# The Prismatic Coefficient

## Units

Unfortunately the data displayed in the boat selection dialog box is a mixture of imperial (ft) and metric (m and kg) units, thus being confusing to everybody. For the majority who are used to the metric system, all-metric measurements would be desirable, while virtual sailors from the USA might want to see the old imperial units. It would be easy to do this with a simple ‘units mode’ switch in one of VS’s preference dialogs. Nautical miles (a minute of arc along a latitude) and knots (nm/h) are exceptions that are universally used and understood.

In what follows, I shall use metric measurements since (a) everyone outside the USA uses them and (b) they are a consistent system for calculations.

## The Prismatic Coefficient

Each floating ship displaces its own weight of water when at rest. In other words the weight of water equal to the volume of its submerged hull is the same as the weight of the vessel. So if we know the weight (mass M) of the vessel from the cfg file we know what the underwater volume (call it V) of its hull should be. Roughly 1 kg of water = 1 litre = 10-3 cubic metres, though the slightly greater density of seawater and its variation with temperature and salinity cause variations of a few %.

A VS cfg file contains two other relevant parameters: the length of the vessel (L) and its cross-sectional area (A). The former may be the length overall (end to end) but generally will not be too much different from the waterline length. The area is presumably the maximum area of any cross-section, or perhaps the cross-sectional area amidships (halfway between stem and stern). We expect that LxA should be bigger than V, but by how much? That is where the prismatic coefficient (call it k) comes in; we define k = V/(LxA).

Suppose a vessel has the same underwater cross-section along its entire length, perhaps like a rectangular punt or a raft. Then V=LxA, and k=1. The underwater shape is mathematically classified as a ‘prism’, hence the name prismatic coefficient. Prismatic coefficients larger than 1 are not possible.

Suppose the vessel is much finer (more ‘pointy’) and is made up of two pyramids butted up. The volume of the two pyramids is LxA/3, so k=1/3=0.33. A Hobie Cat hull approaches this shape underwater. You can’t get lower values of k without substantial concave sections.

Real vessels are in between. Consider a vessel whose underwater hull is parallel (a prism) for its middle half, and has two pyramids stuck on the two outer quarters. A bit of algebra gives k=2/3=0.67.  Actually, most real displacement sailboats have k between 0.49 (a bit fine, will develop a large quarter wave) and 0.55 (probably slow). Large ships usually have k between 0.52 and 0.62; planing powerboats might go up to 0.70. So we have established that a normal vessel probably has a prismatic coefficient of about 0.5-0.7, could be up to 1.0 or down to 0.4 but these are exceedingly rare.

How do VS vessels match up? Look at the survey report for a real shock! These are not real vessels or the data is very wrong.

## FAQ

### Does the prismatic coefficient vary much?

No. Most ships have pretty standard hull shapes. Long in the direction they want to go, and about twice as wide as deep, disregarding little bits sticking out like keels and rudders. Their prismatic coefficients are stock-standard. Extreme prismatic coefficients generally only occur in very small vessels. Punts and rafts at one end of the scale, and canoes and racing catamarans at the other. Even a half-sphere vessel (like an Irish coracle) has k=0.67.

The displacement principle continues to apply to a submerged vessel (submarine or submersible) if they are stopped or cruising slowly at stable depth.

### But small boats don’t seem to have much in the water?

While most vessels rely on displacement to support their weight even when they are moving, some derive lift (just like an aeroplane) from their speed and the shape of their hulls (usually flat or a flattish Vee from midships aft). Examples are racing dinghies, windsurfers, runabouts, and at an extreme hydrofoil boats that lift themselves right out of the water on underwater wings. When any boat supports itself in this way, of course its underwater volume will be much reduced, and the prismatic coefficient becomes less relevant. Without huge power, this is restricted to small vessels.  A hovercraft and a seaplane are special cases too complex to discuss here; they have air lift as well.

### Doesn’t this change with twin hulls? Or three?

Slightly. The hulls still have to support the weight of the vessel and have similar shapes. However twin-hull boats trade off more wetted area (and more frictional drag) in their two hulls for very fine and thin hulls with low wave-making resistance, usually giving them lowish but not abnormal k. Here is an Incat Tasmanian-built wave-piercing catamaran ferry on speed trials past my home. We export them all over the world from Japan and China to the English Channel and the Baltic Sea. One of them holds the Blue Riband for the fastest crossing of the North Atlantic by a commercial passenger-carrying vessel.