Feature

Sinking Yachts and Sea Waves

It's not just Australian yacht designers who want to know what makes boats bend

By Professor Vernon Squire and Dr Michael Meylan

How does a floating vessel respond to ocean waves? This is a question which has occupied the minds of many -- not least the designers of the unfortunate oneAustralia yacht which broke up in heavy seas during the America's Cup challenge races recently.

There is a lot of erudite material in the scientific journals concerned with how things that float bob up and down, move to and fro, and tip this way and that. (In technical language we say they heave, roll, pitch, surge, sway and yaw.) But there isn't a great deal on how much they bend.

Having worked with Antarctic sea ice for nearly a couple of decades, and knowing that ice is generally a pretty stiff material when cold but still flexes in waves, it seemed reasonable to apply some of our work to different materials and to shapes other than frigid, unpleasant ice floes. Perhaps we might even find some way of getting into tropical oceanography for, as much as we love the polar seas, they can be a tad trying at -25C in a 50 knot southerly.

Many people were stunned by the television footage of the oneAustralia yacht visibly flexing under the stresses of the high waves through which it was attempting to sail. Combine those stresses with the typical yacht design tensions which run between mast, keel and stays, and something has to give -- in this case with disastrous results for the Australian team.

Practical Applications

But it's not just fast racing yachts which suffer from this. We also know that big ships such as supertankers flex during passage, and that large structures such as oil and gas rigs can stress appreciably in waves. Both can break up if the strain becomes too intense, with devastating consequences. Thus our goal is defensible in its value to Humankind as well as being challenging and fascinating research.

The trouble with challenging and fascinating research involving mathematics is that it can soon become just a wee bit tricky -- formidable, even -- if a few plausible assumptions are not made at the outset. So we start by simplifying. You know the kind of thing: "assume a perfectly spherical cow". The aim is to produce results which are of value in realistic physical settings by retaining enough of the original fabric of the problem being considered. Standard applied mathematics, but the demanding part. Solving the equations is usually relatively straightforward in comparison.

The year has been spent looking at three different parameterisations of a floating vessel in waves, together with the solution of a rather classical problem but with a new slant. Put simply, the classical problem asks how a rigid vertical barrier, of designated length not necessarily equal to the water depth, affects incoming ocean waves. The barrier can be placed at the surface or on the sea floor, or wherever you wish. Our slant is to make the barrier compliant, so that it sways to and fro with the passing swell.

"Where's the application for that?" you say. Flippantly, we could say we are interested in modelling the hydrodynamics of kelp beds, as we can string a cluster of our pliable sheets together, or perhaps jellyfish. In reality though, our interest was in developing theory to model novel devices for the generation of power from waves, the behaviour of risers, piles and other edifices placed on the sea floor, or even new-fangled forms of breakwaters.

Modelling revealed several interesting features. For example, whereas a free sheet effectively reflects negligible wave energy, a sheet pinned at one end can reflect a sizable fraction. Moreover, wild oscillations can be induced in sheets of a particular length and properties, if the waves are such that resonance occurs. Maybe marine zoologists might be interested after all

Ship Shapes Simplified

The trouble with modelling ships and oil rigs mathematically is that they come in all shapes and sizes. Not to be deterred, we have unpretentiously assumed that they don't. We have worked with relatively simple geometries such as thin and thick plates, and most recently with the circular disk, justifying our results on the basis that many of the phenomena we predict will be seen in the sea-keeping motion and contortions of more complicated vessels. The circular disk has proved to be the most interesting (a word mathematicians tend to use when algebra becomes intimidating) and merits some discussion.

It was pretty clear to us from the outset that finding a solution for the three-dimensional problem of a floating circular disk acted upon by a train of long-crested ocean waves was not going to be simple, because the disk is circular and the waves are planar.

This suggests two natural coordinate frames, rectangular and cylindrical. In both, the three axes point in the direction of motion of the wave, along the wave crest and vertically upwards, but because the waves are fundamentally different, so too is the most obvious coordinate frame. (Long-crested waves have long straight crests like sea waves; cylindrical waves are analogous to the waves produced when a stone in thrown into water.)

Fortunately, the circular is important to vibrational engineering, so its flexural modes have been worked through previously and are known. The flexural modes are important as they are the shapes that an object most prefers to take on as it bends. They lead naturally to resonance.

For us then it was simply a matter of finding out how one mode could couple through the water into the other modes. When done, the problem reduces to solving a large system of algebraic equations, something which is easily done with a fairly modest computer.

Our results show that a circular disk can undergo remarkably intricate motion and flexure, especially when the disk is thick and large in diameter, and is compliant enough to bend, without being so supple as to conform perfectly to the shape of the underlying wave. Places occur where deflection is zero for a given disk location. Also of interest is the motion in the water surrounding the disk, where wave focusing can occur at its rear.

Applications of the circular disk theory are numerous, and again one can think of several possibilities in the life sciences as well as in marine engineering. Engineers, for example, have considered arrays of disks as breakwaters. They again offer possibilities for wave-induced power generation, and large floating structures often have a cylindrical footprint, lending themselves perfectly to the type of analysis we have devised.

We may not have made it into nasty, humid, tropical marine science, but Antarctica and its ice-floes was our rightful love anyway and has provided the warmer latitudes -- and the people who sail in them -- with something to think on.

Dr Meylan is a RSNZ/ FRST Postdoctoral Research Fellow in Otago's Department of Mathematics and Statistics.
Professor Squire holds the Chair of Applied Mathematics at the University of Otago.