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But why? Why does vibration create larger surface tension (assuming that surface tension is what keeps the bubbles bubbly)? You would expect the opposite.

Oh, now I will not be able to sleep tonight.

I can imagine that a flat surface meeting the curve of the droplet makes for a smaller surface area of contact than if the convex curve of the droplet meets a concave curve of the vibrating water. Maybe the weight of the droplet ensures that it will always be in a concave part of the vibrating surface. I don't know enough about surface tension at an atomic level to really have any good ideas on this.

Oh, I just realized that more contact between the droplet and the larger body of water would through attraction probably lead to coalescence. The vibration probably keeps the droplet and water surface from making enough contact for long enough time to join together. Again, I need to know more about the underlying physics.

My guess is that in order for the two bodies of water (the main body and each droplet) to coalesce, their molecules have to agree on a resonating frequency. When there is no other interference, the droplets take on the frequency of the larger body almost immediately. Since the main body is resonating at a frequency above normal, the droplets can't "catch up", and so bounce and wobble along, trying to achieve equilibrium with the larger body, and failing.

Just a guess.

I imagine it's a result of various forces and circumstances (I don't think it's a coincidence that the droplets were soapy water which would increase it's surface tension/bubble strength).

Also keep in mind that a droplets surface would be a mesh of the outermost water molecules held together by their polar attraction. As the sphere bounces and moves its surface would have mini waves and ripples along it that would push against and then move away from the molecules on the water surface below it as the kinetic and polar forces acted.

If you imagine that every sphere of water had portions of its surface moving away from the water surface below and then oscillating back towards the surface while the molecules on the spheres surface that had been touching the water surface below would begin to oscillate back into the sphere.

This would create many points of contact oscillating against and away from the water surface below and thus there might not be enough contact/pressure between the 2 surfaces for it to coalesce at any given time. Imagine bugs whose feet are tiny enough for them to "stand" on water due to surface tension and the principle would be the same. It'd be like an infinite number of these bugs legs jumping up and down on the water at a microscopic level.

Also I'm not familiar enough with how water molecules align themselves while at the surface of something so perhaps the alignment of their atoms helps as well?

Thats all a guess though I'm sure you could google the real answer.