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?