# Water Waves in the Wake of a Boat

As a boy I noticed that the waves from the wake of my model boat would fan out. If I looked at the end furthest from the boat the front had turned so that it was almost running in the same direction as my boat.

The wave also appeared to bend in doing this in exactly the opposite way that I'd expect; waves from a stone thrown in a pond are circular so I'd expect waves from a boat would bend 'backwards' rather than bending to almost run parallel with the boat. I've found a video on You Tube that shows this rather well, if briefly:- https://www.youtube.com/watch?v=PP35lbrx0JQ The example starts at 2'55". I've search the internet in vain for an explanation. I think it may be an illusion because the distance between crests increases with time. Can someone explain this to me please? Also, are the wave crests accelerating? And if so what's causing it? Thanks ever so much if you can.

• Hi John. I've added a diagram that I think represents what you're asking about. I agree that the Youtube video does look as if the wake is curved. If I've got it wrong or you don't like the diagram shout and I'll delete it. – John Rennie Mar 17 '15 at 15:22
• Out of curiosity, if you know the lake well, can you say is it shallower / or deeper at the point where the effect occurs? thanks – user74893 Mar 17 '15 at 18:24
• Hi and thanks for replying! 1. The diagram is just about right except the wake fans out from the front of the boat and curves but not so much as in the diagram. 2. I don't know the lake in the video but the lake where I first noticed this certainly had a flat bottom as I saw it drained, people wade out and I fell in as a lad too! 3. I'm fairly certain I've noticed this with ferry's too. 4. My own theory (wild guess) this that the distance between the crests increases as the wave gets older as a consequence of it's height decreasing if this happens in a non-linear way you'd get the curve? – JohnP Mar 18 '15 at 18:01
• More on wakes: physics.stackexchange.com/q/9482/2451 – Qmechanic Nov 1 '15 at 14:03

First up thanks to all who took an interest especially @irishphysics who stuck with the question for some time.

It turns out that the phenomena was analysed and solved by Lord Kelvin and is known as the Kelvin wave pattern. The pattern itself is the result of a spreading pressure wave which manifests itself as the curved diverging wave crests (the ones I described as bending in my original question) and a set of transverse waves that sort of follow along with (and behind) the boat.

The full (mathematical) explanation is here: (https://www.math.ubc.ca/~cass/courses/m309-01a/carmen/Mainpage.htm).

The final (modelled) wave pattern from the above link looks like this:

which for me is pretty close to my experience.

For those who suggested the depth of water had something to do with it: you were partly right in that the precise angle of the waves from the bow depends on the depth of water rather than on the speed of the boat.

There's also a good explanation of the physics here:- (http://www.prirodopolis.hr/daily_phy/pdf/speed.pdf)

From this link below " Refraction is the bending of waves because of varying water depths underneath. The part of a wave in shallow water moves slower than the part of a wave in deeper water. So when the depth under a wave crest varies along the crest, the wave bends.

So my guess is its due either to variations in water depth or else it's an optical illusion. If you can check it easily, I would bring a measuring rod, and do measurements all around the area below where the wave bends.

Conservation of energy is paramount here (as you know), so no matter what we think we see, if you hold your boat at constant velocity, in reality, that wake is going to dissipate sideways only, never backways toward the boat, over time. So unless there is another force present to give the wave more "kick", the bending must be an optical illusion, no matter how many observers agree that they saw it.

So what possible forces could push the wake back towards the boat? Normally it would spread out wider and wider as the boat moves further away, but to change direction back towards the boat needs a force and unless the change in depth idea is true, see the link above, there is no other force to cause this reversal in direction. If the depth of the lake is constant, that it adds support to the illusion idea.

I have no experience of water wakes, (other being a passenger on ferries and rowboats) but it does seem to me that water waves, like mirages and rainbows, are a much likely cause of optical illusions that anything static, merely because there are just many more variables involved.

So to sum up, unless you can test the depth to ensure it's constant, and make sure that no other forces, such as a wind blowing towards the boat, are involved, I would make put my money on the illusion explanation.

• I don't think this is the case. Ex. i.stack.imgur.com/PaSas.jpg seems to have no asymmetry and the lines of constant phase (or maximums if you prefer) still bend outwards! – user12029 Mar 17 '15 at 18:48
• lovely pic! Still, if's its easily tested ...well ...its easily tested! – user74893 Mar 17 '15 at 18:50
• Bit out of my depth here (sorry, could not resist), but could there be a scaling effect. OP has a toy boat at slow speed, "your" boat looks like is's really moving. Is this plausible as a contributory factor? Total guess, I admit. – user74893 Mar 17 '15 at 19:07
• Hi thanks for your interest. Though I don't know this particular lake the lake where I first observed this didn't vary in depth. I have another observation that may be related. If you drop a pebble in a pond the waves that fan out get shallower and their crests further apart before they disappear. As I understand it that means the wave length increases. But I've no idea if the frequency stays constant or not? As far as the speed of the boat is concerned I've only taken notice of boats that aren't planning and produce a nice smooth sine type wave. The boat in the video has a constant speed. – JohnP Mar 18 '15 at 18:23
• @JohnP With light waves, the waves one I know more about than water waves, the longer the wavelength, the lower the frequency, and the lower the frequency the less energy the wave has. To me that makes sense because the energy of a dropped pebble will eventually go to nothing as it's wave spreads out across the lake. Sure looks likes it's curving in your video. I wonder, if the next time you used the boat, could you ask other people do they see the same thing? It would be best not to tell about the curving first, just ask them to describe the wake. – user74893 Mar 18 '15 at 19:11