This paper is concerned with the detection of a narrow‐band signal using two hydrophones. Specifically, it addresses itself to the problem of finding the “optimal” linear combination of sample spectra, cospectrum, and quadrature spectrum to use as a detection test criterion. The “optimal” detector is defined here to be the one that maximizes the average probability of detection for given probability of false alarm and given observation time, where the average is taken over all possible noise‐phase difference conditions. It is shown that under not unrealistic conditions, the “optimal” detector, among all those that are linear in the sample spectra, co‐, and quadspectrum, is the one that adds the outputs of the detectors and squares the result. However, if one is willing to consider detectors that are not linear combinations of the various spectra, there are detectors that offer significant improvements over the sum and square one and that require no additional information. [Work supported by the Bureau of Ships, U. S. Navy.]