This investigation is aiming to detect and track a moving target using its acoustic emissions and applying a new beam-forming approach. An original algorithm of medium computational complexity is reported that is capable of locating, with good accuracy, the direction of an acoustic source with only a few microphones. This algorithm circumvents the known localisation ambiguities due to the unwanted "grating lobes", by using all spectral content of the acoustic source and progressively and narrowly focusing its beam in the direction of the target. The algorithm starts by locating the direction of the target with low angular resolution, using only low frequency components of the acoustic emissions; in this way, no grating lobes occur even with only a few microphones. Then, the angular resolution is increased by processing higher frequencies of the spectrum, but profiting of the information on the angular position achieved in the previous stages. Some algorithmic means are introduced to increase the performance of the algorithmic against environmental noise. This scheme is described in detail. Some experimental tests are included and discussed.