Theory

History:

In 1908, A.M. Worthington released his book "A Study of Splashes" for which he studied the impact of drops on liquid surfaces. His research prompted a few others to investigate on the matter as well in the next 50 years, but it wasn't till after the second world war before people got really interested behind the reason rain making the sound that it does. There were two reasons for why this happened: First of all, there was the military need of sound detection, by studying the underwater sound of rain, the military could improve both its signal processing and filtering to allow for better detection as well as silence ships. The second reason was due to the use of acoustical measuring instruments for geophysical measurements as it was discovered that several measurements were distorted to something that could only be described as rain falling. 

The source of the sound:

As a (rain)drop falls into a liquid, it simply either produces a sound, or it doesn't. The reasoning as to whether or not it produces a sound depends solely on the impact velocity and the drop diameter, but if we look closely to the drop during impact we can see something interesting. When the conditions for sound are met, a small bubble forms just underneath the impact. We call this bubble entrainment. This entrainment has been studied a lot and it has been found that there is a region where there is a 100% chance of bubble entrainment (and thus a sound!), this can be seen in Figure 1.

Looking closer to the impact of a (rain)drop:

If we take a closer look to the various stages the bubble undergoes when hitting the water and also register the sound that gets created using a hydrophone, you end up with Figure 2. A droplet starts falling (a) and hits the surface (b). The impact creates a bubble under the surface (c, d & e), under which a very small pillar starts forming (f) that seperates more and more from the initial air bubble (g). Until now, no sound has been created yet). Eventually, the small pillar is fully seperated and became its own bubble (h & i). This is the entrainment we talked about before, notice how the hydrophone now picks up a sound. The entrained bubble slowly floats away while a jet is formed and shoots up (j), again without sound. All these frames are taken very close from each other, with only 38.5 miliseconds between frame a and frame j. 

The sounds frequency:

Figure 3 shows the Intensity spectrum level of two different rain events (event a and event b) along a frequency range. The straight solid line depicts the background noise level. As visible in the Figure, the main peak seems to be sitting around 13kHz to 15kHz, however sound can be distinguished from the background noise (at least in case b) starting from around 5kHz all the way to 50kHz.