Adaptive array processing using a quiescent, or conventional, beamformer with adaptive noise cancellation have been implemented for signals based upon plane-wave representations. They have an estimator/subtractor structure and can be interpreted as generalized sidelobe cancellors. One of the useful features of this implementation is that the dimensionality of the adaption space can be reduced from the number of array sensors. While this reduction leads to a partially adaptive processor, it can lead to better performance in applications which are degrees of freedom deficient, or where the number of snapshots is less than the number of sensors, such as large arrays with short-term stationary fields or active systems. Sidelobes in the range/depth ambiguity plane and a sparse number of snapshots are both problems in current matched-field processing. The partially adaptive array leads to an implementation which has both the useful properties of the conventional processor and adaptive sidelobe cancellation in a reduced dimension space.