How to derive a rate (e.g. water dripping) from an audio file
Beat detection is not what you are looking for.

Beat detection aims at looking for a periodic, sometimes implicit pulse in music signals. The events you are interested here are not periodic and every drip counts. What you are looking for is instead "Onset detection", to recognize each percussive event in the audio signal. There are ready to use software solution like aubio or sonic annotator to extract the timestamps at which percussive events occur in an audio file. From there, it is only a matter of running a sliding window over your output, count events within the window and divide by the duration of the window.