Periodic elastic medium in which periodicity is relevant

We analyze, in both (1+1) and (2+1) dimensions, a periodic elastic medium in which the periodicity is such that at long distances the behavior is always in the random-substrate universality class. This contrasts with the models with an additive periodic potential in which, according to the field-theoretic analysis of Bouchaud and Georges and more recently of Emig and Nattermann, 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 hypercubic lattices. The exchange constants are random in a slab of size L(d-1)xlambda and these coupling constants are periodically repeated, with a period 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 approximately L(2/3), L<>L(c) with L(c) approximately lambda(3/2) in (1+1) dimensions, and w approximately L0.42,L<>L(c) with L(c) approximately lambda(2. 38) in (2+1) dimensions.