Wireless Sensor Networks (WSNs) enable the surveillance and reconnaissance of a particular area with low cost and less manpower. However, the biggest problem against the commercialization of the WSN is the limited lifetime of the battery-operated sensor node. Taking this problem into account, a mobile sink is deployed as a robot, vehicle or portable device to only activate the sensor nodes that are interesting to the sink and leaves other nodes deactivated for. This can considerably extend the lifetime of the sensor nodes compared to existing power management algorithm of using all nodes. However, in this environment, the mobility of the sink raises new issues of energy efficiency and connectivity in communications. To solve these issues, we propose a DRMOS (Dynamic Routing protocol for Mobile Sink) method that includes a designated wake-up-zone to make sensor nodes prepare for an incoming sink. The shape of the wake-up-zone is dynamically changing to reflect the past moving patterns of the sinks. Moreover, we present the extensive simulation results and recommend parameters for practical use of DRMOS from the simulation analysis.