Wave polarisation in a dynamic elastic lattice

In an elastic lattice, the continuum notions of pressure and shear waves cannot be employed to describe waves propagating in the discrete medium, especially when the wavelength is comparable with the size of the elementary cell of the periodic system. Here, the concepts of lattice flux and lattice circulation of the displacement field are used to characterise waves propagating in the micro-structured medium. In particular, it is shown that the displacement field can be decomposed into flux-free and circulation-free components, that in the long wavelength limit correspond to shear and pressure waves. The proposed procedure can be applied for any value of the wave vector. A surprising result is that there are lines in the reciprocal space where waves in the lattice have similar features as those observed in the long wavelength limit.