[Mithaldu]

This is correct. The movements match up just fine, but the reaction time there is something above 2 seconds.

[foobarqux]

They need to buffer to get a Fourier transform and to do the autocorrelation.

[tobr]

They say they use an "inaudible high frequency soundwave", so that should be > 20kHz. Shouldn't a buffer of a few milliseconds be more than enough then?

[foobarqux]

Presumably the buffer is longer to make the system more robust by avoiding spurious detections, not because of some fundamental limit like the Nyquist rate. You would need to set the buffer size experimentally.