  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
$L^{d-1} \times \lambda$ and these coupling constants are periodically repeated
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: $w \sim L^{2/3}, L \ll
L_c$ and $w \sim L^{1/2}, L \gg L_c$, with $L_c \sim \lambda^{3/2}$ in
$(1+1)$-dimensions and; $w \sim L^{0.42}, L \ll L_c$ and $w \sim \ln(L), L \gg
L_c$, with $L_c \sim \lambda^{2.38}$ in $(2+1)$-dimensions.
