Midship Area and Prismatic Coefficients
Today we take up two coefficients: Midship Area Coefficient, which we’ll label Cmc, and Prismatic Coefficient, labeled Cp. See the diagrams attached.
Cmc is the ratio of the area of largest midship section of the submerged portion of the hull to the area of the box that bounds it. The dimensions of the box are the Beam at the Waterline at the largest section, Bwl, and the Draft at that section, Do. The largest section area may not be at the exact mid-length of the hull, or at the maximum beam or draft of the hull. See the first figure below. The easiest way to determine the maximum size midship section size is to plot the Sectional Area Curve, and the maximum area will be at the peak of the curve. Unfortunately, I don’t have an example of my own sectional area curve to show, but you can see one in Principles of Yacht Design (Larsson & Eliasson, 3rd ed.) Figure 4.4, pg. 35.
At the fore and aft location of the peak of the sectional area curve, measure the Bwl and the Do. The box bounding area for the largest midship section, therefore, is Bwl x Do. Midship Section Coefficient, Cmc, then is the Actual Midship Section Area divided by the Bounding Box area, as shown in the diagram below. The concept and the equation are shown in the second figure below.
Cmc is used as a gauge to judge the fineness of the midship section. It is useful for comparison between different designs, or to judge how a design is being developed. Say, for example, that you are designing a new hull, and you want the displacement to be within a certain range. On your first pass at developing the lines, you see that the displacement is too large. Maybe the turn of the bilge is a little too sharp. So then you develop a second set of lines and find that the displacement is too low; maybe now the turn of the bilge is not sharp enough. You can compare the shapes of the midship sections, because they are directly related to displacement, and compare the Cmc ratios of each. Analyzing these features will lead you to where you want the final Cmc to be (somewhere in between), and the geometry of the sectional area will show you where to make adjustments in the shape (again, somewhere in between). Cmc is used as an analysis tool, therefore, for developing a new design or comparing two different designs.
This leads us to Prismatic Coefficient, Cp. Cp is like Cmc, except that we take the calculation one step further with a third dimension--length. Cp is a comparison of volumes, not areas. Cp is the ratio of the volume of displacement of the hull divided by the volume of a prism which is the maximum section area multiplied by the Length on the Waterline, Lwl. See the third figure below. The figure shows the equation. Obviously, the prism volume is always larger than the actual displaced volume, so Cp is always going to be less than 1.0, by definition.
Looking at the equation, we see an interesting thing happen. In the denominator, we see Lwl x midship area. We know from the Cmc above, that if we turn its equation around, Midship Area = Bwl x Do x Cmc. So if we substitute these parameters into the Cp equation, we get that Cp = Vol/(Lwl x Bwl x Do x Cmc). But we know the part Vol/(Lwl x Bwl x Do) is equal to Block Coefficient, Cb. At least it will equal it perfectly if Do is also the maximum draft, D. If Do is not the maximum draft, then the calculation will be a little off. But basically, this all reduces to the fact that Cp is the ratio between Cb and Cmc, as shown in the figure. In words, this is: “Prismatic coefficient is the Block Coefficient divided by the Midship Area Coefficient.”
It turns out, after some 150 years or so of analysis, that performance is closely related to Cp. That is, there is an optimum range of Cp for various speeds of the boat traveling through the water. You can see a table of speed/length ratios versus optimum Cp in Skene’s Elements of Yacht Design (by Francis Kinney, 5th ed.) pg.284, which I repeat below:
Speed/Length ratio Cp
1.0 0.52
1.1 0.54
1.2 0.58
1.3 0.62
1.4 0.64
1.5 0.66
1.6 0.68
1.7 0.69
1.8 0.69
1.9 0.70
2.0 0.70
Larsson/Eliasson shows a similar range in their book on page 83, Fig. 5.22, in which they plot optimum Cp against Froude Number. Froude Number is very similar to Speed/Length ratio, and if you convert Froude Number to Speed/Length ratio, you will find that Larsson/Eliasson’s curve is a bit lower than Skene’s curve tabulated above. As is true with many things, therefore, there is some wishy-washiness in the guidelines. Nothing is hard and fast.
Nevertheless, what this tells you is that most displacement boats travel most of the time at Speed/Length ratios of at least 1.0 and slightly above, so you need enough volume to support the hull at those speeds. If volume is either too much or too low—that is if Cp is too big or too small—your hull drag is going to go up. Either the boat is going to have to push too much water out of the way (Cp too big) or it is going to sink into its own waves (Cp too small).
Usually, in sailboat design, the keel and it’s draft is left out of the calculation of volume. This is because, as in the Cb calculation, the keel tends to make Cp less sensitive. So we ignore the keel for calculation of Cp. In powerboat design, we do not do this. If we are designing a trawler or lobster boat, for example, we keep the keel in the calculation because it is a major portion of the hull.
Interestingly, I typically design my sailboats with a Cp of about 0.60. I did the same with my Moloka’i Strait motoryachts. This is just below hull speed, Speed/Length ratio = 1.34. (We can take that up in another post, if you wish). You can also see that approaching planing speeds (Speed/Length ratio => 2.50), Cp reaches 0.70. This goes along with very long and narrow hulls—that is, being still in displacement mode at S/L = 2.0, you need a high Cp. This is why catamarans and trimarans (which have long narrow hulls) have very high Cp ratios.
That’s a lot of material to digest, so I’ll leave it there and wait for questions.
Eric
So that's where tides come from 
