S/s = √L/l
where S and s are called corresponding speeds.
Mr. Froude then established a second law of equal importance, relating to that part of the whole resistance due to wave-making experienced by a ship and a model, or by two models when moving at corresponding speeds.
Mr. Froude’s second law is as follows: If a ship and a model are moving at “corresponding speeds,” then the resistances to motion due to wave-making are proportional to the cube of their lengths. To employ the example given above, let the ship be 250 feet long and the model 10 feet long, then, as we have seen, the corresponding speeds are as 5 to 1, since the lengths are as 25 to 1. If, therefore, the ship is made to move at 20 miles an hour, and the model at 4 miles an hour, the resistance experienced by the ship due to wave-making is to that experienced by the model as the cube of 25 is to the cube of 1, or in ratio of 15,625 to 1. In symbols the second law may be expressed thus: Let R be the resistance due to wave-making experienced by the ship, and r that of the model when moving at corresponding speeds, and let L and l be their lengths as before; then—
R / r = L3 / l3
Before these laws could be applied in the design of real ships, it was necessary to make experiments to ascertain the skin friction of different kinds of surfaces when moving through water at various speeds.
Mr. Froude’s experiments on this point were very extensive. For example, he showed that the skin friction of a clean copper surface such as forms the sheathing of a ship may be taken to be about one quarter of a pound per square foot of wetted surface when moving at 600 feet a minute. This is equivalent to saying that a surface of 4 square feet of copper moved through water at the rate of 10 feet a second experiences a resisting force equal to the weight of 1 lb. due entirely to skin friction. Very roughly speaking, this skin resistance increases as the square of the speed.[17] Thus at 20 feet per second the skin friction of a surface of 4 square feet of copper would be 4 lbs., and at 30 feet per second it would be 9 lbs. Any roughness of the copper surface, however, greatly increases the skin friction, and in the case of a ship the accumulation of barnacles on the copper sheathing has an immense effect in lowering the speed of the vessel by increasing the skin friction. Hence the necessity for periodically cleaning the ship’s bottom by scraping off these clinging growths of seaweed and barnacles.
Mr. Froude also made many experiments on surfaces of paraffin wax, because of this material his ship models were made. It may suffice to say that the skin friction in this case, in fresh water, is such that a surface of 6 square feet of paraffin wax, moving at a speed of 400 feet per minute, would experience resistance equal to the weight of 1 lb. There are, however, certain corrections which have to be applied in practice to these rules, depending upon the length of the immersed surface. The mean speed of the water past the model or ship-surface depends on the form of the stream-lines next to it, and it has already been shown that the velocity of the water next to the ship is not the same at all points of the ship-surface. It is greater near the centre than at the ends. Hence the longer the model, the less is the mean resistance per square foot of wetted surface due to skin friction when the model is moved at some constant speed through the water.
The above explanations will, however, be sufficient to enable the reader to understand in a general way the problem to be solved in designing a ship, especially one intended to be moved by steam-power.
If a shipbuilder accepts a contract to build a steamer—say a passenger-steamer for cross-Channel services—he is put under obligation to provide a ship capable of travelling at a stated speed. Thus, for instance, he may undertake to guarantee that the steamer shall be able to do 20 knots in smooth water. In order to fulfil this contract he must be able to ascertain beforehand what engine-power to provide. For, if the engine-power is insufficient, he may fail to carry out his contract, and the ship may be returned on his hands. Or if he goes to the opposite extreme and supplies too large a margin of power, he may lose money on the job, or else he may again violate his contract by providing an engine and boiler too extravagant in fuel.