© 2020. The American Astronomical Society. All rights reserved. Explaining the observed diversity of supernovae (SNe) and the physics of explosion requires knowledge of their progenitor stars, which can be obtained by constraining the circumstellar medium (CSM). Models of the SN ejecta colliding with the CSM are necessary to infer the structure of the CSM and tie it back to a progenitor model. Recent SNe I revealed CSM concentrated at a distance r ∼ 1016 cm, for which models of SN interaction are extremely limited. In this paper, we assume the concentrated region is a "wall" representing swept-up material, and unswept material lies outside the wall. We simulate one-dimensional hydrodynamics of SNe Ia and Ib impacting 300 unique CSM configurations using RT1D, which captures the Rayleigh-Taylor instability. We find that the density ratio between the wall and ejecta-denoted A 0 or "wall height"-is key, and higher walls deviate more from self-similar evolution. Functional fits accounting for A 0 are presented for the forward-shock radius evolution. We show that higher walls have more degeneracy between CSM properties in the deceleration parameter, slower shocks, deeper-probing reverse shocks, slower shocked ejecta, less ejecta mass than CSM in the shock, and more mixing of ejecta into the CSM at early times. We analyze observations of SN 2014C (Type Ib) and suggest that it had a moderately high wall (10 ≲ A 0 ≲ 200) and wind-like outer CSM. We also postulate an alternate interpretation for the radio data of SN 2014C, that the radio rise occurs in the wind rather than the wall. Finally, we find that hydrodynamic measurements at very late times cannot distinguish the presence of a wall, except perhaps as an anomalously wide shock region.