I have a project that is to model the wave equation in a pond due to a pebble being thrown in. The project is quite simple as all we really have to model is the wave equation with an initial displacement.

However, I was wanting to go a bit more in-depth and try to model it while knowing the initial conditions of the pebble.

I have come up with a simple case of the water being in a perfectly cylindrical pond and the pebble being a perfectly spherical ball, but I have absolutely no idea of how to model the sphere's impact with the water.

I know that the when the sphere initially makes contact with the water, there will be a brief moment of time that the surface tension acts as a normal force and this will cause some displacement of the water and what I assume will be a very fast starting wave speed. I'm stuck on this part of the problem, as the only way I can find to model the surface tension is the Young-Laplace equation, which doesn't even seem like it will work right here.

I also know that after the sphere breaks the surface of the water, there is a period of transition where part of the sphere is in the water and then a point after which the entire sphere is in the water; I do not know how to model either of these. I know once the entire sphere is in the water, it can be modeled with fluid drag (or maybe Navier-Stokes), but I am not sure how that would translate into affecting the wave.

Any ideas on this?


1 Answer 1


In the case of a pebble falling from some height into water, I believe surface tension will be altogether negligible. You should calculate the Weber number to check this, http://en.wikipedia.org/wiki/Weber_number

The dominant effect will be the pressure generated by the displacement of water by the pebble entering. Again, I am not sure that the viscous drag is very relevant for calculating the wave that is generated: it will give you the terminal velocity of the pebble, but initially what you have is an inertial effect, as the pebble transfers some momentum to the water.

Modelling the whole of the process will probably be quite difficult, even in this setting. At sufficient speed, an air bubble is entrained behind the pebble and collapses with a nice Worthington jet, http://dspace.mit.edu/handle/1721.1/67750


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