We present a numerical approach to study solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation on R x T. The dependence on the coordinate x is considered on the compactified real line, and the dependence on the coordinate y is assumed to be periodic. Critical behavior, the formation of a shock in the solutions, is of special interest. The latter permits the numerical study of Dubrovin's universality conjecture on the break-up of solutions to the Kadomtsev-Petviashvili equation. Examples from a previous paper on dKP solutions studied numerically on T-2 are addressed, and the influence of the periodicity or not in the x-coordinate on the break-up is studied.