Underwater acoustic sensor networks (UASNs) have attracted significant interest and been developed for a wide range of applications. Using acoustic signal for transmitting in underwater leads the effects on long propagation delay, low bandwidth, and low data rate. These characteristics result the high end-to-end delay from the sensor to a sink. In this paper, an important property of acoustic signal that the speed of acoustic signal varies with water depth has been considered to explore a new routing protocol in UASNs. This property indicates that the speed of acoustic signal becomes faster when water depth is below 1000 meters. It implies that the shortest end-to-end delay path may not be the shortest path directly form the source to the sink. Based on this property, a per-hop based routing protocol, named depth adaptive routing protocol (DARP), is proposed. DARP takes the acoustic speed in different water depth into consideration. DARP can find a shortest end-to-end delay path to transmit the data. The simulation results also verify that DARP actually reduces the end-to-end delay and outperforms other routing protocols in terms of end-to-end delay.