The search for weak signals with unknown center frequency using energy detectors is a resource allocation problem. This paper considers good strategies for search when the total search time is fixed. The criterion for optimality considered in this paper is maximum probability of detection for specified false alarm probability. The problem is shown to be one in nonlinear programming which has a solution via dynamic programming methods. Several computer simulations are presented for representative cases of interest. A heuristic discussion of these strategies is presented together with suboptimal strategies that require a lower computational burden.