resistance does not increase regularly as the speed increases, but the rate of increase is much more rapid at high speeds than at low speeds. Thus to increase the speed from 7 to 8 knots an extra resistance of 1500 lbs. has to be overcome, whereas to increase the speed from 11 to 12 knots an extra resistance of 6000 lbs. has to be overcome, or four times as much for an increment of 1 knot.

This agrees with our experience. We know how much more difficult it is to increase the speed of a ship by a knot, say, near the top speed than at the lower speeds. Fig. 205 shows the curve of I.H.P. on base of speed of H.M.S. Drake, and the following

shows in tabular form the increase of power necessary for each 2 knots from 10 to 24 knots::

I.H.P..

1,950 3,200 4,800 7,000 10,000 14,800 21,900 31,000

Additional I.H.P. necessary for each increment 1,250 1,600 2,200 3,000 4,800 7,100 9,100 of 2 knots

To increase the speed from 22 to 24 knots requires as much power as is sufficient to drive the ship 17 knots, and to increase the speed from 20 to 24 knots means more than doubling the

horse-power. This great increase of power necessary for high speeds is due to the great increase of resistance.

The rate at which resistance increases as speed increases is therefore a matter of great importance. Mr. Froude found in the Greyhound that up to 8 knots the resistance was varying as the

square of the speed. That is to say, if R, is the resistance at speed V1 and R2 is the resistance at speed V2, then

Mr. Froude found, as the curve in Fig. 204 indicates, that the

resistance varied much faster than the square of the speed at the higher speeds. Indeed, at 12 knots it was varying as the fourth power of the speed, a very high rate of increase. In the case of the Drake the resistance is varying nearly as the third power of the speed between 23 and 24 knots.

Effective Horse-power.—If we know the resistance of a ship which is being towed at any given speed, we can determine the horse-power that is being transmitted through the tow-rope to overcome this resistance. This is the effective horse-power, sometimes called the tow-rope horse-power. This horse-power is a

very different thing from the power exerted by the vessel's own engines or the indicated horse-power (I.H.P.).

In any general case, if R is the resistance in pounds, V the speed in knots (1 knot is a speed of 6080 feet per hour), then

This is the effective horse-power (E.H.P.). Mr. Froude was thus able at once to turn the resistance of the Greyhound at any speed into E.H.P. He found a striking difference between the E.H.P. thus obtained and the I.H.P. which had to be exerted by the vessel's own engines in order to get similar speeds. Thus at 10 knots the E.H.P. worked out to 380, and for this speed the I.H.P. necessary was 786, giving a ratio of E.H.P.I.H.P. of 388 0:42. This was a most important result, showing that of the power exerted at the vessel's own engines the large proportion of 58 per cent. was wasted so far as the ship was concerned. Mr. Froude, on arriving at these results, was led to make further investigations in order to look into the cause of this great loss of power.

786

=

The usual value of this ratio in vessels of the Royal Navy is from 45 to 50 per cent., rising to 55 per cent., or higher in some

The work done by a force (like the strain in the tow-rope), acting through a certain distance, is given by the product of the force and the distance through which it acts. This is independent of the time taken. The power exerted takes into account the time in which the work is done. The unit of power used is the horse-power, which is defined as 33,000 foot lbs. of work performed in one minute.

few cases. This ratio is termed the propulsive coefficient, being the ratio of the horse-power usefully employed to the horse-power actually exerted.

We have been considering above the total resistance experienced by a ship on being towed through water, it is necessary now to inquire how this total resistance is made up. It may be divided into four parts, viz.

1. Resistance due to the friction of water on the surface.

2. Resistance due to the formation of eddies.

3. Resistance due to the formation of waves on the surface.
4. Resistance of the air.

1. Frictional Resistance. This can be directly calculated from data obtained from a series of experiments made by Mr. Froude on boards coated with various surfaces. For an ordinary smooth surface like a vessel coated with paint the resistance in pounds is given by—

R = f.S.V183

where S is wetted surface in square feet;

V is speed in knots;

f is a coefficient.

This coefficient varies according to the length of the surface, being greater for short than for long surfaces. For ships its value does not vary much from 0·009. For short models such as are used in the experimental tank the value of ƒ is greater. We notice that this resistance for a smooth surface varies at a rather lower rate than the square, viz. 183. If, however, the surface is rough, like sand, the coefficient ƒ is twice as great, and the power of the speed rises from 1.83 to 2.0. This illustrates the fact that in order to keep the resistance as low as possible, and so economize horsepower, and therefore coals, it is necessary to keep the bottom clean by periodical docking. On this account, also, ships which are employed on distant service, with the probability of remaining at sea for long periods, are sheathed with copper to prevent fouling.

The frictional resistance is of importance at all speeds, but at low speeds it accounts for the bulk of the resistance. For a torpedo-boat destroyer, which has an abnormally wide range of speed, at 12 knots the frictional resistance is 80 per cent. of the total; at 16 knots, 70 per cent.; at 20 knots, 50 per cent.; and at 30 knots, 45 per cent.

2. Eddy Resistance.-This is due to the eddies formed behind

a blunt ending to the underwater body. Ships built as formerly, with very full sterns and thick sternposts, experienced this resistance to a large extent, but in modern ships of finer form it does not exist to any appreciable degree. Every care is taken to avoid any abrupt terminations which might cause eddy making. One instance of this is seen in the shaft brackets; the section of the arms is made as in Fig. 82, taken to a small radius at the after end, so that no eddies are caused at the rear.

3. Wave Resistance.-It is this form of resistance which becomes of the greatest importance at high speeds, and it is because of the rapid growth of this resistance that it becomes increasingly difficult to obtain these high speeds in full-sized ships.

When a ship is towed through water there are two separate and distinct series of waves brought into existence, viz. those formed at the bow and those formed at the stern. Each of these series consists of (a) a series that diverge with their crests sloping aft, and (b) a series of transverse waves whose crests are nearly perpendicular to the middle line of the ship.

The diverging waves both at the bow and the stern at once pass away from the ship. The transverse waves of the bow series are of the most importance, and the interference of these waves with the corresponding waves at the stern causes considerable variation of the resistance. If a crest of the bow wave series coincides with a crest of the stern wave series there is an increase of wave resistance. A decrease is found to result if a crest of the bow wave series coincides with a trough of the stern series.

When a ship maintains a steady speed, say 15 knots, the accompanying series of transverse waves also has a speed of 15 knots. Such a series has a definite length, viz. 126 ft. from crest to crest, or trough to trough. At 10 knots the length would be 56 ft. It is thus possible to make an estimate of the speed at which a ship is travelling by observing the length from crest to crest of the wave along the side of the vessel. (If V be the speed in knots and L the length of wave in feet, then V = 1·33√L.) As speeds therefore increase, the accompanying wave system gets longer, increasing in length as the square of the speed. In the case of small vessels, like destroyers, travelling at the high speed of 30 knots, say, a wave is created longer than the ship, and she lies on the back slope of a wave of her own creation.

We thus see that as speeds increase, the length and height of the waves formed must also increase very rapidly, and consequently