Why do stones tossed into a pond form circular waves regardless of their shape?

Splash (5)
a great splash captured by Emanuel Szep

Sometime in September I went for a walk around Inverleith Park in
Edinburgh. As I walked past the pond and observed ripples in the
water, I thought about something that I took for granted since I was a child.
For as long as I could remember, stones thrown into water would create waves
that would form concentric rings around the stone…but how exactly did this happen?

As I thought about this I gathered stones of different size and shape. And as
I tossed them into the water, I noticed that the ripples eventually(by the 5th
ripple) converged to circular rings regardless of the shape of the stone. I could
only observe the ripples on the surface but I conjectured that the ripples must
form spherical shells around the location of impact. Now I wondered why this
was so…but I didn’t know anything about fluid dynamics. So I decided to use
the internet to find the answer.

Sure enough, I found that somebody had already asked a similar question on the Physics StackExchange. Here’s my summary of the discussion:

The pond water may be approximated as a homogeneous fluid, and the water
waves are longitudinal waves that travel through the fluid. If we assume that
the stone is released at a trajectory that is normal with respect to the surface
of the water, the 1st wave front should travel at nearly constant speed in all
directions that are normal with respect to the submerged surface of the
stone. And the magnitude of this velocity would be proportional to its
momentum upon impact.


If we measure the difference in radii with respect to the centroid of the stone(\Delta), we would obtain the following inequality: m \leq \Delta \leq M

Now, as the initial wave front travels a larger distance this length becomes
much more important compared to the largest difference in radii, M.
This is why we eventually perceive a circle. As a matter of fact, if we
analyse the distance travelled by the first wave front as a function of
its number of cycles n, we may derive the following ratio:

\frac{Cn+m}{Cn} \leq f(n) \leq \frac{Cn + M}{Cn} and from this we may deduce that \lim_{n \to +\infty}f(n) = 1

This may appear to be superficially similar to the Huygens-Fresnel principle,
but the surface of the pond inside the expanding wave is not perfectly calm
after the main wave has passed through as this principle would require.


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