# Relation between shapes and vibrations

I have observed the following shape on surface of water when the bottle was standing on a cooling water machine. I was amazed by the formation because I had not seen this before. I'm not completely sure if it was caused just by the vibration of the cooling machine.

Specifically, what caused this shape to be appeared? If it is just shaped by vibrations, then what kind and in which frequency made it appear?

• It might be vibrating at a fundamental frequency of the water surface. I am not sure what equations would be best used to describe it though. The closest thing I can think of are solutions to the wave equation on a circular membrane, but I am sure someone knows more than I do here. – Aaron Stevens Nov 12 '18 at 22:55
• en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane – Chris Degnen Nov 12 '18 at 23:02
• – Farcher Nov 13 '18 at 0:02

The surface of the water can be modeled as if it were a slackly-suspended drum head. If the body of water beneath it is shaken, then standing waves will form on the "drum head" when a given frequency component of the shaking source coincides with a resonant standing wave pattern that fits inside the perimeter of the water surface.

Those waves will be capillary waves, in which the compliance arises from surface tension and the inertance is furnished by the mass of the water that is displaced.

The fascinating part of this puzzle is that because the water surface is two-dimensional, the standing wave patterns will be too, and the resulting standing wave resonances can be quite complex- and they will shift around in interesting ways when the driving frequency is changed.