# Why do rain waves form and what is their connection to the texture of the surface they're on?

When it rains and water flows down an inclined street, ripples may form that are carried along with the current. Here's a picture with an example of what I'm talking about

I'd like to know what the fluid mechanical mechanism is for the formation of these waves. A quick google search reveals that they've been identified in some scientific literature as "rain waves" (if you have a jstor subscription you can see http://www.jstor.org/stable/4297089?seq=1) but I've only been able to find phenomenological descriptions of their properties without any details of the physics giving rise to them. In particular, I'm curious if people think that the surface on which the water is flowing must have a particular texture in order to give rise to the waves.

I have a very slightly different answer to that of Bernhard above, but it relies on the same Plateau-Raleigh Instability idea.

The speed of a thin layer of water flowing is going to be slower than a slightly thicker layer. So I expect that thin layer would flow more slowly and water in a thicker film would tend to catch up with the thin layer and build up to form a faster moving 'little wave' or 'ripple'. The wave would leave behind a very thin layer of water wetting the ground before the next 'ripple' comes along.

I guess this may be similar to how a flag flutters.

Ultimately I think if water is flowing down a flat surface the mechanism of flow will change as the flow rate changes from laminar to 'ripples' to some other type of flow and at some point turbulence will be important - this is classical behaviour of a non-linear system where changing a parameter (water flow rate, for example) can change the mechanism of what is going on. By contrast linear systems (e.g. SHM Simple Harmonic Motion) always show the same behaviour (eg for SHM - oscillatory behaviour, but with amplitude and frequency determined by initial conditions/system variables)

This answer will be my hypothesis, I am by no means absolutely sure about this answer, but want to share my thoughts.

Considering the thickness of the liquid film and the inclination of the surface, it may be an interplay between gravity and surface tension. This may be an example of the Plateau-Rayleigh instability, which can for example be seen at the break-up of the stream out of a faucet into droplets.

Except surface tension itself, you also encounter wetting behaviour here, which may be the reason that you see the long vertical streaks. So my intuition would be that gravity controls the speed of the streaks, the wetting/contact line the shape and surface tension controls the width or wavelength.

In other words. If you have a liquid film due to the rain, gravity is the downward force. The wetting behaviour at the front of the film, is a force acting opposite to it. Surface tension now controls that liquid behind it. At some points, the film breaks up, again by surface tension (Plateau-Rayleigh), giving you a new front and the story repeats.

You're asking about the texture of the surface, and the most important aspect of the texture of the surface is in the contact line behaviour. Different textures cause different opposing or breaking forces. I assume rough surfaces (as a street) have higher opposing forces.

• thanks very much for your answer. Your mention of wetting behavior led me to the wikipedia article on how wetting is affected by rough surfaces (en.wikipedia.org/wiki/Wetting#Non-ideal_rough_solid_surfaces) which is very useful. I am less sure about the application of the Plateau-Rayleigh instability and would love to hear other opinions as to whether this is the case. I also wonder whether the ripples are hydraulic jumps (with important contributions from surface tension) or if there is better way to think of them – kleingordon Mar 29 '12 at 1:11
• +1 didn't know about Plateau-Rayleigh Instability, but knew from observation how drops behaved on windows... very interesting to see the science behind it... – tom Jan 8 '15 at 16:55

And perhaps to throw another potential culprit into the ring -- I think these disturbances may be Tollmien-Schlicting waves that arise during the transition from laminar boundary layer flow to turbulent boundary layer flow.

Typically what you begin to see further downstream from these types of waves is a rotational component to the flow. The water appears to "roll up" in the cross-flow direction and you get "bursting". An example of what this may look like is in the lower right of the figure (this is a top-down view of a boundary layer over a flat plate)(source):

Basically the T-S waves grow as they move downstream and eventually burst, causing a cross-flow vorticity and turbulence forms. Similar patterns can be seen in water flowing down a river, or I would argue, thin sheets of water flowing down the pavement.

• +1 - nice to see another answer on this old thread interesting to see this other mechanism in transition from laminar flow – tom Jan 9 '15 at 1:40