First of all, it is not necessarily true that a narrow boat will be less prone to broaching. If a boat is too narrow it will tend to 'root' and that can cause one form of broaching. Like so many things beam is one of those items where moderation is important. Neither excessively beamy nor excessively narrow boats are good from a seaworthiness or motion comfort standpoint. There are lots of reasons why a Pacific Seacraft might be a better choice for you and many reasons why the Wauquiez 33 might be a good choice as well.
As to the Capsize Screen Forula and Motion Comfort Index,
I know that I have explained this on this forum many times before but here it is again, both of these formulas were developed at a time when boats were a lot more similar to each other than they are today. These formulas have limited utility in comparing boats that otherwise are very similar. Neither formula contains almost any of the real factors that control motion comfort or seaworthiness. Neither formula contains such factors as the vertical center of gravity or bouyancy, neither contains weight or buoyancy distribution, and neither contains any data on dampening which really are the major factors that control motion comfort or likelihood of capsize.
I typically give this example to explain just how useless and dangerously misleading these formulas can be. If we had two boats that were virtually identical except that one had a 1000 pound weight at the top of the mast. (Yes, I know that no one would install a 1000 lb weight at the top of the mast.) The boat with the weight up its mast would appear to be less prone to capsize under the capsize screen formula, and would appear to be more comfortable under the Motion Comfort ratio. Nothing would be further than the truth. That is why I see these formulas as being worse than useless.
This is a draft an article written for a different forum but it may help with your original question.
Like so many aspects of yacht design, beam is only a one part of the equation that affects motion, and performance. Like weight wide beam is often singled out as having a particularly large affect on sailing performance. While there are broad generalities that can be assumed about the affects of beam on the behavior of a boat, the realities of the affect of beam on any specific design can vary very widely from the assumed generality.
In general, all other things being equal, a boat with a narrow waterline beam will have less initial stability and a motion marked by softer acceleration and de-acceleration. It will be easier to drive through the water meaning that it should have less drag in high winds and low. Narrow boats tend to have an easier time going to windward especially in a chop. Narrow hulls tend to have better directional stability (helping the boat to hold its course better) but all other things being equal having a larger turning circle. They also tend not to develop as much weather helm from heeling because their heeled waterlines are more symmetrical than those of a beamier boat.
As a broad generality, beamier boats tend to have quicker motions, more wetted surface and therefore more drag. They tend to be slowed by wave action. The have more stability when inverted. They have greater initial stability (stability at small angles of heel) but a smaller angle of ultimate stability (the angle at which the stability becomes negative and the boat would rather invert than come up). For normal coastal conditions they require less ballast for a given stability. They tend to be faster down wind in a moderate to heavy breeze because are more likely to be able to surf or plane. They tend to point upwind better in moderate conditions (a combination of greater stability and more foil length for a given draft). They also tend develop as more weather helm and are more prone to wiping out due to heeling because their heeled waterlines are less symmetrical than those of a narrower boat and wide beam tends to “jack” the rudder and keel out of the water.
Now those were the broad generalities about beam but as I said before boat to boat there are big variations between the normally expected impact of beam and the actual affect of beam. One major reason that there is such a variation is that we tend to talk about the beam of a boat as a single measurement taken at the widest point at the deck. In reality, the actual affects of wide or narrow beam on the performance and comfort of a boat can be altered by a lot of factors. These would include items such as waterline beam, the distribution of beam fore and aft at the waterline and immediately above, the cross-sectional shape of the hull, weight and weight distribution and the amount of free board and height of deck structures.
To really understand the affects of beam you need to understand the affects of form vs. ballast stability. There are three ways that boats develop stability; form (also called stiffness or initial stability), ballast, and dynamic. Of the three, dynamic has the least relevance to displacement boats as they rarely have enough speed to create a useful amount of dynamic stability.
As the name implies, form stability derives from the shape of the hull of the boat. A shallower and wider hull generates more stability than a deeper narrower hull. Visualize a piece of wood lying in the water. On the flat it has a lot of form stability. On edge it has next to none. This is because stability in any boat comes from the distance between the center of gravity (the balance point for all of the weight in the boat including those things that are part of the boat and those things that can be moved around) and the center of bouyancy (the single point that is the center of all of the volumes under water). In a wider, shallower boat (talking about the part of the boat that is in the water) as the boat heels, the center of bouyancy moves more quickly toward the low side for small increases in heel angle.
Form stability has several problems. First of all at large angles of heel, approaching 90 degrees form stability begins to drop off dramatically. This is the point where it can be needed most. Second form stability tends to give the boat a quicker motion as the boat quickly builds stability and the boat snaps from one hard bilge to another. This is less comfortable for the crew in rough conditions and can be more tiring. Lastly, in the extreme conditions of a blue water passage, a boat that depends on large amounts of form stability also tends to be more stable in an inverted position.
Ballast stability has mostly positives associated with it. All other things being equal, the deeper and heavier the ballast the more stability a boat will have. Ballast that is heavier and deeper also gives the boat a slower, more comfortable motion. Of course like most things in yacht design, there are some tradeoffs in this area as well. If the ballast occurs at the end of a deep keel, the boat cannot get into as shallow water. As you start to shorten the keel length you substantially give up performance for the same stability because you either end up with a low aspect ratio foil or you end up with a big bulb or wings also increasing drag. No mater what the ads say, nothing performs like a deep fin keel in all conditions. They do loose some efficiency in heavy seas but not as much as the long keel proponents like to hypothesize that they loose. Of course deep fins have their own compromises but that is not the point of this piece.
There was a lot of discussion of the physics of stability. Ignoring dynamic stability for the moment, the stability of a boat simply a product of the weight of the boat times the horizontal distance from the boat’s center of bouyancy and the boat’s center of gravity. To define the terms the center of gravity is the one point (fore and aft, athwartships and vertically) that is the center of all of the weight on the boat. The center of buoyancy is the single point that is the center of all of the weight of the water displaced by the hull.
When the boat is sitting still and not subjected to outside wind or current forces, the center of gravity occurs directly above or below the center of bouyancy. But as the boat heels, the center of buoyancy moves toward lower heeled over side. Discounting for the moment weight being shifted on a boat (such as crew crossing the deck or fluids in a tank seeking the low side of the tank), the center of gravity does not move relative to the hull of the boat as the center of bouyancy does but it does move relative to its previous position. Depending on the shape of the hull and topsides, the hull either “jacks up” or “rolls out” (typically boast with high form stability, and flared topsides) meaning the center of gravity rises, or “sets in” or “rolls down” (typically boast with slab sides, tumblehome and Vee’d or wineglass hull sections) meaning the hull moves lower into the water and the center of gravity lowers at the boat heels.
The center of gravity also rotates about the instantaneous axis of rotation. (I say instantaneous for because this axis does not remain fixed but shifts as the boat rolls due to the change in shape of the hull as it heels.) If the center of gravity is above this axis of rotation (Lightly or un-ballasted boats, old IOR boats, and some steel boats and some heavy cruisers with heavy rigs, interiors, topsides and decks and low ballast ratios), it will rotate to leeward to leeward the boat heels reducing the potential righting moment of the boat. On boats with the center of gravity below this axis of rotation (modern IMS race boats, boats based on IMS type hulls and traditional wooden heavily ballasted deep draft cruisers, rather than the lighter ballasted heavy-cruisers of the fiberglass era) the center of gravity shifts away from the center of buoyancy increasing the center of gravity aspect of stability. The reason that a deeper or heavier keel helps stability is that it simply lowers the center of gravity of the boat so the separation between the center of buoyancy increasing the center of gravity is greater.
Of course moving weight to one windward, moves the center of gravity to windward, and so helps spread the distance between the center of gravity and center of buoyancy which increases stability. So much so that even on a baot as large as a Volvo 60 (nee Whitbread 60) the crew moves every piece of gear and every crew member (who is not doing a task to leeward) to the windward side of the boat and use hinged stanchions so that the crew can lean out further.
So lets go back and look at the exceptions to the rules above. Normally, beamy boats are assumed to have a quick motion. That is because for a given displacement they are assumed to have a lot more form stability. If you visualize to rectangles of equal area one nearly square and one much wider and shallower obviously the flat rectangle will have more surface area and will rock more quickly from chine to chine. BUT, in the newer wider designs, careful modeling has allowed wider boats with less form stability and more symmetrical heeled waterlines.
If you visualize a boat that had a cross-section that looked like a section of a cylinder as it heeled it would not develop as much form stability and so would have a slower motion and the waterline shape would always remain the same shape. Through computer modeling, the portion of the new beamier boats that are in the water act more like the cylinder model and therefore often develop less form stability as they heel than their beam would suggest and therefore have a more comfortable motion. The center of bouyancy still shifts relative to the center of gravity as they heel and with their wide beam they are still more effective at increasing the distance between the center of gravity and the center of buoyancy and therefore increasing stability.
Another exception to the generalities about the affects comes from the difference between waterline beam and the beam at the deck, as well as, the distribution of beam fore and aft at the waterline and immediately above the water line. As I said above, we almost always look at beam as the number published in the literature about the boat. That beam has absolutely nothing to do with the behavior of a boat in most conditions because the waterline beam can be substantially different than the beam on deck. Because of that a comparatively beamy boat at the deck level can actually have a narrow waterline and behave like a narrower boat in terms of motion and sailing performance. This is often the case with modern IMS and Open class boats.
The shape of the topsides affects how quickly a boat becomes beamy as it heels so a narrow beam boat with highly buoyant topsides can behave like a substantially beamier boat as it heels, rolling out quickly developing form stability and a quick snappy motion. Another factor, which is totally ignored, is the distribution of beam along the waterline. If you visualize two boats with the same displacement, waterline length, cross sectional shape, and waterline beam, but one has a diamond shaped water line plane and the other is a simple rectangle, the boat with the rectangular shape would have more form stability than the boat with the diamond shaped water plane. The rectangle would tend to have a snappier motion as well.
Because of that phenomena a narrow boat that carries its waterline beam to its ends would tend to have more form stability than a beamier boat with more diamond shaped or triangular shaped waterlines. Modern hull forms tend to resemble the diamond model more than the rectangular model and so can have a wider waterline maximum beam than its more traditional models, which tend to push the buoyancy more toward the ends of the boat.
We touched on the affects of the cross-sectional shape of the hull. Basically flat rectangular sections give a lot of initial form stability but with the negative affect of promoting quicker motion. Boats with cylindrical, and deeper Vee’d sections have less form stability and softer accelerations than the flat box model. A beamier boat with cylindrical and deeper Vee’d may actually have a more comfortable motion than a hard chine narrower boat. (That is the principle behind Island Packets and IMS type boats alike)
The weight of a boat comes into play here as well. For a given weight, cross section shape and waterline beam there will be a certain depth of the canoe body. The deeper the canoe body, the lower the form stability. With more weight comes a deeper canoe body and a slower motion for the waterline beam. We also discussed the impact of weight distribution above but in general the lower the vertical center of gravity, the more stability. As a result a light narrow boat can have a lot of stability if the center of gravity is very low (modern race boats for example) and wide boats can lack stability (some lightly ballasted IOR type boats or shoal draft traditional cruisers for example).
Lastly there is a tendency to think of beamy boast as having great stability in the inverted position. Today manufacturer’s (Hunter and Island Packet for example) are getting around that issue by increasing the height of the boat’s freeboard and deck structures. This increases the angle of positive stability and makes the boat more unstable in the inverted position by raising the center of gravity higher above the inverted center of buoyancy. Of course high freeboard brings with it a whole range of negatives that are outside the scope of this already too long discussion.
There are a few other factors at play here as well. Research suggests that the it only takes a breaking wave equal to twice the beam of boat to roll a boat over. In theory this makes a narrow boat more prone to being rolled. In reality there is also the problem that a beamy boat is more likely to dip a piece of its deck into the water and “trip” which somewhat offsets the argument about wave height to beam ratios. A good thing often quoted about a very narow boat is that it less likely to be stable upside down but there are so many other factors that beam alone should not be considered.
In the end the best solution of moderation. Boats like the Saga are too narrow to develop enough initial stability. We were able to easily blow them away with my 28 footer. Boats that are too beamy like Island Packets and the like, also give up a lot for their beam. The newer Finot designed Beneteaus seem to have found a way around the stereotypes but the jury is still out a bit on those.
I hope this helps to answer your question.
Respectfully
Jeff
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Curmudgeon at Large- and rhinestone in the rough, sailing my Farr 11.6 on the Chesapeake Bay