Do ultrasonic transducers generate sound because of current, voltage, or change in either of those?
Piezoelectric crystals appear electrically as capacitors (since they basically consist of two electrodes surrounding a dielectric material. However, at their resonant frequency they behave closer to a pure resistance. In any case, as with any electrical impedance, applying a voltage to them will cause a current to flow. Similarly if you force a current through them, they will develop a voltage equal to the product of the current and their impedance. Usually, though, piezoelectric crystals are considered to be voltage driven because their reaction to electrical stimulation is easier to describe and measure in terms of voltage, rather than current.  The current will vary considerably with frequency as the crystal's resonant frequency is approached. In any case, if you want to generate ultrasound you will need to drive the crystal with an AC signal preferably at or near its resonant frequency to get the best efficiency (its also easier to drive as, mentioned earlier, it appears close to a pure resistor at its resonant frequency). Driving it with the waveform you show is probably not a good idea as only the spectrum of the waveform near the crystal's resonant frequency will be useful. The fast rise and fall times mean high harmonic energy which could excite higher modes in the crystal and cause problems with your drive circuit as the harmonic currents could get large.