Marine Wave Question: Wave Length, Amplitude and Velocity

[ken]

Hi

I've been thinking about the way water flows around, or should I say up, along and down a displacement hull. This has lead me to a hull design idea based on the hull displacing water into a single wave as it passes. The cross section area at any point along the hull would be equal to the cross section of the displaced wave. Disigned properly, for a specific speed, the whole length of the hull would produce just one single wave length instead of the usual bow wave and stern wave.

What I need for this is a formulae relating wave length, amplitude and velocity. Anyone know of one?

I don't really know how this would work out, it's probably been done before, anyone know of that either?

 

[russ_watters]

Originally posted by ken
I don't really know how this would work out, it's probably been done before, anyone know of that either?

Yes, its been done, but its impressive that you figured it out on your own. Here's a decent link.

A displacement hull rides in the water, using normal buoyancy to support it, while a planing hull rides above it. Big ship= displacement hull, speedboat= planing hull.

With a displacement hull, the bow and the stern each produce their own wave. The frequency of these waves is constant based on the properties of water. The wavelength (relative to the ship) depends on the speed of the ship. So the faster the ship goes, the longer the waves get and eventually the bow and stern waves get in sync with each other. When the waves are in sync, the boat rides in a trough between the waves. This speed is the theoretical maximum speed of the ship. To go faster, the ship must start to drive over its bow wave, lifting itself out of the water, obviously requiring an enormous increase in power.

As an example, I was in the Navy on a 434 foot ship. With one engine on, the ship could go 25 knots. With two (double the power), it could only go 29. Even pushing the engines to 110% of rated power still only yieled a fraction over 29 kts.

So, the equation (from the site):

1.34 X the Square Root of the Boat's Waterline Length (This is Theoretical Maximum Hull Speed)

Thats length in feet, giving you speed in knots. For my ship, that equation yields a max speed of 28 kts. I'm sure if you google you can find a derivation of the equation if you really want it.

Now figure out the hull speed of a 1000 ft aircraft carrier

 

[ken]

Hi Russ

Thanks for the reply.
I guess what you are saying is that because max velocity is related to the square root of waterline length, there are diminishing returns for hull length. I have seen that before.

This formulae is not really what I am after. I want the to know how the speed of a wave relates to it's length and amplitude. This will allow me to devlop the hull shape from the wave. I am hopeing that will give the min drag coefficient at maximum displacement speed.

Can I e-mail you a drawing which will make the design idea clear?
I think you will see what I want and why.

I hope it will lead into designing a planing hull to achive min displacement drag coefficient at a velocity just prior to planing.
If I am correct, the WL length will represent a half wave length
at this speed as the transom will be the point of maximum wave displacement when trimmed correctly.

E-mail ken.webster@lands.nsw.gov.au

 

[russ_watters]

Originally posted by ken
This formulae is not really what I am after. I want the to know how the speed of a wave relates to it's length and amplitude.

It doesn't. The speed of the wave is just like the speed of any other pressure wave - its a constant based on the properties of the medium.

I am hopeing that will give the min drag coefficient at maximum displacement speed.

What the equation gives you is a "tip over" point for drag - the point at which it starts to increase radically. If you want a full drag curve, that gets a LOT more complicated. http://www.dt.navy.mil/ip/mfp/Acrobat%20Files/Technical%20Papers/Session%202%20-%20Advanced%20Ship%20Technology/Topic%2011/DGBJapanMay2000Paper.pdf is a more in depth paper on speed:length ratio, drag coefficients, and displacement hulls. The first few pages are the key parts, laying out the theory on displacement hulls. It has a nice drawing of the relationship between wake wavelength and boat speed.

Feel free to email me (russ_watters@lycos.com) but I prefer to keep the discussion here for the benefit of other readers.

 

[ken]

Hi Russ

I e-mailed a .jpg image, is there a gallery on the forum?
Is the design concept clear to you?

I am not after drag curve equations quite yet.
I had wondered if wave velocity in deep water was a constant.

For developing this hull shape, I need to know about any relationships between wave length and amplitude so I can
get the wake rake angle correct. Then, if I can find a
velocity for wake waves, simple trig will give the hull speed.
I guess there may be differenced between fresh and sea water.
Is temperature a factor? I know some aerodynamics but nothing
of fluid interface dynamics.

I have not looked at that last site yet, I wanted to send the image ASAP and communicate a few things. I'll look now though.

 

[ken]

Hi Russ

I read most of the document on that site.

It states:

V = 1.34 * L^0.5

where:

V = velocity of a wave in knots
L = wavelength

I guess that is what you have been trying to tell me.
However seeing this same equation used for hull speed,
I had not related it to wave speed because I thought the
wake was eminating at some angle to the boats movement.

The formula suggests the hull is generating a wave in the
same direction as travel.

As you can see from my drawing, what I really need to work out is the rake angle of the wake. This hull shape results in a max x section about twice the hull mean x section. This will be the x section of the displaced waves above the WL which is a function of wavelength and amplitude. I need the wave length to determine the wake rake angle. Then I can generate the hull sections from the displaced wave at each point along the hull.

So the only thing missing is a wavelength to amplitude function.


You may be wondering about my x section shape and stability.
This is a sea kayak hull and I thought to start by looking at the lengthwise x section area distribution. The deep V is mostly to simplify things but there are other reasons. It provides a deep keel line to resist side drift in wind. Water accelerated up or down the hull, travels a shortest straight line distance to the surface, minimising acceleration forces needed to get it there. I know round bilge has minimum wetted surface area and provides a better fineness ratio as does less sharp ends. Experience tells me that a kayak of deep V section and a more or less diamond WL plan is exceptionally slippery throught the water and offers surprisingly high secondary stability.

My first aim is to generate a shape this way, then do some stability analasys. Stability can be improved by adjusting beam,draft ratio.

I plan to construct some large models for comparison testing, one of a more conventional design.

 

[russ_watters]

Originally posted by ken
I guess that is what you have been trying to tell me.
However seeing this same equation used for hull speed,
I had not related it to wave speed because I thought the
wake was eminating at some angle to the boats movement.

This is because, as I said before, the speed of the wave is determined by the properties of the water and has nothing to do with the ship. As a result, the angle of the wake changes with speed. An interesting side note - water tables are used for supersonic flow visualization: a boat's wake works and looks exactly like a supersonic shock wave in 2d.

There is one issue with the pic you sent me - the point of maximum beam needs to be reached relatively quickly because that's the high point of the bow wave (or nearly so). So sharpening the front of the ship actually reduces its hull speed. That's the reason cargo ships have that funny looking bulb at the water line in the front. HERE is more on that. The bulb itself is underwater and the bow of the ship still "cuts" the water, but the bulb is close enough to the surface to start the bow wave early. Note:

typically giving a 5% reduction in fuel consumption over a narrow range of speed and draft.

Since the bulb is only partially submerged, draft and speed are critical to its effectiveness. That's why it isn't used on all ships. Freighters have a pretty consistent ride.

edit: reading more of that page I linked for you, its apparently more comlicated than I realized. Its not so much that it moves the bow wave forward, it actually kinda creates a second one to partially cancel out the main one.

Now, you're woking on a kayak - that's an important piece of information because what's important changes with the size of the boat, so a lot of the info I've given you thus far may not be relevant. Most importantly - do they plane? How long are they and what is their top speed?

I'm looking on google for wave propagation info (to help find the wake angle), with no luck yet (except a NASA site that says there are two waves - one at a constant 39 degrees and another that depends on the speed of the boat) - I'll keep looking.

 

[ken]

Hi Russ

Single person sea kayaks typically have a LOA of 4.5 to 6 m and a LWL of 4.5 to 5.5 m though there are variants. I started with a LWL of
5.0 m because that seemed a reasonable compromise between speed and monouverability. Another advantage of the deep V I did not mention is slamming in rough conditions, it's a more comfortable ride.

You can see that I am generating the design only on the basis of the wake raked in a straight line from the bow entry. That is going to produce a rearward max beam. Just how rearward depends on the rake angle. It is interesting to me that a 39 deg rake will produce a slightly rearward max beam more typical of crusing and racing kayaks.

Sea kayaks are not typically designed for planeing as you can see from my eliptical keel line. I do not know the typical top speed of sea kayaks. I could determine my speed in my 4 m slalom kayaks but don't have that info at the moment. Most kayaks can surf but can only plane well if specifically designed for that. I suspect a kayak can exceed the max speed suggested by the usual formulae though I should verify this.

Racing kayaks have a max beam rearward from the paddler and above the WL. Do not be fooled though, this a min beam racing rules requirement. The WL plan is quite different.

Kayaks designed for broken water typically have a rearward max beam to increase rearward boyancy for wave piercing. Without this, the stern sinks more and the boat tends to want to surf backwards instead of going through the wave. However the pic has this far more aft than is usual.

Thinking about the 2 wakes idea:
In larger vessils, I have observed the for/aft wave and renonant waves directly astern This seems to be the speed limiting factor described by the formulae. It seems to me, the bow wave is like the stone in a pond ripple effect except the the epicentre is constantly moving with the bow. So I'd expect the bow wave to form an ark around the bow and then rake back in a curve to some angle, Could that be the 39 deg wake angle you mentioned?

It would not be difficult to determine such a curve and base the same wave idea on this instead of the straight line I used in the pic.
That would bring max beam close to midships and look more typical.

I guess in my design, I have been ignoring fore/aft wake. My design is soley aimed at looking at side wake formed by sideways displacement as the hull passes. This may explain the 2 wakes you mentioned and also explain why wake waves are not continuous. As the wakes cross at different angles, their phases either accentuate or cancell. I had wondered about that.

It is difficult to tell what wakes are typical for faster kayaks because these boats create very little noticable wake.

It might be worth checking out these differnt approaches with models. I am familiar with Reynolds number theory but that does not seem to hold good in the light of the wave speed formulae. Do you know the approaches used for marine test models?


The design approaches for ships and boats are different, ships tend to have a long constant x section for ease of loading cargo and construction economy. While this yields a good fineness ratio, small boats seem to perform best with varying x sections despite a worse fineness value. The most effivient WL plans seems to have a concave curve for and aft, reflexing into almost straight lines to a tight convex curve at max beam slightly aft of midships.

That is not too far from the pic so maybe I'm near the right track.
What do you think?