This paper studies caching in (K+L-1) x K partially connected wireless linear networks, where each of the K receivers locally communicates with L out of the K+L-1 transmitters, and caches are at all nodes. The goal is to design caching and delivery schemes to reduce the transmission latency, by using normalized delivery time (NDT) as the performance metric. For small transmitter cache size (any L transmitters can collectively store the database just once), we propose a cyclic caching strategy so that each of every L consecutive transmitters caches a distinct part of each file; the delivery strategy exploits coded multicasting and interference alignment by introducing virtual receivers. The obtained NDT is within a multiplicative gap of 2 to the optimum in the entire cache size region, and optimal in certain region. For large transmitter cache size (any L transmitters can collectively store the database for multiple copies), we propose a modified caching strategy so that every bit is repeatedly cached at consecutive transmitters; the delivery strategy exploits self-interference cancellation and interference neutralization. By combining these schemes, the NDT is optimal in a larger region. We also extend our results to linear networks with heterogeneous receiver connectivity and partially connected circular networks.