We design a new frequency-domain, finite-difference approach, based on a displacement formulation, which correctly describes the stress-free conditions at a free surface. In the conventional, displacement-based finite-difference method, we assign both displacements and material properties such as density and Lamé constants to nodal points (a node-based grid set), whereas in our new finite-difference method, displacements are still defined at nodal points but material properties within cells (a cell-based grid set). In our new finite-difference technique using the cell-based grid set, stress-free conditions at the free surface are described by the changes in the material properties without any additional stress-free boundary condition. By conducting accuracy tests, we confirmed that the new second-order finite differences formulated in the cell-based grid set generate numerical solutions compatible with analytic solutions within the range of errors determined by dispersion analysis; the new, cell-based, weighted-averaging finite-difference method also yields better solutions than the old, node-based, weighted-averaging finite-difference method. The cell-based finite-difference method does not violate the reciprocity.