  This paper addresses fundamental questions arising in the theory of
Bloch-Floquet waves in chiral elastic lattice systems. This area has received a
significant attention in the context of "topologically protected" waveforms.
Although practical applications of chiral elastic lattices are widely
appreciated, especially in problems of controlling low-frequency vibrations,
wave polarisation and filtering, the fundamental questions of the relationship
of these lattices to classical waveforms associated with longitudinal and shear
waves retain a substantial scope for further development. The notion of
chirality is introduced into the systematic analysis of dispersive elastic
waves in a doubly-periodic lattice. Important quantitative characteristics of
the dynamic response of the lattice, such as lattice flux and lattice
circulation, are used in the analysis along with the novel concept of "vortex
waveforms" that characterise the dynamic response of the chiral system. We note
that the continuum concepts of pressure and shear waves do not apply for waves
in a lattice, especially in the case when the wavelength is comparable with the
size of the elementary cell of the periodic structure. Special critical regimes
are highlighted when vortex waveforms become dominant. Analytical findings are
accompanied by illustrative numerical simulations.
