Periodic elastic medium in which periodicity is relevant

We analyze, in both (1+1) and (2+1) dimensions, a periodic elastic medium i n which the periodicity is such that at long distances the behavior is alwa ys in the random-substrate universality class. This contrasts with the mode ls with an additive periodic potential in which, according to the field-the oretic analysis of Bouchaud and Georges and more recently of Emig and Natte rmann, the random manifold class dominates at long distances in (1+1) and ( 2+1) dimensions. The models we use are random-bond Ising interfaces in hype rcubic lattices. The exchange constants are random in a slab of size Ld-1 x lambda and these coupling constants are periodically repeated, with a peri od lambda, along either {10} or {11} [in (1 + 1) dimensions] and {100} or { 111} [in (2+1) dimensions]. Exact ground-state calculations confirm scaling arguments which predict that the surface roughness w behaves as w similar to L-2/3,L much less than L-c and w similar to L-1/2,L much greater than L- c with L(c)similar to lambda(3/2) in (1+1) dimensions, and w similar to L-0 .42,L much less than L-c and w similar to ln(L),L much greater than L-c wit h L(c)similar to lambda(2.38) in (2+1) dimensions.