If you look at the wake behind a duck, or a kayak, or a ship, you might notice two unusual
things: first, the wake isn’t simple (like the perfectly straight shock wake of a supersonic
projectile is) - it’s a fascinating, feathery, ripple-y pattern.
And second, that feathery pattern looks the same - same angle, same repeating pattern
of ripples along the edge, same reverse arcs in the middle - it looks more or less the
same regardless of whether it’s made by a duck, a kayak or a ship, even though they’re
all moving at different speeds and the waves are very different sizes.
The reason water wakes always have this particular shape and pattern is because of the surprising
physics of water waves.
While there's a single speed of light waves, and a single speed of sound waves, there's
no single speed of water waves.
In water, longer waves travel faster, while shorter waves travel slower.
This phenomenon where different wavelength waves travel at different speeds is known
as dispersion, and it makes water waves both interesting and complicated.
Like a boat wake.
To explain the shape of a boat wake, we’ll start by looking at water waves of just a
single wavelength (and speed).
A boat traveling across this simpler water creates a series of circular waves - if the
water waves are faster than the boat, the waves encircle it, but you don’t get a wake.
If the waves are slower than the boat, the boat outruns them and the circles all add
together to create a V-shaped wake.
And if the waves are even slower, the boat will outrun them even more and they’ll add
together to a narrower V shape.
Slower waves make narrower wakes.
Faster waves make wider wakes.
But we need to remember that water waves are waves, that is, they repeat themselves.
So every circular wave is really the first of a series of circular waves.
This means that instead of creating just one v-shaped wake, a moving boat creates a train
of v-shaped wakes that are each exactly one wavelength apart.
The reason a real boat doesn’t make straight v-shaped wakes is that a real boat makes waves
of many different wavelengths.
And because of dispersion, different wavelengths travel at different speeds: the longer ones
travel faster and shorter ones travel slower.
Faster waves create wider wakes, and because faster waves also have longer wavelengths,
that means that wider wakes are further apart.
Similarly, slower waves create narrower wakes, and because slower waves also have shorter
wavelengths, narrower wakes are closer together.
When you add together narrow, closely spaced wakes, with wider, more widely spaced wakes,
with even wider, even more widely separated wakes, and so on, voilá!
The shape of a boat wake!
Look at the beautiful repeating feathery ripples out on the edge, and the wider repeating arcs
inside the wake itself.
If you replace these sharp lines with smoother waves at the appropriate angles and spacings,
you get an even more convincing boat wake.
And we can do the same again in 3D to get a really realistic-looking boat wake.
So in summary: wakes have the shape they do because water waves travel at different speeds.
Slower water waves create narrow, closely spaced V-shaped wakes, and faster water waves
create wider V-shaped wakes that are further apart.
When you add all these different v-shaped patterns together, at the correct angles and
spacings determined by water's dispersion relation, you end up with the unique shape
of a water wake.
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