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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?

Shub
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user541686
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4 Answers4

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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.

mike stone
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  • 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. – JMac Mar 19 '17 at 13:37
  • 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. – Ernie Mar 19 '17 at 15:02
  • I realize that you need to google "roll waveS" plural. In the sigular you get lost of hits on other types of waves. – mike stone Mar 19 '17 at 15:13
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    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: https://www.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". – Jason C Mar 19 '17 at 18:01
  • +1 thank you! Seems to be exactly it :) love the analysis in the notes as well! – user541686 Mar 19 '17 at 19:52
  • @Mehrdad https://en.wikipedia.org/wiki/Rayleigh%27s_equation_(fluid_dynamics)#Background ...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; https://core.ac.uk/download/pdf/4890972.pdf page 23-> – Jokela Jun 26 '17 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.

Mutantoe
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md2perpe
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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.

Ernie
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  • And it's feasible that the rain intensity is varying by a fairly substantial factor, with a period of a couple of seconds? – David Richerby Mar 19 '17 at 10:50
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    @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. – Ernie Mar 19 '17 at 14:17
  • 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. – Vladimir F Героям слава Mar 19 '17 at 19:35
  • 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? – Tim Mar 19 '17 at 19:59
  • @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. – Ernie Mar 20 '17 at 01:34
  • This answer is clearly wrong. You can create roll waves on a slip-n-slide on a sunny day. How did you get 2 upvotes? – user121330 Mar 20 '17 at 16:48
  • @user121330 : You are right that source fluctuation doesn't cause roll waves, but it may have an effect on water flow, and therefore on some aspect of the roll waves. I haven't deleted the answer, because it may provide an area of inquiry if someone notices an anomaly in roll waves that can't be otherwise explained. Even domestic water flow from a pressure-regulated building to your slip-n-slide varies depending on how many how many others use the water, and turbulence in the hose. I edited the answer after reading your comment, to make it clear that this is not the cause of roll waves. – Ernie Mar 20 '17 at 22:20
  • Why anything exhibits a periodic/oscillating behavior is that the conditions are right: the system has some low energy state with a restoring force which drives it back to that state when it is displaced, there is a lack of damping, and there is some source feeding energy into the system. – Kaz Mar 20 '17 at 23:03
  • @Kaz : Thanks for that concise explanation. I'm not saying that source fluctuation causes roll waves or that there is any reason to believe source fluctuations are periodic/oscillating. I edited the answer. – Ernie Mar 21 '17 at 15:16
  • But they approximately are. Not all oscillations are periodic; there can be chaotic ones. I'm not saying that the source fluctuation drives the waves: not any more than, say, a 2000 Hz note from a violin being caused by 2000 Hz signal in the bow! The bow moves smoothly and steadily. The string catches and releases chaotically, causing noise. The resonance of the string filters the noise to certain harmonics. – Kaz Mar 21 '17 at 15:22
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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.

Jens
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Lelouch
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  • I'm not sure why the level ever reduces below the level of the bump... water flowing over a dam is continuous – Tim Mar 19 '17 at 10:03
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    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. – Lelouch Mar 19 '17 at 10:14
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    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... – Tim Mar 19 '17 at 10:18
  • If the depression/bump is elongated, a small crack or something, a wavefront may be formed, after all more than one drop is expected to splash at any instant statistically. – Lelouch Mar 19 '17 at 10:19
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    I will do an experiment and report back – Tim Mar 19 '17 at 10:21
  • I would rather appreciate that, since i built all this up from theory. – Lelouch Mar 19 '17 at 10:23
  • @Tim I don't see why a drop couldn't remove more water than it contained: a small amount of water arriving at high speed could impart its large amount of kinetic energy to a large amount of water moving at low speed (consider throwing a brick into a bathtub). – David Richerby Mar 19 '17 at 10:36
  • I don't see how a dam being overtopped by a few raindrops would cause visible wave-fronts multiple centimetres across and multiple millimetres high. The amount of water in one of these waves is a huge number of raindrops. – David Richerby Mar 19 '17 at 10:37
  • Mmh the experiment was pretty inconclusive- but in the real world I can't imagine a smooth enough surface such that it would flow in one smooth wavefront - normally there would be a couple of channels of water from lower points in the ridge. @DavidRicherby the kinetic energy of a raindrop is very small - around one milijoule, despite the significant height it has fallen (I think they have a fairly low top speed?). The brick in your example has around 10 joules. – Tim Mar 19 '17 at 11:01
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    @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. – David Richerby Mar 19 '17 at 11:43