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?!

  • $\begingroup$ Also if anyone knows any way I can get a copy of this paper I would be very grateful as apparently even my University doesn't have a subscription. $\endgroup$ Mar 23, 2013 at 13:13

1 Answer 1


I think that your difficulty arises from the fact that you are trying to think about this in terms of a static problem, whereas any "extreme" properties of metamaterials result from the essentially dynamic phenomena. I.e., you cannot really have a material with a negative mass or a negative module (although, admittedly, there are some ingenious mechanisms that engineers can construct for you to achieve the semblance of either).

The extreme properties of metamaterials are usually manifested in the vicinity of inclusion resonances. Suppose that you have a doubly-periodic composite. You can take an external excitation (think time-harmonic for the sake of simplicity) and tune it to excite a mode in one of the corners of the Brillouin zone. This would result in a standing wave, with all inclusions oscillating, some perfectly in phase, some perfectly out of phase. If you now de-tune your excitation a little bit, you will obtain a strong beating motion, which can have an extremely long characteristic wavelength. If you are interested in modelling the envelope of this beat, then you will find that its “effective properties” can often be described by the models with very unusual properties, such as the ones you described in your question. However, if the entire motion is considered, your mass remains reassuringly positive (and constant) and your modulae satisfy all of the usual constitutive requirements.

All interesting properties of (not necessarily acoustic) metamaterials are just interference phenomena. This situation is a bit similar to the famous experiments on superluminal light propagation through the strongly absorbing media: http://www.nature.com/nature/journal/v406/n6793/abs/406277a0.html


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