Fluid mechanics

Waves and ripples: maths applied on holiday

Half way through a hike (well, more of a long walk, really) in the mountains of the beautiful La Cumbrecita, Córdoba, we spotted a little cove that looked the ideal place for a well-deserved rest (it may be winter here, but it was 30 degrees with relentless sun). After creeping past the cows and sliding down the stones we found ourselves at a mountain beach with clear, almost perfectly still, waters. The urge to throw pebbles into the stream was instinctive. I no longer needed to nap; I only wanted to make a splash! I believe Billy Connolly once said “Never trust a man who, when left alone with a tea cosy, doesn’t try it on”. I’m sure something similar (albeit less witty and revealing) can be said about people with small stones and still ponds.

I filmed some ripples and waves on my phone and I was reminded of the beautiful underlying mathematics…

The water is initially calm with some depth – this is its equilibrium state – and the introduction of the pebble (which plops into the pond) causes disturbances (waves and ripples) in the water.  The force of gravity tries to restore the equilibrium state  (the density of the water is greater that the density of the air above) and the disturbances propagate in all directions.  I am assuming that gravity is the only significant restoring force on the larger, prominent, waves (those that are clearly visible in my poor filming), so the ripples that form and spread are examples of surface gravity waves (I am ignoring surface tension here, but smaller capillary waves can be seen at the edge of the outer ring, if you look close enough – this is justified below).

The  linear theory of deep and shallow water waves can be found in most good books on fluid mechanics. My personal favourites are by Patterson and (of course) Batchelor, but there are many others. Without going through the details, the end result of some fun modelling and manipulation with PDEs and potential flow theory is the dispersion relation for linearised gravity waves


where \omega is the angular frequency of the wave, k is the wavenumber (which is 2\pi/\lambda, where \lambda is the wavelength), g is gravity, and h_0 is the depth of the undisturbed water.

The water in the video may not look very deep (and it isn’t – I got in and it only came up to my knees), but its not its absolute value that’s important but its size in comparison with other important measurements. Here the depth (a length) is a lot bigger that the wavelength (the distance between successive peaks). Since h_0\gg\lambda , we consider these waves to be deep water waves. Now, the wavenumber k=2\pi/\lambda, so we can say that kh_0\gg 1, and hence \tanh{(kh_0)}\approx 1. Therefore, the dispersion relation for the prominent waves in the film reduces to


Gratifyingly elegant! The speed of a wave, or the phase velocity, is the velocity a wave crest is travelling at and is given by c=\omega/k=\pm\sqrt{g\lambda/(2\pi)}. This equation tells us that longer waves (those with larger \lambda) move faster that shorter ones. This is demonstrated in the video above (filmed in slow motion on my phone – nothing technical!), where we can see that the outer waves are larger (have longer wavelengths) than the smaller ones they encapsulate. This was quite a basic experiment (more of a game than a test!) but I find the link between pebble throwing and mathematical fluid mechanics quite pleasing – yes, I am a proud nerd!

(It’s worth noting in passing that if \lambda\gg h_0 the waves are said to be shallow water waves. In this case, \tanh{(kh_0)}\approx kh_0 and c=\pm\sqrt{gh_0}. From this we can see that the wavespeed in shallow water depends on the depth of the water, but is independent of the wavelength. Thus, waves in shallow water do not have frequency dispersion.)

I then threw two stones, one after the other, into the centre of the pond (see the video above). I think I can just about see the large wavelength, fast moving, outer waves caused by the second stone catch up and overtake the shorter wavelength inner waves of the first splash. Perhaps not the most detailed and controlled experiment, but I am on holiday, after all!

Are they waves or are they ripples?

Well, I wasn’t certain at first.  Ripples in water are often capillary waves, meaning their behaviour is dominated by surface tension.  Their wavelength is typically less than the order of a centimetre and their wavespeed more than 20 centimetres per second. From the videos, the prominent waves seem to fall outside of these ranges while the smaller disturbances outside the “rings” appear to be capillary waves, or ripples. A more convincing argument is based on looking at the group velocity of the waves.

The group velocity, c_G is the velocity with which a group, or packet, of waves propagates. In the video, the packet of the prominent waves is the expanding ring. Within this group we can see several individual waves, each moving with their wavespeed (or phase speed). For deep water gravity waves, the group velocity is  c_G=c/2. So the group velocity is smaller than the phase velocity. This means that gravity waves at the back of a packet will catch up with the front (the outer ring) before vanishing. This seems to happen here. If we try to track an individual crest in the slow-motion video (at the top of the page) we find we lose track of it when it is at the front and notice other waves are behind it. So it seems reasonable to conclude that there are surface gravity waves. Yippee.

If you have a very keen eye, you should be able to see that the smaller ripples in front on the outer ring behave differently. These ripples move slower with increasing wavelength, meaning the ones at the front of a packet of ripples end up at the back.  Furthermore, individual ripples seem to pop-up at the font of the packet and vanish at the back. This suggests the group velocity is faster than the phase velocity – a characteristic of capillary waves and ripples.