10.3390/sym14081524The Stability of a Hydrodynamic Bravais Latticehttp://orcid.org/0000-0002-4667-6829CouchmanMiles M. P.Dr.Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UKmc2277@cam.ac.ukhttp://orcid.org/0000-0001-9944-9198EvansDavis J.Dr.Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAdjevans@mit.eduhttp://orcid.org/0000-0002-7936-7256BushJohn W. M.Dr.Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAbush@math.mit.edu2607202214821072022e1524We present the results of a theoretical investigation of the stability and collective vibrations of a two-dimensional hydrodynamic lattice comprised of millimetric droplets bouncing on the surface of a vibrating liquid bath. Using the theoretical model, we derive the linearized equations of motion describing the dynamics of a generic Bravais lattice, as encompasses all possible tilings of parallelograms in an infinite plane-filling array. Focusing on square and triangular lattice geometries, we demonstrate that for relatively low driving accelerations of the bath, only a subset of inter-drop spacings exist for which stable lattices may be achieved. The range of stable spacings is prescribed by the structure of the underlying wavefield. As the driving acceleration is increased progressively, the initially stationary lattices destabilize into coherent oscillatory motion. Our analysis yields both the instability threshold and the wavevector and polarization of the most unstable vibrational mode. The non-Markovian nature of the droplet dynamics renders the stability analysis of the hydrodynamic lattice more rich and subtle than that of its solid state counterpart.Division of Civil, Mechanical & Manufacturing Innovation2154151© 1996-2022 MDPI (Basel, Switzerland)2022This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly citedbouncing dropletsFaraday waveslattice instabilitynormal-mode analysisphononsSymmetrySymmetry2073-8994MDPI