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In acoustic metamaterials we have simultaneously negative bulk modulus, $\beta$, and effective mass density, $\rho$.

I understand how one can obtain a -ve $\rho$ by constructing a solid-solid system with vastly different speeds of sound, as this can be considered as a mass-in-mass system connected by springs see here.

But how to get a negative bulk modulus eludes me, I am aware that in doubly negative acoustic metamaterials the negative bulk modulus is achieved by having a sphere of water containing a gas i.e. bubble-contained-water spheres. But I can't see why this would result in a negative composite bulk modulus.

I mean obviously water has an extremely high bulk modulus and is practically incompressible, whilst air is highly compressible so external pressure on the system would fail to compress the water however the pressure would be transmitted to the gas which would compress, thus creating an extremely low-pressure region in between the gas and water (the water wouldn't expand due to it's large bulk modulus right?) and thus the gas would then re-expand and possibly exert pressure on the water - I suppose if this pressure exceeded the external pressure then the sphere might expand resulting in an expansion of the system upon application of external pressure and thus a negative bulk modulus???

I believe the answer may lie in this paper and I will attempt to get my University vpn to work to see if I can access it.

Any information about acoustic metamaterials would be greatly appreciated.

P.S: How is metamaterials not an existing tag?!