# Wakes

Discussion in 'Hydrodynamics and Aerodynamics' started by Anthony Appleyard, Apr 24, 2014.

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### Anthony AppleyardAnthony Appleyard

https://en.wikipedia.org/wiki/Wake

This Wikipedia page seems to contain a good mathematical discussion about the usual V-shaped boat wakes; but I still need some information; each V is composed of an imbricated succession of parallel waves waves successively starting and finishing; but what wavelength are those waves (measured at right angle to that arm of the wake V) for each speed in knots? Does that wavelength vary as the square root of the speed of the boat, or what?

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### DCockeySenior Member

The distance between the waves in a boat's wake depends only on the speed of the boat, not on the size of the boat.

The distance in the direction of the boat's travel between waves is 2 * Pi * U^2 / g
where U is the speed of the boat and g is the gravitational constant. This relationship is for deep water. In shallow water the water depth is a factor.

Is this as general interest question, or are you creating computer graphics/illustrations or ????

Last edited: Apr 24, 2014
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### Leo LazauskasSenior Member

David has given you the correct formula for the wavelength of deep-water
"transverse" waves, i.e. those perpendicular to the ship's track.
You need to multiply by [cos(theta)]^2 for other angles, where theta=0
is in the direction of the track and theta=90 degrees corresponds
to waves travelling parallel to the track.
So, for theta=0, you recover the "transverse" wave formula, for theta=90
the wavelength is equal to zero.

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### Anthony AppleyardAnthony Appleyard

In this case, I am making computer graphics.

Thanks :: I realized that wavelength varies as square of speed; but, for somewhere to start from, please, if the boat is going at (say) 10 knots, what is the distance between waves in meters as measured parallel to the direction of the boat?

Wikipedia says "Each wake line is offset from the path of the wake source by around 19° and is made up of feathery wavelets angled at roughly 53° to the path". In the image at this link (https://en.wikipedia.org/wiki/File:Fjordn_surface_wave_boat.jpg) the wake lines start offset at a much narrower angle and gradually seem to settle to 19deg; each wake line is made of large waves (which I would not call "feathery"), which seem to be at a much smaller angle than 53deg to the boat's direction, if the photo is taken from straight above.

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### RurudyneSenior Member

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### jehardimanSenior Member

There is no easy answer to that, but Google Kelvin wakes for a start. You might need to get your head turned around before you can continue with this, so start this way...Think of a pebble being thrown into a still pond...now for a 10 knot vessel think of a pebble being thrown every 1/10 of a second 1.6878 ft further along the boats track. Here is a nice example of that concept

Now to look at the case of a boat...

1) The water ahead of the stem is not moving and has no kinetic energy, only potential and internal energy. (a lot of CFD screw this up).
2) The celerity of a pressure disturbance in a fluid (nominally an ideal fluid) is just a function of the energy input, gravity, and the fluid density. (basic gravity wave theory and the basis for far field wake analysis).
3) As the vessel moves forward, it imparts energy into the fluid, each del x of advance imparting a del E over a del t. Now here is where Froude number and thin ship theory come in. The vessel is advancing, and due to the instantaneous slope of its boundary...
a) the boundary is advancing transversely faster than the disturbance celerity...
b) the boundary is advancing transversely slower than the disturbance celerity...

In case a) the energy is additive (up to a breaking crest or cavitation point...the near field effects) and perhaps for very fast boats only the transom matters. For case b) the pressure disturbance waves spread out at a differential speed relative to the vessel and interact with each other. Since a normal ship-shaped hull has 4 major points of boundary inflection (stem, bow quarter, stern quarter, and transom), from these points there are actually generated the 4 main wave trains of a vessel wake (see Wigley hull and Froude powering humps and hollows)...

So therefore the actual distances between the true wake wave crests is determined by vessel speed and geometry, however once propagated, the relationship between them is maintained by the properties of the fluid.

Edit to add, water depth also changes wave celerity, but the assumption is usually "deep water"...just to cut down on confusing the concept

Last edited: Apr 24, 2014
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### Leo LazauskasSenior Member

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### Anthony AppleyardAnthony Appleyard

Thanks everybody :: but please, one thing, as a starting point: if something on deep water in Earth gravity makes plain sine-type waves, and their only frequency component is one per second, how fast do those waves travel, in meters per second?

Presumably the algebra in Wikipedia assumes that every frequency component is present up to the longest-wavelength and thus fastest possible.

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### Anthony AppleyardAnthony Appleyard

Herewith a CGI attempt at making boat wakes.

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### jehardimanSenior Member

That's the problem, an item dropped into water doesn't make a sine shaped wave. While a wake may look like a continious wave train, it is not. It is a linked set of impluse waves. An impluse wave is a single event where the energy fluxes through the fluid as a transition from potential energy to kinetic energy back to potential energy. At any single point in time (at some small time after the initial disturbance) approximately 1/2 the energy is in kinetic disturbance and 1/2 in the potential. The speed (i.e. length of time) at which the distrubance occured sets the propagation speed (i.e. celerity) and the amount of energy sets the initial amplitude so that energy is maintained.

If you are really trying to get into wave mechanics, as opposed to general hydrodynamics, I would recommend you get a copy of Oceanographical Engineering by Wiegel (chapters 2 through 4). Otherwise for a general theoritical text use Hydrodynamics by Lamb (chapter IX).

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### Anthony AppleyardAnthony Appleyard

Please, are these units in meters and seconds, or what? In those units, please what is g?

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### jehardimanSenior Member

Doesn't matter as long as you are consistant....pi is unitless... U^2 is (dist/time)*(dist/time) = (dist^2/time^2)... g is gravational acceleration = (dist/time^2)...

so doing the math;

(dist^2/time^2)/(dist/time^2) = distance

which is what you are after.

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### Leo LazauskasSenior Member

Are you and your client (or lecturer) happy with that illustration? If so, there is
no need to do any more work on it.

If you want something a bit more sophisticated you could use my ZGREEN
program with one source at the bow and a one sink at the stern. That would
give you a "feathery" looking wave pattern. The wave heights are saved in a
simple text file which should be easy to use with your graphics package, even if
you have to massage it a bit.

jehardiman's book suggestions are probably way beyond your mathematical
capabilities at this stage.

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### Anthony AppleyardAnthony Appleyard

Thanks for the book references.

There is no client for my image :: I made it for my own interest.

Can ZGREEN cater for boats moving and waves reflecting off things?

I have Visual C++ and I have written several Windows applications including a CGI mesh-editor. (Free downloads including program source text, not for money.)

https://en.wikipedia.org/wiki/Cycloid#Related curves

I had the idea that, with a simple steady train of waves on water, each molecule goes in a vertical circle, with the circle biggest at the water surface, and with the phase angle of where it is on the circle different by 360deg for each wavelength, giving a wave shape like a curtate cycloid (see link). When wave height reaches 1/pi of the wavelength, the wave shape becomes a plain cycloid. If the wave tries to become any higher for the same wavelength, the cycloid becomes prolate, and some water tries to go through other water, and the wave breaks and develops a white foam top. Is this correct?

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### Leo LazauskasSenior Member

It can certainly produce wave patterns similar to those of moving ships.

The examples only take a couple of minutes to run on a PC.
I suggest you try running Example 2a and then the wave plotting program to
produce a plot like the first attached. Or Example 2b to produce the 2nd
attached plot which is at a different spacing.
Note the difference in the transverse waves.

Of course, you will need to put a representation of the ship so it fits
between the two "wavemakers", but I think it will be easier and more realistic
than what you are trying to do now.

Wave reflections are not explicitly catered for, but you can set up arrays
of singularities to simulate reflection from solid barriers.

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