Time dependence of the x and y components of the force for the various initial sine wave conditions considered in this article, as indicated in the panels. In each row, the output time, traced by the parameter Δ in Eq. (53), increases from left to right, as indicated in each panel, with in the right panels, where Δ_sim corresponds to the output time in the simulations discussed in Sect. 5 (see also Table 1). Note that in this last case, the value indicated in the figure is not Δ but Δ_sim, but the differences are small. The exact solution given by Eq. (14) applied to the LPT displacement (dashes) is compared to the analytical formula (51) for F_x and with the LPT acceleration (15) for F_y (solid lines). The two top and two bottom rows use 40LPT and 15LPT, respectively, to compute the displacement field and its acceleration. In the two bottom rows, we assume z = 0.