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Underwater acoustic (UWA) sensor network deployments may be used in many applications for environmental, scientific, military, and commercial purposes. Several topologies are in use, but the most widely used topology is linear. Indeed, placing nodes on a single line offers more opportunities in terms of large coverage and high-rate services. The UWA channel is a shared medium. Thus, a medium access control (MAC) protocol is necessary, primarily to regulate and coordinate nodes' access. MAC protocol design should take into consideration large propagation delays to favor better network throughput. Performance of most developed protocols in linear topologies does not exceed 1 in terms of normalized network throughput, or equivalently, channel utilization. We explore transmission schedules in three important contexts. Single collision domain with unicast traffic. In an N-node network, we develop transmission schedules achieving a normalized network throughput of 2 - (2/N). This is the best that can be done in such a context, as demonstrated using a general greedy approach combined with an exhaustive search for small-size networks. Single collision domain with broadcast traffic. We propose a periodic per-node fair schedule with the shortest period. Achievable throughput in such conditions is close to N/2. Likewise, we prove that the throughput is upper bounded by N - 1 under the per-node fairness constraint. Partially overlapping collision domains with unicast traffic. We consider a simple illustration of such a configuration. The proposed transmission schedule depicts a scenario where messages originate at one end of the network, and are sequentially relayed node by node (i.e., hop by hop) in the direction of the final destination located at the other end of the network. Furthermore, for all three discussed contexts, we build up computationally efficient algorithms that generate transmission schedules regardless of network size. We explore the idea of exploiting nonzero propagation delays for linear topologies to improve network throughput. In recent UWA sensor networks, the linear topology is a fundamental component that may be used to build more complex network configurations. This study would then serve as a base for future research into this area.