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Identifying acoustic sources with high spatial resolution and precision is significant for machinery noise control. Off-grid compressive beamforming is an effective method for creating acoustic maps with a high spatial resolution. In this study, a two-dimensional off-grid compressive beamforming method via weighted block ℓ 1 norm regularization (WBS-OCSBF) is proposed. The weighted ℓ 1 norm offers a more uniform sparsity penalty compared to the ℓ 1 norm. This method can enhance the performance of off-grid compressive beamforming. Additionally, it is easily implemented on mature linear programming solvers and converges with only a few iterations. Numerical simulations are conducted to analyze the effects of several major parameters on the performance of the proposed method. The simulation results demonstrate the superiority of the proposed method over some traditional methods. The proposed method is then validated using a loudspeaker noises localization experiment. A gearbox transmission system and a transformer are used as two specific objects to explore the industrial application value. The proposed method can effectively localize the gear meshing noise and transformer discharge faults with a high spatial resolution. • Compressive beamforming via weighted Block L1 Norm is proposed in this paper. It enhances the spatial sparsity of the reconstruction results. • Simulation and speaker positioning experiments verify the effectiveness of the method. • Two industrial application experiments proved the potential industrial application value of the method proposed in this paper. [ABSTRACT FROM AUTHOR]