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A while ago it was raining and I noticed that, on sloped pavement, water was flowing in very regular consistent periodic waves, as you see below.

However, I realized I had no idea why this should be happening. There was nothing uphill actually creating these waves, and they continued down as far as the pavement went, despite the rain that was falling on them along the way. Why wasn't the water flowing down smoothly, or irregularly?
What causes the noticeable wavelike patterns? Is there a name for this phenomenon?

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  • $\begingroup$ Does the wavelength depend on the inclination of the pavement? $\endgroup$
    – user27542
    Commented Mar 19, 2017 at 10:23
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    $\begingroup$ Is wind ruled out as a possible cause? If not, I speculate that the pattern is caused by gusts of wind. $\endgroup$
    – aroth
    Commented Mar 20, 2017 at 12:30
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    $\begingroup$ is this NOT a duplicate - physics.stackexchange.com/q/22897 $\endgroup$
    – user66043
    Commented Mar 21, 2017 at 2:29
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    $\begingroup$ Possible duplicate of Why do rain waves form and what is their connection to the texture of the surface they're on? $\endgroup$
    – John Rennie
    Commented Mar 21, 2017 at 9:45
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    $\begingroup$ I think the top answer here really needs to be migrated to this original. $\endgroup$
    – JMac
    Commented Mar 21, 2017 at 15:20

4 Answers 4

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These waves are called "roll waves." They are due to an instability in shallow shear flows. The analysis is much too complex for a short answer, but if you google "Roll Wave" you will find more images and links to technical articles. If you are not bothered by a little mathematics you will find a discussion of the cause of the instability starting on page 259 in these online lecture notes: https://courses.physics.illinois.edu/phys508/fa2016/amaster.pdf

After the waves have formed due to the instability, the actual form -- a series of breaking waves -- is due to the non-linear propagation effect described by md2perpe -- the deeper the water the faster the wave.

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  • $\begingroup$ This seems to be the only answer that isn't speculation and gives a clear reason. I'm glad to see there is a lot of analysis on this even if I can't follow most of the math. $\endgroup$
    – JMac
    Commented Mar 19, 2017 at 13:37
  • $\begingroup$ Great answer! Thank you for the link to your lecture notes, which I will read with great interest. It may be, however, that micro-changes in source flow may be at least a contributing factor to this phenomenon. I may change my mind after reading your notes. $\endgroup$
    – Ernie
    Commented Mar 19, 2017 at 15:02
  • $\begingroup$ I realize that you need to google "roll waveS" plural. In the sigular you get lost of hits on other types of waves. $\endgroup$
    – mike stone
    Commented Mar 19, 2017 at 15:13
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    $\begingroup$ In my Google searches (there's a lot of info pollution, try "physics roll waves", "fluid dynamics roll waves", "roll waves shear flow", etc.) I came across a nice paper on the topic: math.ubc.ca/~njb/Research/mandre.pdf - Which even goes so far as to explicitly mention "[Roll] waves are also common occurrences in shallow, laminar fluid films flowing on street gutters and window panes on rainy days". $\endgroup$
    – Jason C
    Commented Mar 19, 2017 at 18:01
  • $\begingroup$ @Mehrdad en.wikipedia.org/wiki/… ...At the critical layers Rayleigh's equation becomes singular. This is the whole raleygh "instability". These systematic roll waves can yet be easily defined with a solid mathematics without too much complexity; ie; core.ac.uk/download/pdf/4890972.pdf page 23-> $\endgroup$
    – Jokela
    Commented Jun 26, 2017 at 12:39
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I think that the explanation is that waves of different size run with different speeds. This makes the faster waves run up to the slower which make them stack up, or constructively interfere.

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This as a speculative explanation, which describes source fluctuation that might affect the volume of water flowing down the street. It's not the cause of the roll wave phenomenon (see Mike Stone's excellent answer), but it may affect it by increasing water volume sufficiently to disrupt the roll waves. I've observed micro-variations in flows in mountain streams that are not characterized by shallow shear flows.

When a cold front passes through warm air, it pushes the warm air up and over the advancing cell of cold air. Moving cold air forms a wedge and creates an inclined plane that the warm air climbs. Likewise, when a warm front pushes cold air, it climbs over the cold air, but since cold air hugs the ground more tenaciously than warm air, the inclined plane has a more gradual slope. In either case, warm air climbs into the atmosphere gradually, not straight up.

Clouds form when moist warm air reaches the condensation altitude, where rising warm air is cooled to the dew point. As the rising warm air cools and expands, the dew point changes as the pressure inside the cloud changes.

When the rising conveyor belt of moist air acquires the temperature of surrounding air it stops rising, and the dew point stops falling. Droplets begin to condense, and when they are large enough for the force of gravity to take them off the stalled conveyor belt of moist air, they fall as precipitation.

The process doesn't affect all the air in the cloud simultaneously. As saturated layers of warm air climb the incline of cold air, time provides opportunity for changes in micro climate that may vary the amount of precipitation falling off the top of the conveyor belt.

The roll waves you see on smooth pavement might be affected as the volume of flow varies with air currents and temperature conditions at the top of the atmospheric conveyor belt, where pressure, temperature, and dew point undergo micro-changes.

This could be verified by observing a strictly controlled release of water down the same surface. If characteristics of the waves that appear during a strictly controlled release differ from waves that appear during rainfall, it may indicate that source fluctuation has an effect.

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  • $\begingroup$ And it's feasible that the rain intensity is varying by a fairly substantial factor, with a period of a couple of seconds? $\endgroup$
    – David Richerby
    Commented Mar 19, 2017 at 10:50
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    $\begingroup$ @DavidRicherby : I've observed similar micro-variations in flow in mountain streams during and immediately after rainfall. I changed the answer, labeled it speculative, and proposed an experiment that might help. $\endgroup$
    – Ernie
    Commented Mar 19, 2017 at 14:17
  • $\begingroup$ Do you have any reason to believe that this precipitation has anything to do with any front or conveyour belt? Even if yes, any fluctuations caused by the structure of the conveyor belt would be on much larger scale and less regular. On can often see that the rain intensity varies in pulsew, but mostly it is just turbulence. It may be series of weak microburst-like features, but the scale would be much larger than. $\endgroup$
    – Vladimir F Героям слава
    Commented Mar 19, 2017 at 19:35
  • $\begingroup$ I've observed the same in a waterfall. I think it's possibly just the water increasing and decreasing in airation - the flow is constant? $\endgroup$
    – Tim
    Commented Mar 19, 2017 at 19:59
  • $\begingroup$ @VladimirF : Rainfall great enough to produce this flow likely came from a front and rising warm moist air. However, absent a better explanation for source fluctuations, you are correct that there's no reason to expect them to be regular. Large scale turbulent fluctuations might increase water volume so much that shallow shear flow may be overwhelmed entirely, which is a reason you may be correct to doubt their effect on this phenomenon. $\endgroup$
    – Ernie
    Commented Mar 20, 2017 at 1:34
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I may have an explanation. Such a behavior is not to be expected (perhaps) in an ideal situation (uniform surfaces etc.). So, if it's due to an irregularity, the following may be true.

Suppose that we have a small bump/irregularity/cavity on the road (very small, perhaps a cm in height, and a bit longer along the road). Such things can be created on the natural surface of the road due to non-uniform eroding. So when rain falls on the slope these bumps/cavities act like a small dam/reservoir respectively. However, very soon When the rainwater level rises beyond their heights, water flows, and stops again when the level goes under. Due to falling raindrops, water in these cavities/bumps splashes, causing more water than equilibrium height to move out. This creates periodic flow of water down the slope.

Basically, we need a mechanism to disrupt the (expected) continuous flow. Any other ideas?

Any questions may be asked.

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  • $\begingroup$ I'm not sure why the level ever reduces below the level of the bump... water flowing over a dam is continuous $\endgroup$
    – Tim
    Commented Mar 19, 2017 at 10:03
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    $\begingroup$ You're forgetting the mechanism in which the level rises. Drops falling is not a very smooth cause of rising water. They splash, taking more water out when its already at the brim, causing the water level to drop again. $\endgroup$
    – Lelouch
    Commented Mar 19, 2017 at 10:14
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    $\begingroup$ I would need to do some tests but I don't think splashes do cause more water to leave than enters, and even if it did it would not cause this long uniform wave... $\endgroup$
    – Tim
    Commented Mar 19, 2017 at 10:18
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    $\begingroup$ I will do an experiment and report back $\endgroup$
    – Tim
    Commented Mar 19, 2017 at 10:21
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    $\begingroup$ @Tim The brick wasn't a great example but I was just trying to quickly show that a falling object can splash "a lot" relative to its own size. Even a falling water drop can lift quite a lot of water, but I think the fundamental problem here is one of scale: it's hard to imagine how a few splashing raindrops cause such a large wave-front. $\endgroup$
    – David Richerby
    Commented Mar 19, 2017 at 11:43

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