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Author Topic:   Porpoising in 15-Foot Hull
Sunseekr3 posted 05-31-2004 08:42 AM ET (US)   Profile for Sunseekr3   Send Email to Sunseekr3  
One more question for the resident Whaler experts. My 1991 SUPER SPORT 15 runs with an extremely bow high attitude. It is powerd by a Yamaha 70-HP. It also tends to porpose quite a bit when running around 25-MPH or higher. Has anyone else had this same experience? I am considering putting a Doel-Fin on the motor's lower unit. I have installed them on smaller dinghys in the past and they have helped push the nose down and with getting on plane more easily. [Give your] opinion.
Striper 15 posted 05-31-2004 06:36 PM ET (US)     Profile for Striper 15  Send Email to Striper 15     
This has been a hot topic. It may be that the engine is too low. I have the same problem with my 15' Striper. I found that my engine AV plate was about 1" lower than my transom so I raised it one hole. I just did it yesterday so I haven't tested it yet.

User JohnJ80 seems very knowledgable on the subject (see post below).

http://continuouswave.com/ubb/Forum4/HTML/002622.html

Sunseekr3 posted 05-31-2004 09:03 PM ET (US)     Profile for Sunseekr3  Send Email to Sunseekr3     
Thanks for the reply. Let me know how raising the engine worked out. I have the feeling that that is what I have to do also.
skred posted 06-01-2004 08:35 AM ET (US)     Profile for skred  Send Email to skred     
Bow-high attitude can be caused by several factors: excess weight in the stern, engine height (as you mentioned), or (often) trim angle. The engine should normally be trimmed all the way down (in) when starting out, and gradually trimmed up (out) as you increase speed on-plane. At the point where porpoising occurs, you have trimmed up too much. My former 15CC with its original 70 Yamaha (fairly heavy motor) and myself (also fairly heavy at 245) and one passenger would plane beautifully and show no signs of porpoising when trimmed up about 15%. I always started in the "full down" position on the motor. I also had shifted my battery and fuel tank to the center console for a better balance.
Florida15 posted 06-01-2004 09:18 AM ET (US)     Profile for Florida15    
I had a 1987 15' with a Johnson 50 on it that seemed to be slow to plane and was really bow high for a time. I didn't have a speedo on it so I took my GPS out and did some experimenting. I found that the bow high problem was most prevalent when doing between 18 and 22 mph. So, to fix the problem I just powered right through that range. Problem solved.
Striper 15 posted 06-01-2004 12:36 PM ET (US)     Profile for Striper 15  Send Email to Striper 15     
skred - I can agree with you that weight distribution, motor height and surely trim angle all contibute to porpoising. I am reluctant to agree that trim angle adjustment is a preferred fix for the problem. To me, if you give up trim angle, you give up speed and efficiency and the limiting factor on trim angle should be motor slip, not porpoising.

I will be disappointed if I am unable to correct my porpoising problem with weight distribution and motor height adjustment.

skred posted 06-01-2004 03:38 PM ET (US)     Profile for skred  Send Email to skred     
Striper15: I agree. Last resort should be trim, if it reduces your trim range. I have always trimmed all the way in to start, and adjusted out for maximum performance. If I had to run at full "down" (or nearly) trim all the time, I'd be looking elsewhere for the problem.
Sunseekr3 posted 06-01-2004 07:17 PM ET (US)     Profile for Sunseekr3  Send Email to Sunseekr3     
Thanks for the replies everyone.

Trimming the engine out makes the boat porpose quite badly. When accelerating onto plane I always trim the motor to the full down position. Once on plane I trim the motor out to try to get rid of the V-spray and to get a bit more speed.

From reading quite a few posts I'm begining to think that the motor needs to be raised at least 2 holes and possibly more. I did contact the place that makes the foil that many people talk about on this site. The man I spoke with seems to think that the motor needs to be raised as well.

At present the boat is in the marina and I will not be able to get to it until the weekend so hopefully the problem can be solved by raising the motor as well as bolting on one of those foils. I will keep you informed as to the progress.

acseatsri posted 06-01-2004 08:08 PM ET (US)     Profile for acseatsri  Send Email to acseatsri     
Re the Doelfin and raising motor- if your cavitation plate is out of the water, a Doelfin will be ineffective. Also figure on losing about 3-4 knots of top end with the fin. I think you guys put too much emphasis on raising the engine to stop porpoising- It's more a combination of trim and load distribution. If you can power through the range where it porpoises without beating up yourself and the boat, good luck.
JohnJ80 posted 06-02-2004 09:42 PM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
If you have your motor properly set, the AV plate and/or the fin should be riding at the water's surface. The whole idea is that that the lift generated by the AV plate or the fin is sufficient to get the plate to ride at the surface. Then should some force cause the stern of the boat to be depressed, the lift is once again generated and it lifts again to the surface. Since air is considerably less dense than water, it goes no further than the surface (otherwise you'd be airborne or flipped).

I'd wager that if you had a boat set up with the AV plate completely out of the water on a plane, it would either be highly unstable or a gross mismatch of the boat to the transom (like a 20" shaft on a 25" transom or worse).

In my experience there is no one single cause of porposing in all boats. The thing is that it an oscillating mechanical system and there are potentially lots of causes for porpoising. The ones I can think of and I have seen in engineering literature are weight distribution, angle of thrust, lever arm length to thrust source, various hull issues, sea conditions, loading, rigidity and flex in parts of the hull, etc...

In an oscillating system you need to find the particular variable that needs to be damped to remove the oscillation.

Here is a good paper, written by some Japanese engineers, which shows that be playing with damping forces, you can increase the stable region of craft operation. There is also a good bibliography.

http://ns.marine.osakafu-u.ac.jp/~katayama/katayama_PP/ICHD'98.pdf

So, if your read through this sort of paper you find that there are many potential reasons for porpoising that are usually beyond any of us to find except experimentally - by changing one thing and observing. For some that means moving weight forward, others it means moving it aft. It can mean moving a motor up OR down, it can be adding a fin. Each of these effects change the mechanical system and its components that lead to oscillation.

There is no one fix that works for every boat every time, but there is probably one fix that primarily does it FOR A GIVEN BOAT. That is why everyone has anecdotal evidence about how to fix porpoising and the advice very often conflicts with the next guy's recommendation.

So, a fin properly applied on a motor properly height adjusted so that it planes at the surface when the boat is planning will help to damp out the oscillation. It definitely will contribute to making it more difficult for the boat to try to push the stern deeper in that half of the porpoising bow amplitude cycle - that is damping.

It is possible to pretty much remove porpoising (except in extreme and usually ridiculous configurations) by (a) adjusting vertical height on motor, (b) adding a fin, (c) making a prop change, (d) load adjustments, (c) trim tabs.

That all being said, it is typically necessary to raise the motor on most boats from the position that the motor is installed by the dealer.

Most dealers have not a clue about how this all works and will almost invariably mount a motor more towards the down side than up because it is easier for them since there are less chances the motor will overheat from lack of cooling water and that the prop will blow out in turns. That does not mean that a motor all the way down is properly mounted.

Even Whaler takes a conservative approach and recommends 0-3/4" for the AV plate above the bottom of the hull. Most people that get into this find that something like 1-1.5" is better. That all being said, it is very difficult to measure the plate height accurately to the bottom of the hull since the hull is not really all that flat and its angles vary depending on where on the bottom you measure it. As well, the AV plate moves through an arc so it is not usually straight (parallel) with the bottom for most of its range. So, where you measure is somewhat of a judgement call - and less resolution than the 3/4" each hole provides. So, it is generally a question of experimentation again.

J

jimh posted 06-04-2004 09:24 AM ET (US)     Profile for jimh  Send Email to jimh     
John--Thank you for the fine article you have contributed, but it has me in conflict. Let me explain.

There are two recurring themes in the on-line literature of internet boat web sites: the Japanese are the real experts at building an outboard motor, and the $29.95 plastic Doel-Fin cures all problems. You come along with this reference to an engineering research article that is in complete conflict with this.

Misters Toru Katayama and Yoshiho Ikeda of the Department of Marine System Engineering at Osaka Prefecture University report that they cannot find a good explanation or cure for porpoising. In one single research paper they violate both of these important tenants of on-line boating forum advice.

At its most simple analysis, they clearly do not know of the Doel-Fin and its power. This calls into question the validity of the "Japanese as expert" theme. Now if we assume they do know about the Doel-Fin yet they to not affirm it as the solution to the problem, this calls into question the validity of our second theme, the Doel-Fin cures all problems.

At a single stroke, your brush has painted over the two most influential movements in all of modern marine on-line literature.

Can this be true?

JohnJ80 posted 06-04-2004 10:45 AM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
Jim, you are just a rascal! ;-)

But you raise a good point. The Japanese don't know the answer either and fins don't cure all problems.

Certainly if you listen to the fin guys (mfgs), fins cure every problem including acne, falling arches, and relationship problems. That is, of course, not true either.

Also, if you search this forum it is probably 50-50 on the japanese question you raise (but I just don't want to go there).

Actually if you search this forum, I would guess it to be about 50-50 or 60-40 (fin/no fin). For some it works and some it doesn't. The fact remains that the problem is just more complex than that and this paper proves it. IMHO, both sides are equally wrong in a general sense.

The important points on solving porpoising problems are:

1. It is an oscillating mechanical system. It can be sent into oscillation by an external stimulation - like a wave. Damping, of some sort, is required to remove or reduce the oscillation.

2. There are a whole lot of contributing factors that may lead to conflicting solutions on a boat by boat basis. What works on one boat may or may not work on another. There is no one silver bullet for every problem.

3. Measurement of the causes of oscillation are difficult at best and impossible at worst.

4. Solutions are found, both practically and realistically, by an experimental process - changing only one thing at a time and observing the result. Intuition is not going to cut it to get to the answer.

5. This problem knows no national boundries and engineers worldwide have equal problems in determining the root cause of the porpoising problem (this for Jim's benefit).

6. Any boat can be made to porpoise by removing the factors that contribute to damping.

J

jimh posted 06-07-2004 08:13 AM ET (US)     Profile for jimh  Send Email to jimh     
[Changed TOPIC; was "Bow high running atttude 1991 15 SS".]
jimh posted 06-07-2004 08:21 AM ET (US)     Profile for jimh  Send Email to jimh     
Authors Toru Katayama and Yoshiho Ikeda (see hyperlink above) include an interesting bibliography in their article on porpoising. Note the date of their first reference work:

[1] Day, J. P. and Haag, R. J. (1952), “Planing Boat Porpoising –A Study of the Critical Boundaries for a Series of Prismatic Hulls-, Thesis submitted to Webb Institute of Naval Architecture.

Here we are now, fifty-two years later, and the cause of porpoising is still something of a mystery.

Their fine article also documents how the hydrodynamic forces that contribute to pitching and heaving change significantly with boat speed, can even reverse direction, and are non-linear.

JohnJ80 posted 06-07-2004 11:02 PM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
Jim,

Did you actually find the 1952 paper or a link to it? That sounds like even more interesting reading.

J

kingfish posted 06-08-2004 02:30 PM ET (US)     Profile for kingfish  Send Email to kingfish     
Re: Remarks above from acseatsri and JohnJ80-

1. acseatri - The DoelFin will be ineffective at high planing speeds if the cavitation (anti-ventilation) plate is out of the water, and that is as it should be, as the a. v. plate *should* be out of water at high planing speeds. That doesn't mean that the DoelFin won't be effective in helping to get on plane more quickly with less bow rise, and holding plane at lower speeds; it will do those things.

2. JohnJ80 - Setting up a boat with the a. v. plate out of water when on plane is neither a cause for instability nor is it a gross mismatch of the boat to the transom (a 20" lower unit on a 25" transom would put enough of the prop *behind* the transom that you probably wouldn't get up onto plane to begin with). In fact, it is a commonly accepted tenet of proper set-up that the motor be hung on the transom in such fashion that the a. v. plate is above the extended line of the lowest point (keel) of the hull bottom between 1" and 1-1/2", which will put the a. v. plate above the water at high planing speeds, by design. There are always exceptions to the rule, but any time the a. v. plate is in the water at high planing speeds by design (or otherwise), there is some loss of speed, fuel efficiency and general performance.

JohnJ80 posted 06-08-2004 05:44 PM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
It is certain that the very end of the hull is out of the water, so any AV plate that contacts the actual transom is out of the water. Rapidly, as you move back from the stern, water come ever higher eventually forming the wake behind the boat. Even a few inches can make a large difference in water level (why jack plates are used - get the motor up higher in the wave of water coming off the end of the hull).

My point was that if you adjust your AV plate so that it is grossly higher than the bottom of the hull will most likely lead to equal instabilities as if it is grossly low. I probably misspoke and used a little too much hyperbole.

Usually, because of either the AV plate or the fin, it is at the surface of the water at a WOT plane simply because of its very purpose - it provides lift to the stern. A fin does not have a large enough surface area to go higher - it would have to be an airplane wing or at least a foil that would operate in a less dense medium (foam and water filled air instead of solid water).

I agree that you want the fin to be at the surface very early in the planing range of the boats operation. if it is forced too low at too high of a speed, I would expect that it would exacerbate the oscillation rather than help it.

I'm not exactly sure of the terminology, but it is something like the more you get the direction of thrust pushing horizontally through the center of gravity (or center of something, maybe center of effort (?) - there has to be a marine architecture term for this) the more efficient you are and the less likely to oscillate. If you look through this forum there are instances where people found that they could make their boat oscillate (porpoise) by putting too much weight forward (it was a trolling motor, I believe in onw case).

The USUAL culprits though are too much weight aft requiring more lift in the stern, and motors hung too low on the transom requiring raising.

My main point in all of this is it is still a process of experimentation and that each boat requires somewhat different solutions but that it is usually ONE factor that resolves the problem for a particular boat. However, that does not imply that that one problem will solve it for EVERY boat - most specifically, not for even every boat of the same model and motor. There are other factors like loading and seaway that come into play.

J

Richard Quinlivan posted 06-08-2004 10:41 PM ET (US)     Profile for Richard Quinlivan  Send Email to Richard Quinlivan     
Johnj80

Your insights are good. I have done a similar research and have considerable experience in the stability and conrtol of aircraft. In fact a good deal of my career was in the area of correcting dificient aircraft responses. Generally these techniques are feedback control systems involving angular rate and linear acceleration feedbacks. In my search I think the reason that there is little real engineering data on porpoising is that there is no real $ in it. Aircraft are a different story.

In my research I think that the difference in the instability in planing hulls from that of aircraft is that as the planing hull rotates in pitch and heaves in the vertical is that the center of pressure of the lift force moves forward and aft. If that force moves from forward of the cg to aft of the cg the pitching moment changes sign and causes the instability. What is not so clear is why this situation is closely tied to trim angle as the literature clearly says it is.

the reason raising the moter lets you trim out more to align thrust to velocity is that the moment produced by the motor is reduced as the lever arm is reduced. You therefore can get the motor trimmed out further before the trim angle gets to the critical force. The effectiveness of planing aids and trim tabs is that they inhibit the high trim angles and also they geberate damping forces when they move through the water.

The way that this works with my boat is that my porpoising problem occurs at 25 to 35 mph and the planing aid (Turbolift in my case) is still in the water. At 40+ mph the boat is higher in the water and attitude flattens out because the center of pressure is moving aft. This gets the Turbolift out of the water so it has no effect one way or another.

All that being said; lift the motor, move weight forward, and if that doesn't work try a planing aid. A SS prop is useful because it allows more trim range without breaking loose.

Dick

JohnJ80 posted 06-08-2004 11:14 PM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
Dick - excellent explanation. It makes sense.

I would think that another complication in the marine application over the aircraft one is that the vehicle has to operate at the interface to two different viscocity "fluids" - i.e. water and air and that the boundry is virtually never flat or stable (turbulent).

So, then if you go further with the aircraft analogy, as you add trim tabs you get even more control of the alignment of the velocity and thrust vectors - but you can do it by making the control of the trust vector (trim) more independent of the attitude of the hull relative to the water. So, you can place the hull where you want it and then align the thrust accordingly. As well you can do this at the higher planing speeds when the fin is out of the water.

The trim tabs become more analagous to the trim surfaces on an aircraft wing that are used to control the attitude of the aircraft as it passes through the air. With trim tabs, you get much the same sort of control. Correct?

So, also, the proper trim on the engine is really not a matter of much movement at all. You are adjusting it through maybe a 15 degree arc or so to just align it with the velocity of the boat when on a plane. Without trim tabs, problem is that when you adjust the thrust vector (trim) you are also adjusting the attitude of the boat to the water.

Getting obsessed with this as I am, I added trim tabs to my boat last year. I wanted to see what they really do. I also needed to manage the ride of my boat so that I didn't pound through large boat wakes. When I added the tabs, I found that I could really adjust all steering resistance right out of the boat. I can pretty much set up the trim on my boat so that I can turn the wheel with a single finger.

Does that all make sense?

J

Richard Quinlivan posted 06-09-2004 12:08 AM ET (US)     Profile for Richard Quinlivan  Send Email to Richard Quinlivan     
Indeed the trim tabs give an extra degree of freedom that lets you adjust boat trim and thrust alignment independently. I don't have tabs so I don't have any practical experience with them.

The big difference between a wing and a planing hull is that the wing is totally immersed in the fluid whereas the boat is on the interface. The other factor is that if we consider smooth water; as the boat goes faster the area of the wing or hull is reduced in proportion to the square of speed. Lift at any speed is a function of planing area and angle of trim. If you trim higher ihe lift due to angle of trim (angle of attack in a/c terms) goes up and the planing area goes down. The significance of this is what trim angle minimises drag. Lowering drag increases speed! The rotton part of this is according to the literature I have seen the minimum drag occurs at an angle of trim higher then the porposing limit. To maximise speed increase trim until you start to porpoise!

The other interesting factor is that to go fast you need the cg aft so you can get the trim angle up. If you do that you have a lot of trouble getting on plane. This is a great argument for trim tabs. Some racing boats have water bladders forward to pump water in and out of so they can independently adjust cg location.

Dick

Sunseekr3 posted 06-16-2004 05:42 PM ET (US)     Profile for Sunseekr3  Send Email to Sunseekr3     
Just to bring everyone up to date; I raised the motor so that the ventilation plate is about 1.5" above the lowest point of the transom. The result is about 50% less porposing and much more ability to trim the motor out. At present I have a turbolift on order from Grand Island Marine and hope to have it sometime next week. Once I bolt it up I will let everyone know the results. Thanks again for all the input.

Sean

davej14 posted 06-17-2004 04:35 PM ET (US)     Profile for davej14  Send Email to davej14     
Sean,

What prop are you running? Brand, dia and pitch? Are you getting any ventilation in turns? I'm looking for input because I am running my 70 merc at the same height with a turbolift on a Dauntless 14 and get quite a bit of ventilation with a Black Max 19 pitch. I'm considering another prop or will need to lower the motor a hole.

JohnJ80 posted 06-18-2004 03:58 PM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
On my Dauntless 15 - which I believe is fairly similar, I have a FICHT 75HP motor, a Turbo Lift and the motor is ALL the way up. I don't ventilate unless I make a high power, almost to the stops turn with the boat (not normal operation). In normal operation, I don't ventilate at all. I can trim the motor up at least up to 40% depending on speed and conditions.

Note: The purpose of triming is, as Dick has pointed out, is to align the thrust vector with the velocity vector. Simply stated, this means that you want the thrust off the prop to be parallel to the direction of travel. So, you don't want to trim all the way up or all the way down, but to the point where they align.

It is possible, with most boats, to get them to porpoise at some point by ridiculous configurations of trim and loading. So, it is probable you will always be able to make your boat porpoise.

Sunseekr3 questions for you:

1. Did the ride improve - i.e. less harsh in chop or waves?
2. Did your bow come down somewhat, especially when trying to go to a plane?
3. Any other ride characteritics that you noted that changed?

J


Sunseekr3 posted 07-19-2004 06:18 PM ET (US)     Profile for Sunseekr3  Send Email to Sunseekr3     
Hi Everyone,

I have been going around and around with Grand Isle Marine in Fla. I think that I heard every excuse in the book as to why they could not ship my Turbolift to me. Finally I recieved it last Thursday...only to find that it was the wrong size! I am going to send it back and get a refund, (Hopefully), as they don't seem interested in making a and shipping a unit for my motor in any sort of a timely manner.

In the mean time I bought a Doel-fin and mounted it. THe change was quite dramatic. The bow comes down quickly and the porposing is all but gone. The boat settles much more nicely comming off of plane and I can get more trim range than before.

To answer a previous question I am running a Yamaha prop with a 13x19 pitch. I do not have any ventilation issues with the prop either with or without the Doel-fin. The motor is still mounted at maximum height.

To say the least I am very diappointed with the Turbolift people. The unit looks great but it simply does not fit the outdrive. Both the 70 and 90 hp circa 1992 Yamaha outboards use the same lower unit, according to Yamaha. The turbolift that I recieved was much too narrow for the cavitation plate on the engine. Briefly I thought about having it redrilled but decided against it.

Take care All,

Sean

JohnJ80 posted 07-19-2004 11:46 PM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
Sorry to hear about your experience with Grand Island Marine. I had the opposite experience but no one gets it right all the time.

Glad to hear that you are well along the way to solving your problem. This is a game of inches - each change gets you a little bit more towards a solution.

So, are you happy with your boat's performance now? Did this all work out for you?

J

davej14 posted 07-20-2004 07:50 AM ET (US)     Profile for davej14  Send Email to davej14     
My experience with the turbolift peole was also a good one. They are clearly a small operation, but they sent the fin right out as promised. When I received it I also thought it was too small and called them to discuss. They convinced me to try it and although the bolt pattern looked too narrow and the slot for the fin on the cavitation plate looked like it wouldn't fit, when I bolted it on it was perfect.

Sorry to hear of your experience, but glad the doel fin works for you AND you saved a few $$.

JohnJ80 posted 07-20-2004 10:52 PM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
You know, dave, that was exactly the same experience I had. I thought it wouldn't work and actually wound up calling them on my cell phone while installing it. The trick was you need to remove or significantly loosen the steering trim skeg to make it fit. I had forgotten that.

When you look at it, it seem to be pretty simple to install (it is), but there is a trick to it. I wonder if that was the problem?

Incidentally, thanks to all on this thread. this is a great thread loaded with lots of good information!

J

jflots posted 07-22-2004 10:12 AM ET (US)     Profile for jflots  Send Email to jflots     
Simply put small classic whaler hulls aren't efficient. New smaller hulls are better but still need weight forward to run right. Look at how far forward the new 130 sport rear seat is. If you can put all gear and equipment(including fuel tank and battery) forward in the boat. If you can't then keep the passenger on the front seat.
JohnJ80 posted 07-23-2004 08:38 AM ET (US)     Profile for JohnJ80  Send Email to JohnJ80     
Incidentally, neither are the Post Classics. I notice that my Dauntless 15 makes a bigger wake at slow speeds (hard to do 'no wake') than other boats. I also notice that I can go out in much worse conditions and have a better ride than pretty much anyone else out there.

They have a very high bow rise with most of the reserve floatation well aft as well as the Center of Gravity. That makes them have great seakeeping capabilities but it also gives them a propensity to porpoise.

The porpoising can be handled through a variety of means. I'll take that tradeoff any day and twice on Sunday for the ability to handle tough conditions.

Any boat is a collection of compromises. You an also get a boat that is great in flat water - fast, smooth etc... But, that boat will probably be not much fun in chop.

We used to have a 15' tri hull. That think was wonderfully stable - you could walk side to side and it wouldn't heel hardly at all. In flat water it just flew due to the flat aft and mid section. but put that baby in any chop and your next stop was to the dentist to replace all your now loosened fillings.

J

dan99 posted 08-15-2011 04:14 AM ET (US)     Profile for dan99  Send Email to dan99     
[Seven years later...] [Revives this discussion--actually twice with duplicate articles--in a somewhat imcomprehensible manner, apparently talking about a problem with a non-Boston Whaler boat.]
Russ 13 posted 08-16-2011 11:35 PM ET (US)     Profile for Russ 13  Send Email to Russ 13     
The easy way to reduce de porpusing: move weight forward, in the boat. Too much stern weight is most of the problem.
jimh posted 08-17-2011 08:13 AM ET (US)     Profile for jimh  Send Email to jimh     
I hope that helps Sunseekr3. Also, you should write to those researchers at the Institute (see 1952 scientific study above) to let them know of your discovery. I know it has been 59-years since they published, but they may still be studying the problem.

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