Crunching Numbers: Displacement/Length Ratio

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Concordia yawl under sail

We’ve already discussed the basic concept of a boat’s displacement or weight. We’ve also discussed waterline length and how it relates to a boat’s theoretical maximum hull speed. Considered separately, however, length and displacement yield only a general notion of what a boat is like. If you blend the two values you can make a much more nuanced evaluation. The displacement/length ratio (or D/L ratio) is the tool yacht designers have created to do this.

To find a boat’s D/L ratio, you first calculate its displacement in long tons (DLT), with 1 long ton equaling 2,240 pounds. Then take the boat’s load waterline length (LWL), multiply it by 0.01, and cube the result. Finally, take this result and divide it into DLT. The complete formula is as follows: D/L = DLT ÷ (0.01 x LWL)³.

As an example, to find the D/L ratio of a 12,000-pound boat with a load waterline length of 28 feet, you first divide 12,000 by 2,240 to find the boat’s displacement in long tons: 12,000 ÷ 2,240 = 5.36 long tons. Then multiply 0.01 by 28 (0.01 x 28 = 0.28) and cube the result (0.28³ = 0.022). Then divide the DLT by this number to find the D/L ratio: 5.36 ÷ 0.022 = 243.6.

A boat with a D/L ratio below 100 is considered ultralight; a D/L value between 100 and 200 is light; 200 to 300 is moderate; 300 to 400 is heavy; and over 400, by modern standards, is very heavy. For a boat of a given length the lower its D/L ratio, the less power it takes to drive the boat to its nominal hull speed and the more likely it is the boat can exceed its hull speed. The 12,000-pound boat in our example above, with its D/L ratio of 244, falls almost exactly in the middle of the range; it needs a moderate amount of power to reach its nominal hull speed of 7.09 knots (1.34 x √28 = 7.09) and stands a reasonable chance of exceeding that speed in some situations.

The higher a boat’s D/L ratio, the more easily it will carry a load and the more comfortable its motion will be. Depending on the sort of cruising you do, these factors may be more important than how fast you are going. Boats with moderate characteristics are generally best suited for cruising, but a “moderate” coastal boat should be lighter than a “moderate” offshore boat. Coastal cruisers carry less gear and supplies and normally sail shorter distances in more protected water. They also benefit more from incremental increases in speed, as they are more likely to sail on a tight schedule and normally seek a safe harbor every night. For this type of sailing I recommend a D/L range of 150 to 300.

Conversely, bluewater cruisers carry more gear and supplies, are sometimes subject to extreme motion in open water, and are less likely to be sailing on a tight schedule. For this type of sailing I recommend D/L ratios between 250 and 400. You can, of course, fiddle these ranges upward or downward according to your own preferences.

When using D/L ratios to evaluate boats, you need to bear in mind that the ratio varies a great deal depending on the displacement value used to calculate it. This is why you need a reasonably realistic displacement number to work with. As we discussed in the displacement post, to get a realistic number you usually need to correct the displacement figure published by the boat’s builder upwards by quite a bit to account for the load the boat normally carries when sailing.

You’ll note that our hypothetical 12,000-pound boat, with its moderate D/L ratio of 244, quickly becomes less moderate as we load it for a cruise. Add Nigel Calder’s minimum recommended displacement correction for light coastal cruising (12,000 + 2,500 = 14,500 lbs.) and the boat’s D/L ratio becomes 294, which nearly qualifies it as a heavy boat. Up the ante even more by loading the boat for heavy coastal or moderate bluewater use (12,000 + 3,750 = 15,750 lbs.) and it moves well into the heavy range with a D/L ratio of 320.

These are not just theoretical increases. Many is the sailor (myself included) who has purchased an empty boat and has reveled in its sprightly performance, only to be demoralized upon discovering how much less sprightly it is when loaded for a serious cruise. The lighter a boat is to begin with (particularly if it is a catamaran), the more dramatic (and demoralizing) this transformation will seem. Most cruisers soon forget how much better their boat sailed before they loaded it with stuff. But if you are devoted to performance you should be draconian when loading your boat and should closely monitor your D/L ratio.

Once you have figured out a boat’s D/L ratio, you can also use that figure to make a more accurate estimate of the boat’s maximum hull speed. I’ll show you how in the next Crunching Numbers post.

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4 Responses
  1. Michael Ossipoff

    I’ve read that the bow-wave is generated when water is pushed up by the advancing hull’s cross-section. Then why not rate a sailboat’s estimated speed-capability by sail-area (SA) divided by its underwater hull-cross-section?

    A hull’s average underwater hull-cross-section is proportional to W/L, where W = weight, and L – LWL.

    …and a hull’s max underwater cross-section should be reasonably or roughly proportional to its average underwater cross-section.

    So then why not rate a sailboat’s estimated speed-capability by:

    SA * L / W

    Regarding the explanation that the stern is about to fall into the trough of the bow-wave, making the boat have to sail uphill–Isn’t an object’s weight what makes it hard to drag it uphill? And isn’t the hull’s LWL a quantity that postpones that problem, lessens it for a while?

    Then that, too suggests:

    SA * L / W.

    Of course, as I mentioned, the hull’s underwater cross-section is what is pushing-up and generating the bow-wave that’s causing that uphill problem, and influencing the height of that bow-wave and the depth of its trough.

    So this is just based on what I’ve read about the cause of wave-making drag.

    The above-stated formula agrees well with the order of the various sailboat racing-class’s compensation-ratings for multi-class races.

  2. Michael Ossipoff

    I should add that SA * L / W is intended as a rough measure of a boat’s ability to reach and exceed its hull-speed.

    But if one wanted something that tries to roughly suggest a plausible speed for a displacement-boat, then SA * L / W could be multiplied by the square-root of L, giving:

    SA * L^(3/2) / W

    …where L^(3/2) means the 3/2 power of L.

  3. Michael Ossipoff

    Charles–

    I want to reply to your answer to my comment about the need for sail-area in a speed-comparison-rating formula, but now I can’t find that article & comment-space.

    So, is it alright if I reply here?

    Here is what I want to say:

    First, thank you for your reply.

    Yes, after I posted that, I realized that that article was just referring to the *hull’s* contribution to top-speed, so it wouldn’t be reasonable to expect it to mention sail-area.
    .
    But of course, if you want SLR, the thing that you multiply sqrt(L) by to get a top speed estimate, as a rating-number for comparing different boat’s typical top-speeds, then you need a formula that includes sail-area. Alright, I understand that a power-function of Displacement/L^3 is the way Dave suggested for the hull’s role in the formula. I was just saying that a top-speed estimation rating formula must take sail-area into account too. The part about the hull is only half of the formula.
    .
    ….but yes, it’s the hard part.
    .
    Dave & I were doing the same thing: Representing the hull’s contribution to the speed-rating formula as a power-function of a hull-parameter.
    .
    Dave used a power-function of Displacement/L^3. I used the hull’s maximum submerged cross-section. (…which I approximated by the hull’s *average* submerged cross-section, because that can be determined from the available specifications. I assume that the max cross-section is proportional to the av cross-section.
    .
    The average cross-section is: Displacement/L. Dividing the Sail-Area by that, gives:
    .
    SA*L/W
    .
    (I use Weight for displacement, because W is what is given in the specifications, and we don’t know if the boat will be used in salt or fresh water, and their density-difference is too small to matter, given the other inevitable errors in speed-ratings.)
    .
    So, where Dave uses a power-function of Displacement/L^3, I use a power-function of SA*L/W.
    .
    (SA*L/W)^p
    .
    When I posted last year, I used p = 1.
    .
    Now I prefer p = .5
    .
    That latter value seems more plausible, for reasons that I state below.
    .
    But, ideally, of course, a value for p should be found that gives least-squares error, over a variety of boat DPNs.
    .
    More about that in a minute. But, for now, which hull-parameter is better? I’ve only looked at the match of Sqrt(SA*L/W) to the boats’ DPNs. I don’t know how good the result is with Displacement/L^3)^p.
    .
    If the latter gives a better match to the actual DPNs, then I can’t argue with that.
    .
    The disadvantage of D/(L^3) is that of course there are drastically-different hulls with the same value of that parameter. Maybe the boat is very wide-beamed, but rapidly comes to a sharp poingt at the ends. Or maybe its max cross-section is less, but it’s a vertical-fronted rectangular barge.
    .
    But, in that parameter’s defense, whichever of those two ways the hull is getting a high displacement/(L^3), either way it’s going to increase the wave-drag.
    .
    But does it make a great difference which it is? I don’t know.
    .
    The disadvantage of using max submerged hull cross-section (approximated by average submerged hull cross-section) is that it doesn’t tell whether the hull is sharp, coming quickly to a point at the ends, or whether, instead, it’s a straight cylinder with a flat front that’s perpendicular to the hull’s longitudinal-axis.
    .
    But not many boats are like that latter description.
    .
    The advantage of SA*L/W is that, though some boats have wider beam for their length, they all try to have as good a form as possible, given that length/beam ratio. And so, likely there isn’t much variation, & so maybe the sharpness & streamlinedness of the hull can be disregarded, since it varies little.
    .
    …unless the boat is a pram or a barge-scow. That would be a problem that would make a boat look faster than it is, in displacement-sailing. On the bright side, maybe such a boat is intended for planning anyway, and so these formulas aren’t even applicable to it.
    .
    Would the (Displacement/L^3)^p do a better job with the barge-scow, and thereby even do a better job overall? I don’t know.
    .
    So I must admit that I don’t know which hull-parameter is better.
    .
    Anyway, here’s my description of my suggestion:
    .
    I’ve read that submerged hull cross-section is a causative-factor for wave-drag, because pushing that cross-section through the water is what raises the waves and incurs pressure-resistance.
    .
    As for the uphill-sailing model, the pressure under a wave at some depth is proportional to the wave’s height above that depth, & that dynamic pressure is proportional to speed squared, but not area of the boat cross-section. The height of that bow-wave above the stern-trough, divided by the length of the boat is the sine of the angle of the slope that the boat is climbing. Multiplying that by the boat’s weight gives the force required for that climb.
    .
    So, because the average submerged cross-section is proportional to the boat’s weight/length, and because the dynamic-pressure’s rearward component is acting on the boat’s submerged maximum cross-section (approximated by its average cross-section), then (Sail-area & Length/ Weight) is relevant either way you look at it.
    .
    …as pressure-drag, or as sine of hill-steepness X Weight.
    .
    That SA*L/W being a ratio of two areas (sail & average submerged hull-cross-section), I’ll abbreviate it as “A”, because it’s important & I write it a lot.
    .
    Because A has everything to do with the speed of a submerged object against pressure-drag, then it seems very relevant to how well a boat can achieve or exceed the 1.34 SLR.
    .
    So, last year, I proposed assuming that SLR is proportional to A.
    .
    Now I feel that it’s probably better to assume that SLR is proportional to sqrt(A), because, for a submerged object, speed is proportional to sqrt(driving-force/cross-sectional-area). …suggesting that sqrt(A) is likely more relevant to the boat’s SLR than A would be.
    .
    And, in fact, the assumption that SLR is proportional to sqrt(A) agrees very well, remarkably well, with the DPN ratings of various boats. Alright, the agreement isn’t perfect, just astonishingly good.
    .
    And how good agreement could one expect anyway?
    .
    The faster boat, & the bigger ones, are crewed by more expert sailors.
    .
    …and the ones with crews of 2 or 3 can be more efficiently-sailed than the ones crewed by only 1.
    .
    …and different boats are more predominantly raced in different locations, where the different wind-conditions would favor different boats. …i.e. San Francisco Bay vs Southern California.
    .
    …and the smaller the boat, the more likely that its sailor is younger, & therefore lighter.
    .
    All of those things prevent any formula from matching the ratios of the boats’ DPNs very well.
    .
    For all those reasons, perfect agreement with the DPNs would be impossible. The agreement between SLR proportional to sqrt(A), and the DPNs, then, is about as good as could be expected for any speed-comparison rating-formula.
    .
    Still, there are other things to try, and one should always try everything, in these matters.
    .
    …such as, for the general formula A^p, find the value for p that best agrees with one or more comparisons of the DPNs of two boats. Or, better yet, find the value of p that gives the lowest least-squares sum for the error of A^p …its least-squares error with respect to the DNPs of many boats.
    .
    …preferably doing that least-squares optimization separately for various different categories of sailboats, such as single-hander boats that aren’t primarily for racing.
    .
    …or, instead of just A^p, use A^p + K, where K could be positive or negative. Then the least-squares optimization is for two variables instead of just one.
    .
    …or, instead of SLR proportional to A^p + K, maybe SLR equal to K1*A^p + K2. (K2 positive or negative). Then the least-squares optimization is for 3 variables. The other way, about proportionality, & optimizing only 2 variables would surely be much easier…but not nearly as easy as just SA*L/W, with just one variable to determine.
    .
    Another thing that I didn’t check was how well just plain A, to the 1st power (as opposed to sqrt(A)), would do, with respect to the DPNs. I doubt very much that it would do as well as sqrt(A), but everything should be tried.
    .
    Sqrt(A) was a simple, natural, obvious thing to try, and it proved remarkably consistent with the DPNs.
    .
    But completeness compels us to also check the least-squares optimization of the value of p for A^p.
    .
    ……& those other functions, with p, k1 & maybe k2 least-squares optimized to for best agreement with the sailboats’ DPNs.
    .
    Michael Ossipoff

  4. Michael Ossipoff

    Sorry, I didn’t intend a follow-up comment, but I should emphasize that, when using the assumption that SLR is proportional to sqrt(SA*L/W), then of course the whole speed-comparison-rating formula would be:

    sqrt(SA*L/w) * sqrt(L)

    = sqrt(SA*(L^2)/w)

    = L * sqrt(SA./W)

    =

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