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In wireless sensor networks (WSNs), asynchronous sleep-wake scheduling protocols can be used to significantly reduce energy consumption without incurring the communication overhead for clock synchronization needed for synchronous sleep-wake scheduling protocols. However, these savings could come at a significant cost in delay performance. Recently, researchers have attempted to exploit the inherent broadcast nature of the wireless medium to reduce this delay with virtually no additional energy cost. These schemes are called “anycasting,” where each sensor node forwards the packet to the first node that wakes up among a set of candidate next-hop nodes. In this paper, we develop a delay-optimal anycasting scheme under periodic sleep-wake patterns. Our solution is computationally simple and fully distributed. Furthermore, we show that periodic sleep-wake patterns result in the smallest delay among all wake-up patterns under given energy constraints. Simulation results illustrate the benefit of our proposed schemes over the state of the art.