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Suppose we have a spherical cow... I mean stone, and we throw it to a lake.

Can we tell anything about the velocity (speed and angle) at which it falls based on the sound it makes?

And based on a picture of the waves it makes in the water?

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As you can imagine, the impact of any object with a water surface is a highly nonlinear system. As such the evolution of the system is hardly ever solvable in closed form. Hence, approximation methods are necessary, and depending on the degree and focus of the approximation, you arrive at different results.

There is an interesting book, Leighton - the acoustic bubble, which is focused on the sound that is generated by bubbles in water (which may, of course, not necessarily be produced by an impact on the surface). I have been able to create a software simulation of the sound of rain/rivers/brooks from the equations contained in the book (namely in chapter 3.2 the pulsating spherical bubble), which I find remarkably authentic. You may find some more information there about your specific problem.

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I would suggest simplifying the problem. A nice problem would be to determine if the velocity of a drum stick tip can be determined from the sound it makes when it hits the center of a snare drum set to a specific tension. Setting up an experiment to determine this would not be too difficult and the math is far simpler. The qualitative results could be applied to your original problem.

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