What happens when a raindrop hits a puddle?

By | March 31, 2024

Have you ever walked through the rain on a warm spring day and seen that perfect puddle? You know, the one where the raindrops seem to land at just the right pace, creating a dance of disappearing circles?

Even before I entered the field of fluid flow research almost fifteen years ago, I was fascinated by the waves that appear after a raindrop hits a puddle.

When I focused on the study of unstable waves in liquid plates – aimed at reducing unwanted waves in industrial coating and atomization processes – my fascination with puddle waves turned into an obsession. What is going on? Where does the pattern come from? Why does the impact of rain in a puddle look different than when rain falls elsewhere, such as in a lake or the ocean?

It turns out it all has to do with something called dispersion.

In the context of water waves, dispersion is the ability of waves of different wavelengths to each move at their own individual speed. When we look down on a puddle, we see a collection of such waves moving together like one ripple in the water.

When a raindrop hits the ground, imagine it as a “thing” on the surface of the water. This thing can be idealized as a packet of waves of all different sizes. After the raindrop falls, the pack waves are ready to start their new life in the puddle.

However, whether we see those waves as ripples depends on the body of water on which the raindrop lands. The number and distance between the rings you see depends on the height of the puddle. This has been verified in some very cool ripple tank experiments, where a drop falls at the same speed into a tank of water at different depths.

Shallow puddles allow ripples because they are much thinner than they are wide. The balance between the surface force – between the pool of water and the air above it – and gravity tips in favor of the surface force. This is critical because surface force depends on the curvature of the water surface, while gravity does not.

An initially shallow puddle becomes curved on the surface after the impact of the raindrop. The surface force is different for long waves than for short ones, causing waves of different sizes to break into ripples. In shallow pools, the long waves move slowly away from the point of impact, while the short waves move quickly, and the really short waves move very quickly, becoming tightly packed at the edge. This creates the enchanting pattern we see.

Raindrops may react differently in other situations. Imagine rain hitting a lake or an ocean – or those deep puddles that require overshoes. Here the raindrop hits the water, but the force due to gravity becomes more important. It moves waves of all shapes and sizes at the same speed, which can overpower the wavy effect due to the surface force.

The combination of teaching partial differential equations at the undergraduate level while continuing to explore liquid sheets led to what I call the “puddle equation.” When the equation is solved, it creates an animated simulation of what happens after a raindrop hits a puddle. It is a simplified version of an equation from one of our group’s more recent research efforts, but it is also consistent with the classical description of ripples.

I use this approximate description of pool waves as a way to get students excited about math by relating it to the world around them.

A model of waves in a scattered puddle after a raindrop has fallen.  The top three figures show what happens after a droplet hits the puddle, with arrows indicating the passage of time.  The bottom figure shows a cross-section through the pool, highlighting that the initial wave beam caused by the raindrop splits into waves of different sizes.  Large waves in the center move slower than small waves at the edge.  Nate BarlowA model of waves in a scattered puddle after a raindrop has fallen.  The top three figures show what happens after a droplet hits the puddle, with arrows indicating the passage of time.  The bottom figure shows a cross-section through the pool, highlighting that the initial wave beam caused by the raindrop splits into waves of different sizes.  Large waves in the center move slower than small waves at the edge.  Nate Barlow

The study of surface force-driven waves is important for applications such as coating processes involved in making batteries and solar cells.

Such waves also appear as a result of the leg movement of a water warrior insect, but research has shown that the water warrior is not specifically looking to create those waves to enable travel.

The beauty of puddle waves is in itself no small thing. By connecting nature to its primordial language – mathematics – we gain access to its control panel, allowing us to observe every little detail and reveal all its secrets.

This article is republished from The Conversation, an independent nonprofit organization providing facts and trusted analysis to help you understand our complex world. It was written by: Nate Barlow, Rochester Institute of Technology

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Nate Barlow does not work for, consult with, own stock in, or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

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