  We investigate thin-slit diffraction problems for two-dimensional lattice
waves. The peculiar structure allows us to consider the problems on the
semi-infinite triangular lattice, consequently, we study Dirichlet problems for
the two-dimensional discrete Helmholtz equation in a half-plane. In view of the
existence and uniqueness of the solution, we provide new results for the real
wave number $k\in (0,3)\backslash\{2\sqrt{2}\}$ without passing to the complex
wave number and derive an exact representation formula for the solution. For
this purpose, we use the notion of the radiating solution. Finally, we propose
a method for numerical calculation. The efficiency of our approach is
demonstrated in an example related to the propagation of wave fronts in
metamaterials through two small openings.
