Skipping Stones

My MSc dissertation can be viewed here.

animated skipping stone

Splashes are a very interesting area of fluid dynamics. My MSc looked at the some problems relating to the impact of solid bodies on fluids. The most fun application of this is standing at the beach playing skipping-stones (of course there are more serious examples such as maritime structures and bouncing-bombs). Let us look at a stone hitting the water surface,

skipping stone

The equations of motion for a circular stone are,

\( {dx \over dt} = V_x\)

\( {dy \over dt} = V_y\)

\( {dV_x \over dt} = - { \rho K V^2 S \sin (\alpha + \beta) \sin \alpha \over 2m } \)

\( {dV_y \over dt} = { \rho K V^2 \sin ( \alpha + \beta ) \cos \alpha \over 2m } -g \)


\( K = \{ 1 \text{ if } y\leq0 \)   or   \( 0 \text{ if } 0 \lt y \} \)

\( S=r^2 \Biggl[ \cos^{-1} \Bigl( 1- {d \over r \sin \alpha} \Bigr) - \Bigl( 1 - {d \over r \sin \alpha} \Bigr) \sqrt{1- \Bigl( 1- {d \over r \sin \alpha} \Bigr)^2 } \Biggr] \)

and \( \rho \) is the density of the water.

It is interesting to see under which conditions, for the parameters, we can achieve the stone skipping effect. An example of this is shown below.

skipping graphs

Clearly though, the only question anybody would be interested in is how to get more skips! Fixing the tilt of the stone to be about 20° (since this gives a good range of throwing angle) I investigated. I created the graph below which seems to show that the number of skips is (approximately) a linear relationship with the initial speed the stone is thrown at. This means if you want to achieve more skips then throw it with a bit more speed! Common sense really.

number of skips