Crown shape in water waves? Watching this slow motion video of waves caused by a droplet falling on water, I am puzzled by the first wave shape.

The first wave rises with circular symmetry as I expected, but is not even in all directions, it rather has peaks with certain angular periodicity, much like a crown. Each of these peaks give rise to small droplets. 
After this the outgoing waves move away symmetrically and with constant elevation through all angles. 
Is this phenomenon (crown shape) studied and described already? How to explain such angular difference in amplitude, and how come is not present in the following waves?
 A: 
Is this phenomenon (crown shape) studied and described already?

Wetzel (1987) describes and tries modeling the splash process, including the crown. (There's a link to the PDF.)
A reference to it may be found on p. 208 of Surface Waves and Fluxes, Volume II — Remote Sensing, Editors: Geernaert, G.L., Plant, W.J. (Eds.). (Google might show you the preview.)

Crown-forming instability phenomena in the drop splash problem,
Rouslan Krechetnikov, George M. Homsy, 2009 (warning: pdf):

With the help of a set of experiments and new theoretical un- derstanding we were able to gain new insights into the nature of the instability responsible for the crown spike formation, as well as into the peculiarities of the crown evolution. In particular, we discovered that there are three major types of crowns — axisym- metric, regular, and irregular — and their selection is done at the very early stages of ejecta formation. Through estimates of growth rates and our kinematic measurements, the Richtmyer–Meshkov instability mechanism is found to play a dominant role at short times. The crown dynamics also exhibits a nontrivial bifurcation behavior, which includes frustration and irregular crown phenom- ena.
  The above observations require new theoretical advances both at the linear and nonlinear levels of description in order to achieve a complete fundamental understanding of the crown formation and a quantitative comparison with experiments.


Additional information from an incomplete review of publications
This phenomenon is very known and still receives attention, as can be seen from the number of recent publications (see below for an incomplete list). 
It was first photographed and described in 1908 by A. M. Worthington in his work A study of splashes.
The dynamics and relevant physics is clear and checked experimentally. The important elements have been identified and used to clasify the different forms of the phenomenon. For a short covering see this PSE answer or this paper that also tries to identify different regimes. That last paper shows that the "crown shape" is formed in the initial state of the collision and that it is at this moment that its features are determined, and hints on the origin being the Richtmyer-Meshkov instability.
A later paper shows that the phenomenon has several elements in agreement with calculated values from a model based on the Rayleigh-Plateau instability.
Finally, after a chase over recently published articles, however an incomplete and inexhaustive one, I have not find a consensus on the origin of the phenomenon, neither a theoretical model that explains it. Below there are some of the articles found:
Origin of ejecta in the water impact problem
 Krechetnikov, R.  2014 
Physics of Fluids
Numerical analysis of droplet impact onto liquid film
 Shetabivash, H., Ommi, F., Heidarinejad, G.  2014 
Physics of Fluids
Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth
 Lombardini, M., Pullin, D.I., Meiron, D.I.  2014 
Journal of Fluid Mechanics
Stability on time-dependent domains
 Knobloch, E., Krechetnikov, R.  2014 
Journal of Nonlinear Science
Splash wave and crown breakup after disc impact on a liquid surface
 Peters, I.R., Van Der Meer, D., Gordillo, J.M.  2013 
Journal of Fluid Mechanics
A: It seems to be an effect of the interaction between the liquid and the gas surrounding it, though the mechanism is not understood. You can see videos here of droplets falling in slow motion at atmospheric pressure, and low pressure. The crown vanishes completely at low pressure. The authors of the study claim

The mechanism by which the gas affects splashing remains unknown. Recently, we used high-speed interference imaging to measure the air beneath all regions of a spreading viscous drop.  Although an initial air bubble is created on impact, no significant air layer persists until the time a splash is created. This suggests that splashing in our experimentally accessible range of viscosities is initiated at the edge of the drop as it encroaches into the surrounding gas - not by air trapped beneath the drop at impact.

A: This phenomena appears tp be related to the Saturn's hexagonal aurora (storm, ?) 
nytimes video , youtube 
As the sphere approaches the surface the air is compressed and it happens a speedy radial exit , that's why with no atmosphere the effect is not present (as seen on the video of the answer of Bardamu).   
explanation on nytimes

In laboratory experiments with rotating fluids, they were able to
  reproduce the six-sided shape, providing reassurance that there is
  nothing supernatural going on at Saturn.
Scientists on Earth have been pondering what causes the vortex to take
  such an unnatural-looking shape.
In 2010, Ana Aguiar of Lisbon University and colleagues pointed out
  that the position of the hexagon on Saturn corresponded to the
  latitude of a narrow and very speedy jet stream. They suggested that
  friction with slower-moving atmosphere on either side of the jet
  stream would create eddies, miniature hurricanes, that would push the
  jet stream north and south as it went around the planet, resulting in
  a wave shape.

A very recent study "Origin and dynamics of vortex rings in drop splashing" (Nature Jul/2015) shows in detail the formation of the vortexes but noting on the crown.  
To show the similarity of the crown effect and the Saturn hexagon I toke a  from the video linked in the question,   and from . 
