  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.
