In this paper, two generalized likelihood ratio (GLR) detectors are presented for the case of multiple snapshots of test data to detect the presence of an underwater acoustic source in the deep ocean. The two GLR detectors are termed the eigenvalue detector (EVD) and the constant false alarm rate eigenvalue detector (CFAR EVD), respectively. Theoretical analysis and numerical results show that for a given input signal-to-noise ratio (SNR) of the array, the GLR detectors achieve higher output SNRs when the spatial dimension of test data decreases. To further enhance the detection performances of the GLR detectors, we propose a dimension-reduced (DR-GLR) method based on array sampling of modal information. This DR-GLR method combines the characteristics of sound propagation and array receiving. According to normal mode theory, acoustic signals emitted from the acoustic source lie in the modal space spanned by the sampled modal information of the array. Resulting from the restriction of the array size, it often occurs in deep ocean when the dimension of " effective modal subspace” is less than that of the test data which is equivalent to the number of hydrophones. Based on this phenomenon, we reconstruct the modal information by merely retaining the " effective modal subspace” to formulate the dimension reduction matrix. The DR-GLR test statistics is deduced by employing the dimension reduction matrix when using the vertical linear array (VLA) and the horizontal linear array (HLA), respectively. The DR-GLR detectors when using an HLA require more computational amount than when using a VLA. Simulation experiments are conducted to analyze the detection performances of the two GLR detectors, and verify the performance improvement effects of DR-GLR detectors. The numerical results show that the CFAR EVD presents good robustness to the uncertainty of the noise power and the DR-GLR detectors outperform the GLR detectors in detection performance. It also turns out the acoustic signals received by the HLA lie in a lower-dimensional " effective modal subspace” than by the VLA, and thus when using an HLA the DR-GLR detectors present higher detection probabilities than using a VLA. Moreover, the smaller the dimension of the " effective modal subspace”, the better the performance improvement of the DR-GLR detectors will be. The dimension of the " effective modal subspace” increases with hydrophone spacing and/or the source frequency increasing.