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In this paper we present an energy efficient route discovery method for reactive protocols in wireless ad hoc networks. In these types of protocols, the route discovery process typically uses expanding ring search discovery method. While expanding ring search method is efficient as compared to simple flooding, we present a method which further improves energy consumption and reduces route discovery overhead. The proposed method reduces the broadcast cost and as a result decreases the overall energy consumption. The expanding ring search method is modified and a new approach, which uses disks instead of rings, is presented. In this method, the node discovery process uses regions similar to disks, which have variable dimensions, and as a result, reduction in time and energy consumption is accomplished. The expanding disk search is suitable for various types of implementation scenarios ranging from time to energy critical applications. In addition, the analysis for the optimal number of disks in the expanding disk search algorithm is also presented.