows the two modes similar to those in (b) at two different frequencies ( $5158.3 + 81.583 i$ Hz and $5280.4 + 71.982 i$ Hz) and the corresponding dipole and quadrupole modes in the unit cell. The simulated asymmetric lateral beaming in a Willis medium at 5305 Hz is shown in (f).
Reuse & Permissions
Figure 4
Measurement results. (a) shows the three-dimensional (3D) printed Willis scatterer; (b) shows the measurement setup of the Willis media placed in a 2D waveguide (two 11/16-in.-thick acrylic plates of $48 \times 48$ -in. size separated by 2.75 cm in parallel); the 32 scatterers are arranged in a square lattice pattern with a lattice constant of $d = 6.466$ cm; the open necks of the scatterers are oriented towards the $+ x$ direction; sound pressure fields were measured in three $12.5 \times 9.5$ -cm areas indicated by dashed rectangles (12 cm away from the scatterers). (c) is the simulated sound pressure field at 5305 Hz for a normally incident Gaussian beam along the $+ y$ direction; (d)–(f) are the measured sound pressure fields at 5305 Hz corresponding to the three areas in (b) and (c) marked by dashed rectangles.
Reuse & Permissions
Figure 5
Transmission properties through the Willis medium. (a) shows the transmission coefficients calculated from simulation results along the $+ y$ and $\pm x$ directions ( $| T_{+ y} |^{2}$ and $| T_{\pm x} |^{2}$ ) and the reflection coefficient along the $- y$ direction ( $| T_{- y} |^{2}$ ). A maximum transmission along the $+ x$ direction of 47.2% occurs near 5360 Hz in the simulation. (b) shows the asymmetry ratio, defined as the ratio between the sound intensity through the two dashed rectangles along the $+ x$ and $- x$ directions in Figs. 4 and 4; the solid curve represents the simulation results, while the circles represent the measurement data. The maximum asymmetry ratio between the $+ x$ and $- x$ directions ( $| I_{+ x} | / | I_{- x} | = 9.27$ ) occurs at 5305 Hz in the measurements.
Reuse & Permissions

Physical Review Research