The first step to building this was to figure out how it worked – the paper was a great help here.  Each transducer needed to be fed a 40kHz wave form – but with a phase relative to the other transducers calculated to ensure that the high pressure areas converged at a specific point in 3D space.  Moreover, the signals from the downward-firing transducers had to be 180 degrees out of phase with the ones firing upwards.  

The original Nature paper lists a formula that allows the calculation of the phase delay φ_T for each transducer given a focus point p:

Where N represents the number of discrete phases, k the wave number (k=2 pi/ lambda), p_t represents the position of the transducer and the function d calculates the Euclidean distance.  The function takes the focus point as an argument and for each transducer, calculates the distance from the focus point to the transducer, multiplies that by the so-called “wave-number” that converts the distance into a phase and then limits the resulting phase to an integer number from 0 to N-1.  Modulo adding N/2 to the result gives the phase for the downward firing transducers.