Theory and experiments on surface acoustic waves at a lossy metasurface: complex dispersion relation and surface slow sound, by Logan Schwan (Laboratoire d’Acoustique de l’Université du Maine, UMR 6613 CNRS)

DENORMS Action’s Workshop “Modelling of high performance acoustic structures Porous media, metamaterials and sonic crystals”, Rome, 24-25th January 2017

Website of DENORMS Action

Programme of the Workshop

Acoustic metamaterials (i.e. subwavelength absorbers), metasurfaces and sonic crystals for audible sound manipulation

Speaker: Logan Schwan (Laboratoire d’Acoustique de l’Université du Maine, UMR 6613 CNRS)

Abstract: 

The emergence of surface waves at the boundaries of impervious media when structured with resonators has paved the way to novel applications in imaging techniques: the dispersion relation of the surface waves can be tuned by designing the properties of the surface microstructures.

While the dispersion relation is usually presented for real wavenumbers, the surface wavenumber is actually a complex quantity, the imaginary part of which accounts for the attenuation of the waves induced by the losses. These viscothermal losses are unavoidable in practice and govern the magnitude of resonances which can be responsible for bandgaps. If methods to theoretically predict the real part of the surface wavenumber, and retrieve it experimentally are now sufficiently robust, retrieving the imaginary part of it can be challenging (Geslain et al., 2016). In particular, it requires to distinguish the attenuation of the wave induced by the losses from that induced by the geometrical spreading of energy from the source.

Here, the propagation of Surface Acoustic Waves (SAWs) at a metasurface is investigated both theoretically and experimentally. The objective is to predict and retrieve experimentally the real and imaginary parts of the complex SAW wavenumber in the presence of the viscous and thermal losses in the resonators.

The metasurface consists of the two-dimensional and periodic arrangement of circular boreholes resonators. The theoretical model relies on a boundary layer approach using plane wave expansion, which provides the effective metasurface admittance governing the complex dispersion relation of plane SAWs. The admittance model is first validated experimentally for plane wave reflection at normal incidence by impedance tube measurements on a single cell. The complex dispersion relation of SAWs is then derived using the effective admittance and the results are compared with the data from semi-anechoic chamber measurements.

In the experiments, the metasurface consists of 29x40=1160 resonators and is excited by a speaker at one edge. The pressure field has been measured along a line in front of the speaker and the complex dispersion relation is retrieved using the Spatial Laplace Transform for Complex Wavenumbers (SLaTCoW). Due to the point-source excitation from the speaker, the geometrical spreading of the wave is accounted for using the Green function of the point-source above the admittance surface.

Results show that, the significant admittance contrast around the resonance leads to the deviation of the SAW dispersion relation from the air-line, with a group velocity smaller than the sound speed. However, the capability to guide waves along the surface with such a “slow sound” is rapidly hindered by the SAW attenuation as the dispersion curve approaches the Bragg limit: as the group velocity approaches zero, the SAW attenuation strongly increases with a characteristic attenuation length of the order of the SAW wavelength. That prevents the effective propagation of the wave at the surface, and localizes the sound field near the speaker. Experimental and theoretical results are in good agreement, which suggests that the complex dispersion relation is related more to the metasurface admittance and less to the nature of the excitation.