  The aim of this article is to study the attenuation of transient
low-frequency waves in 2D lattices in both plane and antiplane problems. The
main idea of this article is that analytical solutions to problems of mechanics
of discrete periodic media can be obtained by a method of asymptotic inversion
of the Laplace and Fourier transforms in the vicinity of the quasi-front of
infinitely long waves; moreover, in this method it is possible to take into
account the contribution of short waves. Using this method, we obtain
asymptotics of perturbations in lattices in plane and antiplane formulations
under a local transient load. Besides, we show that equations describing 2D
plane motion of a square lattice can be represented in the form of two linearly
independent wave equations, each of which contains one unknown function only.
By analogy with the theory of elasticity, one equation describes the
propagation of shear waves in the lattice, while the other equation describes
the propagation of longitudinal waves. As a result, it is shown that, in a
homogeneous infinite lattice, a load can be specified in such a manner that
either predominantly longitudinal or predominantly shear waves are formed. The
problems under study are also solved by a finite difference method. The
qualitative and quantitative correspondence of asymptotic and numerical
solutions is shown.
