When I took Hullform from its initial "amateur" format to the first professional version, one of the first items on the list of additions was a capacity to estimate the drag of a designed hull.

I was already familiar with hull drag principles, from preparing a set of lectures on the Science of Sailing, so I knew the issues which had to be addressed. The program had to be able to represent drag which originated from skin friction, from the forcing apart of the volume of water through which the hull moved (form, or profile, drag), and from the generation of waves on the water surface. Of these, the one which gave the schemes I used most problems was wave drag.

Essentials of Wave Drag

There are three main sources of the drag on a boat hull, namely skin friction (due to the roughness of the hull surface), form drag (due to the effort required to force the flow apart, as the hull moves through the water) and wave drag.

The connection between waves and drag is very different from that for other two sources. Skin friction and form drag can both be measured by the loss of energy to turbulence. However, wave drag is due energy which is radiated away, as the waves generated by the hull propagate outward.

Obviously, the lengths and speeds of waves produced by a hull are affected by its length and speed. Wave theory shows us that the length and speed of a wave are closely related; when we add this point into the drag equation, things can become quite messy!

Skipping to the later part of the mess, the basic equation of a surface wave can be written as:

Here, "g" is acceleration due to gravity, and "p" is the ratio of circumference to diameter of a circle (3.14159…). The exact numerical formula is not all that important for now - what is important is that the long waves move faster.

A slow moving boat will therefore sustain short waves, and as it speeds up, the waves will get longer. But there will always be a range of waves generated, with the smaller ones carrying very little energy, and the larger ones able to carry more. To keep the analysis simple, we need to focus on the longest wave produced.

To understand this long wave, we must think of what happens to the water flow around the hull. Up front, the force which the bow applies to the water, to push it out of the way, makes the water "pile up". This forms the bow wave.

At the sides, the flow around the hull accelerates, and so the level drops (due to Bernoulli's Principle - we won't linger on that here).

At the stern, the flow slows again, and the level rises again.

This rise, fall and rise again of the water is the long wave, which is very small at low speeds, but becomes dominant as speed increases.

It has a range of options - it can move forward with the hull, at the hull speed, or move out at an angle, when its speed will be a fraction of the hull speed, and its length a similar fraction of the hull length:

However, only at most one speed and length combination will obey the speed-wavelength law. At low speeds, we do find that somewhere in the range of angles from 0 to 90 degrees, there is one valid combination.

As the speed of the boat increases, this angle reduces to zero. This occurs when the speed of the boat matches the natural speed of the long wave. Beyond this point, the long wave is being dragged at a speed greater than it would naturally have. Many changes occur in the flow to handle this condition, and they all mean large increases in wave drag.

Before we finish, there is one loose end to be tied up. The short waves may not contribute much to overall hull drag, but they combine to form a readily-seen wedge shape in the hull's wave pattern. There is a lot of physics and geometry to this pattern, but it doesn't help us understand wave drag, so we can happily skip it.

Calculating the Drag Force

We can calculate roughly what the wave drag term looks like, with only a little physics and mathematics - and it doesn't take long for the message to become clear.

For simplicity, we consider only the longest wave along the hull. Its height relates to the pressure applied by the bow of the hull, to the water through which it moves. Bernoulli's Law tells us that the pressure - and so the height of the water which forms the wave - depends on the square of the hull speed. As a result, the height of the wave depends on the square of hull speed.

The period the wave takes to pass corresponds closely with the period the hull takes to pass - so varies inversely with hull speed. The speed of particles in a wave crest is determined by the ratio of its size to its period - so by the cube of hull speed. And the energy of these particles depends on the square of their speed …

So far, we've tracked down factors making the drag of the long wave depend on something like the sixth power of hull speed - and we haven't got to the end of it, by any means. Let's not go any further, but simply bear in mind that, like a sixth-power curve, wave drag rises steeply when the long-wavelength term becomes dominant.

Naval architects use schemes based on the formal equivalents of the above analysis, to estimate the drag of their designs. Below is one example, for a heavy-displacement, 10 metre long hull, showing the relationship between drag force and hull speed. (This was calculated using Hullform's "Gerritsma" drag scheme)

The vertical shaded line is drawn at

and illustrates that the steep rise of drag force occurs when hull length and wave length match.

You may also see small oscillations in the curvature of the graph, due to the changing effects of shorter waves as hull speed changes.

Controlling Wave Drag

Before you worry about the wave-induced drag of your design, you should firstly decide whether the hull is operating in the regime where wave drag becomes important. Smaller vessels often are, but large ships never are.

Having decided that wave drag will be an issue, you must minimise the effect of the hull-length wave. This means you must minimise its size. However, your options here may be limited.

The wave is basically a result of the water pushed aside as the hull moves through. For a given hull displacement, there is not much we can do about the amount of water pushed aside. We can, however, have some effect on the amount of the flow distortion which goes into the long wave, and how much goes into shorter waves.

This point is best introduced by considering the waves generated by a barge. When it moves, a large bow wave and stern wave are generated, but little occurs in between. Obviously, a barge is not a low-drag hull form, but it does what we want - if, perhaps, in the wrong way. Short waves are generated bow and stern, but with straight sides along most of the hull, there is little to generate the long wave. In short, for high-speed operations, we need a "full hull".

We express the "full hull" property by the prismatic coefficient, which is the ratio of volume displaced to the product of waterline length and maximum cross-sectional area. A craft like a barge has a prismatic coefficient close to 1.0, while one with very slender ends can have a value of about 0.5 or less.

The two simple hull-like forms below show what prismatic coefficient means in terms of hull shape. The top one has a coefficient of 0.67 (unusually large), produced by maintaining the draft of the hull from stem to stern, with very rounded plan form. The lower form has a more common value of 0.55, and shows more tapered stem and stern, with greatest draft amidships.

By way of illustration, the design used for the previous drag curve had a fairly large prismatic coefficient, of 0.62. A modification with thinner ends giving a prismatic coefficient of 0.53, showed lower drag below the limiting hull speed, but larger drag above:

The benefit of the Gerritsma drag calculation scheme is that it estimates the best prismatic coefficient and the best centre of mass position, for any operating speed. Two crucial terms needed to get the best performance out of any displacement hull become available "up front".

Putting It All to Work

For a 10-metre hull, the Gerritsma scheme tells us that the best prismatic coefficient and centre of mass positions vary according to:

Because of the relation between wave length and wave speed, lengths scale with the square of the speed - for example, you can double all speeds for a 40 metre hull, halve them for a 2.5 metre hull. There's little in the centre of mass variation, but from 5 to 8 knots we go from a very skinny-ended hull, to a pretty fat one. And as you can see by the graph above, there are real drag and speed benefits from getting the prismatic coefficient right, when the hull is operating just below its wave-limiting speed.

Shallow Water Effects

A point of particular interest in yacht racing is that the speed of a wave decreases in shallow water. This means that the limiting hull speed is less.

There are two effects. The obvious one is that there is more drag at speeds just below the normal limiting hull speed. Less obviously, light displacement yachts can reach planing conditions at a lower speed. This means that the slower increase of drag with speed in the planing mode starts earlier.

The condition where this occurs corresponds to water depth about one sixth of hull length - for example, one metre for a large, six-metre dinghy. While this is hardly likely to be a depth of concern for offshore sailors, those using inland waters will very often encounter such depths.