Beams  and  Ships

If you look at a Viking ship, or the Great Britain, or even a small boat, you can see that vessels often comprise a shell, stiffened by frames and stringers, with beams at intervals along the length, provide greater strength. The whole thing is not unlike the fuselage of a wooden glider. The stiffening is required because the shell is thin compared with its diameter, a topic discussed in the page about tubes. An exception is the  dug-out boat, made from a single log. The dug-out is at one end of the spectrum of compromises between simple construction and efficient disposition of material.

A large ship steaming through calm waters is a magnificent spectacle, and she looks impregnable, but anyone who has seen seas reaching over ten metres high will have had other thoughts about this. Joseph Conrad's book, "Typhoon", gives a long description of the struggle of ship and men to survive in a situation where the future is the nest few seconds. J. M. W. Turner, in "Snow Storm", depicted a similar scene including the bending of the mast by the wind.

What forces does a ship have to bear? She has to retain her shape, strength and stability for all legal loadings, with a safety factor. The real test comes from the sea. Waves can come from any direction, because they are moving across the sea, and the ship is moving too, usually in some other direction, and the arrival of these waves can cause the ship to pitch, roll and yaw. Any combination of these motions is possible, and all occur, often in unpleasant ways. The power of the water is clearly signified by the thudding, pounding, hammering and vibration which runs through the ship in heavy seas.

The picture below shows some "random" waves running past a rectangle which represents a ship. Relative to the ship, which may be moving quickly, the waves may look quite different from the view seen by a stationary observer. Waves may look relatively innocuous, but variable forces can create the insidious effect known as metal fatigue. Over a period of time, changing forces, individually too weak to damage a structure, can slowly build up changes in the internal structure of the metal, until eventually an extra large input of force is enough to take the material over the edge, and it cracks. The ship that fails is not the same ship that left the yard - the metal has changed. What has happened is that the material has changed so much that its yield point and breaking stress are much less than they were when it was new.

It was a crack that did for the Comet 1 airliner, starting at a corner of a square hole, and propagating with tremendous speed, powered by the excess pressure in the fuselage. Ships too, have suffered cracks which have started at the corners of hatches, and have travelled slowly to the edges of the decks, superstructure and even down the sides. Luckily, if a crack meets a stiffening member, it may get no further, and indeed, large structures are often designed to stop the propagation of cracks. So headlines such as "Hairline cracks found in aircraft" may not signify anything as dramatic as they suggest.

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These waves are relatively innocuous, but look at the ones in the next pictures. Here we see waves that are quite long relative to the ship. When the amplitude is large, and the wavelength is similar to the length of a ship, the support of the ship by the sea can be limited to relatively small areas, either amidships, or near the bow and the stern. In extreme cases, if the ship is not strong enough, she may break her back.

One might ask why all ships are not designed to withstand the largest possible waves. This question will be addressed in the next paragraph.

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Addition of Many Waves

Each of the next three pictures shows the sum of ten sine waves with randomly chosen amplitudes and wavelengths.

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The variable amplitudes are the source of the "every seventh wave" idea. The sea is criss-crossed by waves going in all directions, with many different heights and wavelengths. The result is usually an apparently patternless motion. The chance of three waves being in phase at a given point at a given time is clearly less than for two. The chance of four adding up is less than for three, and so on. Seven is just a convenient way of saying "not very often". If these waves had been continued long enough,  much greater amplitudes would have been seen, and eventually one where all ten waves more or less add up.

The fourth picture above is the spectrum of wave heights for 500000 waves. To make this picture, ten sine waves were added, all with the same amplitude.  The wavelengths were set at successive multiples of the golden ratio (1.618033989 . . . .), to make all the ratios irrational. A peak was defined as the highest point between two successive zeroes. The width of the graph runs from zero to the maximum possible height, obtained when all ten waves add together. The height of the graph corresponds to 1275 events  in a  bin. The graph shows that the chance of even seven of the ten waves adding up is very small.

Click here to download or run in place a program which adds six waves of equal amplitude. The wavelengths in pixels are 523, 521, 313, 311, 241 and 239, which are all prime numbers. How long will it take for the pattern to repeat? How long will you have to wait to see the six waves add up in phase and fill the green box.

This illustrates the difficulty of building an absolutely unsinkable ship of any size. The larger the wave, the more unlikely it is and the more it costs to design for it. The designer has to estimate the probability of a ship being struck during its useful life by a wave of a given size.

Given the number of ships at sea, it might be impossibly expensive to design so that no ship is ever overwhelmed. The same idea affects defences against floods - the higher the water level, the less often it happens, and the more expensive it is to resist.