Waves caused by bodies falling in water

[webdreamer]

I've been trying to find equations that model the effect on water of something falling onto it (a pebble for example), without much success. Do you know them or where I can find them? It's such a common thing, it suppose it's not that hard, I just don't know where to look. It's of course a wave, but the wavelength gets smaller as it gets away from the "epicenter", sort of speak.
I also would like to know how it starts propagating, I mean, does the amplitude of the wave grows at first, as the pebble goes deeper, does it send several waves?

I'm trying to simulate the effect of bodies falling on water for a game.

 

[LightHero]

The most comprehensive equation for describing the motion of a wave propagated from a source in a two-dimensional fluid is the Navier-Stokes equation, which describes the conservation of momentum in the fluid. However, this equation is generally too complex to be solved analytically, and must usually be solved numerically.The simplest way to approximate the effect of something falling into water is to use the impulse-momentum theorem, which states that the change in momentum of an object is equal to the impulse (force over time) applied to it. This can be used to calculate the effect of a falling object on the water, assuming the force of the object's impact is known.Assuming the falling object is small enough, its effect on the water may be modeled by a linear wave. The initial amplitude of the wave will be determined by the impulse of the object, and the initial wavelength will be determined by the size of the object and the density of the water. The amplitude of the wave will decrease with distance from the source, and the wavelength will also increase as the wave spreads out. The rate of decay of the wave's amplitude with distance will depend on the properties of the fluid. For small objects, the effect of the wave on the water will likely be localized; however, for larger objects, the wave may spread out to create multiple waves of various wavelengths, amplitudes, and directions.