An analytical solution for the dynamics of a square lattice under general boundary conditions

The dynamics of a finite simple square lattice is studied analytically by solving the dynamical equation as a partial difference equation with two discreate cell indices under physically significant general boundary conditions. The Rosenstock-Newell lattice model, with nearest-neighbor central and shear harmonic forces only, is used. The solutions contain terms representing all the possible reflections of phonons from the boundary, and the resulting phonon frequencies have explicit dependence on the size of the lattice as well as the boundary conditions imposed. (Author)