Existing works have widely studied relay-aided underwater acoustic networks under some specialized relay distributions, e.g., equidistant and rectangular-grid. In this paper, we investigate two fundamental problems that under which conditions a relay should be deployed and where to deploy it if necessary in terms of the energy and delay performance in linear underwater acoustic networks. To address these two problems, we first accurately approximate the complicated effective bandwidth and transmit power in the logarithm domain to formulate an energy minimization problem. By analyzing the formulation, we discover a critical transmission distance, defined as open distance, and explicitly show that a relay should not be deployed if the transmission distance is less than the open distance and should be otherwise. Most importantly, we derive a closed-form and easy-to-calculate expression for the open distance and also strictly prove that the optimal placing position is at the middle point of the link when a relay should be introduced. Moreover, although this paper considers a linear two-hop relay network as the first step, our derived results can be applied to construct energy-efficient and delay-friendly multi-hop networks. Simulation results validate our theoretical analysis and show that properly introducing a relay can dramatically reduce the network energy consumption almost without increasing the end-to-end delay.