My Tek 2212 DSO isn't the latest technology, as can be seen from the sparse dots of the waveform, but, with a microphone connected to it, it does reveal that the pulses rapidly rise and exponentially decay, falling to half their peak level within 3mS. Sound travels at approximately 340m/s, but, since it has to go out and come back before the echo is received, its speed is effectively halved to approximately 170m/s for echolocation in terms of distance to the object being perceived. That means the approximately 3mS pulse effectively occupies about half a meter of echolocation range.
The pulse waveform oscillates at around 2.3kHz, perhaps as a result of a resonant frequency of the piezoelectric transducer. I have ordered more transducers and plan to employ them in different ways and possibly modify them to alter any resonances they may have.