We propose a simple mechanical system that can be used as a toy model for calculating the effective density of acoustic metamaterials. Through analytical calculations, it gives a better understanding of how the effective density can become negative, when the system responds elastically instead of inertially. We show that this toy model can reproduce, qualitatively, the acoustical behaviour of some real acoustic metamaterials.

Liquid foams are known to be highly efficient to reduce the amplitude of acoustic or even blast waves. More surprising, this last years we have shown that liquid foams, which are natural isotropic materials, act as « acoustic metamaterials » exhibiting a negative effective density for wavelength larger than the bubble size. In this range of frequency, acoustic propagation becomes evanescent and the sound is fully blocked. This observation breaks with a widespread opinion : to efficiently dissipate acoustic energy one needs an open-cell material in order to permit large-scale motions of air in the whole structure. Indeed, a liquid foam is intrinsically a closed-cell structure, its bubbles of gas are shaped by thin films attached to a liquid skeleton, and far from being a limitation for soundproofing, these thin deformable liquid films allow a strong coupling between air and liquid motions and lead to the extraordinary acoustic properties. Very recently, we have pushed forward our investigations on the origin of acoustic attenuation in liquid foams and we have revealed that losses are due to both thermal and viscous losses in audible.