Directionality of generalized acoustic sensors of arbitrary order

For several decades there has been a great deal of interest in acoustic sensors that can make multiple measurements at a single point in the ocean. The order of such sensors has been defined by linking it to the order of the Taylor series approximation of the pressure field at that point. Following this definition, the pressure, vector, and dyadic sensor is of order zero, one, and two, respectively. For this theoretical study, a multichannel three-dimensional spatial filter is derived for a directional acoustic sensor of arbitrary order. Explicit formulas are found for the filter coefficients that maximize the array gain (directivity index) of the filter as well as an explicit expression for the maximum array gain (directivity index). This process is repeated for the case of a first-order null placed in the direction opposite to the look direction of the multichannel filter. Finally, an example is presented which tracks the array gain and beamwidth of a third-order acoustic sensor as the order of the null is assigned values 0, 1, 2, and 3.