CHAPTER II

.

WAVES AND RIPPLES MADE BY SHIPS.

It is impossible for the most careless spectator to look at a steam-vessel making her way along a lake, a boy’s boat skimming across a pond, or even a duck paddling on a stream, without noticing that the moving body is accompanied in all cases by a trail of waves or ripples, which diverge from it and extend behind. In the case of a steamer there is an additional irregular wave-motion of the water caused by the paddle-wheels or screw, which churn it up, and leave a line of rough water in the steamer’s wake. This, however, is not included in the true ship-wave effect now to be discussed. We can best observe the proper ship-wave disturbance of the water in the case of a yacht running freely before the wind when the sea is fairly smooth. The study of these ship-waves has led to most important and practical improvements in the art of ship-designing and shipbuilding, and no treatment of the subject of waves and ripples on water would be complete in which all mention of ship-waves was omitted.

In order that we may explain the manner in which these waves are formed, and their effect upon the motion of the ship, and the power required to move it forward, we must begin by a little discussion of some fundamental facts concerning liquids in motion.

Every one is aware that certain liquids are, as we say _sticky_, or, to use the scientific term, _viscous_. A request to mention sticky liquids would call up the names of such fluids as tar, treacle, gum-water, glycerine, and honey. Very few people would think of including pure water, far less spirits of wine, in a list of sticky, or viscous liquids; and yet it is quite easy to show by experiment that even these fluids possess some degree of stickiness, or viscosity. An illustration may be afforded as follows: We provide several very large glass tubes, nearly filled respectively with quicksilver, water, alcohol, glycerine, and oil. A small space is left in each tube containing a little air, and the tubes are closed by corks. If we suddenly turn all the tubes upside down, these bubbles of air begin to climb up from the bottom of the tube to the top. We notice that in the quicksilver tube it arrives at the top in a second or two, in the water tube it takes a little longer, in the oil tube longer still, and in the tube filled with glycerine it is quite a minute or more before the bubble of air has completed its journey up the tube. This experiment, properly interpreted, shows us that water possesses in some degree the quality of viscosity. It can, however, be more forcibly proved by another experiment.

To a whirling-table is fixed a glass vessel half full of water. On this water a round disc of wood, to which is attached a long wire carrying a paper flag, is made to float. If we set the basin of water slowly in rotation, at first the paper flag does not move. The basin rotates without setting the contained water in rotation, and so to speak slips round it. Presently, however, the flag begins to turn slowly, and this shows us that the water has been gradually set in rotation. This happens because the water sticks slightly to the inner surface of the basin, and the layers of water likewise stick to one another. Hence, as the glass vessel slides round the water it gradually forces the outer layer of water to move with it, and this again the inner layers of water one by one, until at last the floating block of wood partakes of the motion, and the basin and its contents turn round as one mass. This effect could not take place unless the water possessed some degree of viscosity, and also unless so-called _skin friction_ existed between the inside of a glass vessel and the water it contains.

We may say, however, at once that no real liquid with which we are acquainted is entirely destitute of stickiness, or viscosity. We can nevertheless imagine a liquid absolutely free from any trace of this property, and this hypothetical substance is called a _perfect fluid_.

It is clear that this ideal perfect liquid must necessarily differ in several important respects from any real fluid, such as water, and some of these differences we proceed to examine. We must point out that in any liquid there may be two kinds of motion, one called _irrotational_ motion, and the other called _rotational_ or _vortex_ motion.

Consider any mass of water, such as a river, in motion in any way; we may in imagination fix our attention upon some small portion of it, which at any instant we will consider to be of a spherical shape. If, as this sphere of liquid moves along embedded in the rest of the liquid, it is turning round an axis in any direction as well as being distorted in shape, the motion of that part of the fluid is called _rotational_. If, however, our little sphere of liquid is merely being stretched or pulled into an ovoid or ellipsoidal shape without any rotation or spinning motion, then the motion of the liquid is said to be _irrotational_. We might compare these small portions of the liquid to a crowd of people moving along a street. If each person moves in such a way as always to keep his face in the same direction, that movement would be an irrotational movement. If, however, they were to move like couples dancing in a ball-room, not only moving along but turning round, their motion would be called rotational. Examples of rotational, or vortex motion are seen whenever we empty a wash-basin by pulling up the plug. We see the water swirl round, or rotate, forming what is called an _eddy_, or whirlpool. Also eddies are seen near the margin of a swiftly flowing river, since the water is set in rotation by friction against objects on the banks. Eddies are likewise created when two streams of water flow over each other with different speeds. A beautiful instance of this may be viewed at an interesting place a mile or two out of the city of Geneva. The Rhone, a rapid river, emerges as a clear blue stream from the Lake of Geneva. At a point called _Junction d’eaux_ it meets the river Arve, a more sluggish and turbid glacier stream, and the two then run together in the same channel. The waters of the Rhone and Arve do not at once mix, but the line of separation is marked by a series of whirlpools or eddies set up by the flow of the rapid Rhone water against the slower Arve water in contact with it.

Again, it is impossible to move a solid body through a liquid without setting up eddy-motion. The movement of an oar through the water, or even of a teaspoon through tea, is seen to be accompanied by little whirls which detach themselves from the oar or spoon, and are really the ends of vortices set up in the liquid. The two facts to notice

## particularly are that the production of eddies in liquids always

involves the expenditure of energy, or, in mechanical language, it necessitates _doing work_. To set in rotation a mass of any liquid requires the delivery to it of _energy_, just as is the case when a heavy wheel is made to rotate or a heavy train set in movement. This energy must be supplied by or absorbed from the moving solid or liquid which creates the eddies.

In the next place, we must note that eddies or vortices set up in an imperfect fluid, such as water, are ultimately destroyed by fluid friction. Their energy is frittered down into heat, and a mass of water in which eddies have been created by moving through it a paddle, is warmer after the eddies have subsided than before. It is obvious, from what has been said, that if a really perfect fluid did exist, it would be impossible by mechanical means to make eddies in it; but if they were created, they would continue for ever, and have something of the permanence of material substances.

[Illustration: FIG. 27.—The production of a vortex ring in air.]

A vortex motion in water may be either a terminated vortex, in which case its ends are on the surface, and are seen as eddies, or whirls; or it may be an endless vortex, in which case it is called a _vortex ring_. Such a ring is very easily made in the air as follows: A cubical wooden box about 18 inches in the side has a hole 6 inches in diameter made in the bottom (see Fig. 27). The open top of the box is covered tightly with elastic cloth. The box is then filled with the white vapour of ammonium chloride, by leading into it at the same time dry hydrochloric acid gas and dry ammonia gas. When quite full of dense white fumes, we give the cloth cover of the box a sharp blow with the fist, and from the round hole a white smoke ring leaps out and slides through the air. The experiment may be made on a smaller scale by using a cardboard box and filling it with the smoke of brown paper or tobacco.[13] If we look closely at the smoke ring as it glides through the air, we shall see that the motion of the air or smoke particles composing the ring is like that of an indiarubber umbrella-ring fitted tightly on a round ruler and pushed along. The ring turns itself continually over and over, the rotation being round the circular ring axis line. This rotatory motion is set up by the friction of the smoky air against the edge of the hole in the box, as the puff of air emerges from it when the back of the box is thumped. A simple but striking experiment may be made without filling the box with smoke. Place a lighted candle at a few feet away from the opening of the above-described box, and strike the back. An invisible vortex ring of air is formed and blows out the candle as it passes over it. Although it is quite easy to make a rotational motion in an imperfect fluid, and in fact difficult not to do it, yet of late years a very interesting and valuable discovery has been made by Professor Hele-Shaw, of a method of creating and rendering visible a motion in an imperfect liquid like water, which is irrotational. This discovery was that, if water is made to flow in a thin sheet between two plates, say of flat glass, not more than a fiftieth of an inch or so apart, the motion of the water is exactly that of a perfect fluid, and is irrotational. No matter what objects may be placed in the path of the water, it then flows round them just as if all fluid friction or viscosity was absent.

This interesting fact can be shown by means of an apparatus designed by Professor Hele-Shaw.[14] Two glass plates are held in a frame, and separated by a very small distance. By means of an inlet-pipe water is caused to flow between the plates. A metal block pierced with small holes is attached to the end of one plate, and this serves to introduce several small jets of coloured water into the main sheet. In constructing the apparatus great care has to be exercised to make the holes in the above-mentioned block very small (not more than ¹⁄₁₀₀ inch in diameter) and placed exactly at the right slope.

The main water inlet-pipe is connected by a rubber tube with a cistern of water placed about 4 feet above the level of the apparatus. The frame and glass plates are held vertically in the field of an optical lantern so as to project an image of the plates upon the screen. The side inlet-pipe leading to the pierced metal block is connected to another reservoir of water, coloured purple with permanganate of potash (Condy’s fluid), and the flow of both streams of water controlled by taps. The clear water is first allowed to flow down between the plates, so as to exclude all air-bubbles, and create a thin film of flowing water between two glass plates. The jets of coloured water are then introduced, and, after a little adjustment, we shall see that the coloured water flows down in narrow, parallel streams, not mixing with the clear water, and not showing any trace of eddies. The regularity of these streams of coloured water, and their sharp definition, shows that the liquid flow between the plates is altogether irrotational.

The lines marked out by the coloured water are called _stream-lines_, and they cut up the whole space into uniform _tubes of flow_. The characteristic of this flow of liquid is that the clear water in the space between two coloured streams of water never passes over into an adjacent tube. Hence we can divide up the whole sheet of liquid into tubular spaces called tubes of flow, by lines called stream-lines.

If now we dismount the apparatus and place between the glass a thin piece of indiarubber sheet—cut, say, into the shape of a ship, and of such thickness that it fills up the space between the glass plates—we shall be able to observe how the water flows round such an obstacle.

If the air is first driven out by the flow of the clear water, and then if the jets of coloured water are introduced, we see that the lines of liquid flow are delineated by coloured streams or narrow bands, and that these stream-lines bend round and enclose the obstructing object.

The space all round the ship-shaped solid body is thus cut up into tubes of flow by stream-lines, but these tubes of flow are now no longer straight, and no longer of equal width at all points.

They are narrower opposite the middle part of the obstruction than near either end.

[Illustration: FIG. 28.]

At this point we must make a digression to explain a fundamental law concerning fluid flow in tubes. Suppose we have a uniform horizontal metal tube, through which water is flowing (see Fig. 28). At various points along the tube let vertical glass pipes be inserted to act as gauge or pressure-tubes. Then when the fluid flows along the horizontal pipe it will stand up a certain height in each pressure-tube, and this height will be a measure of the pressure in the horizontal pipe at the point where the pressure-tube is inserted. We shall notice that when the water flows in the horizontal pipe, the water in the gauge-pipes stands at different heights, indicating a _fall in pressure_ along the horizontal pipe. We also notice that a line joining the tops of all the liquid columns in the pressure-pipes is a straight, sloping line, which is called the _hydraulic gradient_. This experiment proves to us that when fluid flows along a uniform-sectioned pipe there is a uniform fall or decrease in pressure along the pipe. The force which is driving the liquid along the horizontal pipe is measured by the difference between the pressures at its extreme ends, and the same is true of any selected length of the horizontal pipe.

It will also be clear that, since water is not compressible to any but the very slightest extent, the quantity of water, reckoned, say in gallons, which passes per minute across any section of the pipe must be the same.

[Illustration: FIG. 29.]

In the next place, suppose we cause water to flow through a tube which is narrower in some places than in others (see Fig. 29). It will be readily admitted that in this tube also the same quantity of water will flow across every section, wide or narrow, of the tube. If, however, we ask—Where, in this case, will there be the greatest pressure? it is certain that most persons would reply—In the narrow portions of the tube. They would think that the water-particles passing through the tube resemble a crowd of people passing along a street which is constricted in some places like the Strand. The crowd would be most tightly squeezed together, and the pressure of people would therefore be greater, in the narrow portions of the street. In the case of the water flowing through the tube of variable section this, however, is not the case. So far from the pressure being greatest in the narrow portions of the tube, it can be shown experimentally that it is precisely at those places it is least.

This can be demonstrated by the tube shown in Fig. 29. If water is allowed to flow through a tube constricted in some places, and provided with glass gauge-pipes at various points to indicate the pressure in the pipe at those places, it is found that the pressure, as indicated by the height of the water in the gauge-glasses at the narrow parts of the tube, is less than that which it would have at those places if the tube were of uniform section and length, and passed the same quantity of water. We can formulate this fact under a general law which controls fluid motion also in other cases, viz. that _where the velocity of the liquid is greatest, there the pressure is least_. It is evident, since the tube is wider in some places than in others, and as a practically incompressible liquid is being passed through it, that the speed of the liquid must be greater in the narrow portions of the tube than in the wider ones. But experiment shows that after allowing for what may be called the proper hydraulic gradient of the tube, the pressure is least in those places, viz. the constricted portions, where the velocity of the liquid is greatest. This general principle is of wide application in the science of hydraulics, and it serves to enable us to interpret aright many perplexing facts met with in physics.

We can, in the next place, gather together the various facts concerning fluid flow which have been explained above, and apply them to elucidate the problems raised by the passage through water of a ship or a fish.

Let us consider, in the first place, a body totally submerged, such as a fish, a torpedo or a submarine boat, and discuss the question why a resistance is experienced when an attempt is made to drag or push such a body through water. The old-fashioned notion was that the water has to be pushed out of the way to make room for the fish to move forward, and also has to be sucked in to fill up the cavity left behind. Most persons who have not been instructed in the subject, perhaps even now have the idea that this so-called “head resistance” is the chief cause of the resistance experienced when we make a body of any shape move through water. A common assumption is also that the object of making a ship’s bows sharp is that they may cut into the water like a wedge, and more easily push it out of the way. Scientific investigation has, however, shown that both of these notions are erroneous. The resistance felt in pulling or pushing a boat through the water is not due to resistance offered by the water in virtue of its inertia. No part of this resistance arises from the exertion required to displace the water or push it out of the way.

The Schoolmen of the Middle Ages used to discuss the question how it was that a fish could move through the water. They said the fish could not move until the water got out of the way, and the water could not get out of the way until the fish moved. This and similar perplexities were not removed until the true theory of the motion of a solid through a liquid had been developed.

Briefly it may be said that there are three causes, and only three, for the resistance which we feel and have to overcome when we attempt to drag a boat or ship through the water. These are: First, _skin friction_, due to the friction between the ship-surface and the water; secondly, _eddy-resistance_, due to the energy lost or taken up in making water eddies; and thirdly, _wave-resistance_, due to energy taken up in making surface-waves. The skin friction and the eddy-resistance both arise from the fact that water is not a perfect fluid. The wave-resistance arises, as we shall show, from the unavoidable formation of waves by the motion of the boat through the water.

In the case of a wholly submerged body, like a fish, the only resistance it has to overcome is due to the first two causes. The fish, progressing through the water wholly under the surface, makes no waves, but the water adheres to its skin, and there is friction between them as he moves. Also he creates eddies in the water, which require energy to produce them, and whenever mechanical work has to be done, as energy drawn off from a moving body, this implies the existence of a resistance to its motion which has to be overcome.

Accordingly Nature, economical on all occasions in energy expenditure, has fashioned the fish so as to reduce the power it has to expend in moving through water as much as possible. The fish has a smooth slippery skin. (We say “as slippery as an eel.”) It is not covered either with fur or feathers, but with shiny scales, so as to reduce to a minimum the skin friction. The fish also is regular and smooth in outline. It has no long ears, square shoulders, or projecting limbs or organs, which by giving it an irregular outline, would tend to produce eddies in the water as it moves along. Hence, when we wish to design a body to move quickly under the water, we must imitate in these respects the structure of a fish. Accordingly, a Whitehead torpedo, that deadly instrument employed in naval warfare, is made smooth and fish-shaped, and a submarine boat is made cigar-shaped and as smooth as possible, for the same reason.

If the floating object is partly above the surface, yet nevertheless, as far as concerns the portion submerged, there is skin friction, and the production of eddy-resistance. Hence, in the construction of a racing-yacht, the greatest care has to be taken to make its surface below water of polished metal or varnished wood, or other very smooth material, to diminish as far as possible the skin friction. In the case of bodies as regular in outline as a ship or fish, the proportion of the driving power taken up in making eddies in the water is not large, and we may, without sensible error, say that in their case the whole resistance to motion is comprised under the two heads of skin friction and wave-making resistance. The proportion which these two causes bear to each other will depend upon the nature of the surface of the body which moves over the water, and its shape and speed.

At this point we may pause to notice that, if we could obtain a perfect fluid in practice, it would be found that an object of any shape wholly submerged in the fluid could be moved about in any way without experiencing the least resistance. This theoretical deduction is, at first sight, so opposed to ordinary preconceived notions on the subject, that it deserves a little attention. It is difficult, as already remarked, for most people who have not carefully studied the subject, to rid their minds of the idea that there is a resistance to the motion of a solid through a liquid arising from the effort required to push the liquid out of the way. But this notion is, as already explained, entirely erroneous.

In the light of the stream-line theory of liquid motion, it is easy to prove, however, the truth of the above statement.

Let us begin by supposing that a solid body of regular and symmetrical shape, say of an oval form (see Fig. 30), is moved through a fluid destitute of all stickiness or viscosity, which therefore does not adhere to the solid. Then, if the solid is wholly submerged in this fluid, the mutual action of the liquid and the solid will be the same, whether we suppose the liquid to be at rest and the solid to move through it, or the solid body to be at rest and the liquid to flow past it.

[Illustration: FIG. 30.—Stream-lines round an ovoid.]

[Illustration: FIG. 31.—Tube of flow in a liquid.]

If, then, we suppose the perfect fluid to flow round the obstacle, it will distribute itself in a certain manner, and its motion can be delineated by stream-lines. There will be no eddies or rotations, because the liquid is by assumption perfect. Consider now any two adjacent stream-lines (see Fig. 31). These define a tube of flow, represented by the shaded portion, which is narrower in the middle than at the ends. Hence the liquid, which we shall suppose also to be incompressible, must flow faster when going past the middle of the obstacle where the stream-tubes are narrow, than at the ends where the stream-tubes are wider.

By the principle already explained, it will be clear that the pressure of the fluid will therefore be less in the narrow portion of the stream-tube, and from the perfect symmetry of the stream-lines it is evident there will be greater and equal pressures at the two ends of the immersed solid. The flow of the liquid past the solid subjects it, in fact, to a number of equal and balanced pressures at the two ends which exactly equilibrate each other. It is not quite so easy to see at once that if the solid body is not symmetrical in shape the same thing is true, but it can be established by a strict line of reasoning. The result is to show that when a solid of _any_ shape is immersed in a perfect liquid, it cannot be moved by the liquid flowing past it, and correspondingly would not require any force to move it against and through the liquid. In short, there is no resistance to the motion of a solid of any shape when pulled through a perfect or frictionless liquid. When dealing with real liquids not entirely free from viscosity, such resistance as does exist is due, as already mentioned, to skin friction and eddy formation. In the next place, leaving the consideration of the movement of wholly submerged bodies through liquids whether perfect or imperfect, we shall proceed to discuss the important question of the resistance offered by water to the motion through it of a floating object, such as a ship or swan. We have in this case to take into consideration the wave-making properties of the floating solid.

We have already pointed out that to make a wave on water requires an expenditure of energy or the performance of mechanical work. If a wave is made and travels away over water, it carries with it energy, and hence it can only be created if we have a store of energy to draw upon. If we suppose that skin friction is absent, and that the ship floats upon a perfect fluid, it would nevertheless be true that, if the moving object creates waves, it will thereby reduce its own movement and require the application of force to it to keep it going. We may say therefore that if any floating object creates waves on a liquid over which it moves, these waves rob the floating body of some of its energy of motion. The creation of the waves will bring it to rest in time, unless it is continually urged forward by some external and impressed force, and wave-generation is a reason for a part at least of the resistance we experience when we attempt to push it along.

Accordingly, one element in the problem of designing a ship is that of finding a form which will make as little wave-disturbance as possible in moving over the liquid. It is comparatively easy to find a shape for a floating solid which shall make a considerable wave-disturbance on the water when it is pulled over it, but it is not quite so easy to design a shape which will not make waves, or make but very small ones.

If we look carefully at a yacht gliding along before a fresh breeze on a sea or lake surface which is not much ruffled by other waves, it is possible to discover that a ship, when going through the water, creates _four distinct systems of waves_. Two of these are very easy to see, and two are more difficult to identify. These wave-systems are called respectively the oblique bow and stern waves, and the transverse and rear waves. We shall examine each system in turn.

The most important and easily observed of the four sets of waves is the oblique bow wave. It is most easily seen when a boy’s boat skims over the surface of a pond, and readily observed whenever we see a duck paddling along on the water. Let any one look, for instance, at a duck swimming on a pond. He will see two trains of little waves or ripples, which are inclined at an angle to the direction of the duck’s line of motion. Both trains are made up of a number of short waves, each of which extends beyond or overlaps its neighbour (see Fig. 32).

[Illustration: FIG. 32.—Echelon waves made by a duck.]

Hence, from a common French word, these waves have been called _echelon waves_,[15] and we shall so speak of them.

[Illustration: FIG. 33.—Echelon waves made by a model yacht.]

On looking at a boy’s model yacht in motion on the water, the same system of waves will be seen; and on looking at any real yacht or steamer in motion on smooth water, they are quite easily identified (see Fig. 33).

The complete explanation of the formation of these bow or echelon waves is difficult to follow, but in a general way their formation can be thus explained: Suppose we have a flat piece of wood, which is held upright in water, and to which we give a sudden push. We shall notice that, in consequence of the inertia of the liquid, it starts a wave which travels away at a certain speed over the surface of the water. The sudden movement of the wood elevates the water just in front of it, and this displacement forms the crest of a wave which is then handed on or propagated along the surrounding water-surface. If two pieces of wood are fastened together obliquely, as in Fig. 34, and held in water partly submerged, we shall find that when this wood is suddenly thrust forward like a wedge, it starts two oblique waves which move off parallel to the inclined wooden sides. The bows of a ship, roughly speaking, form such a wedge.

[Illustration: FIG. 34.]

Hence, if we consider this wedge or the bows of a ship to be placed in still water and then pushed suddenly forward, they will start two inclined waves, which will move off parallel to themselves.

If we then consider the wedge to leap forward and repeat the process, two more inclined waves will be formed in front of the first; and again we may suppose the process repeated, and a third pair of waves formed. The different positions of the ship’s bows are shown in the diagram at 1, 2, and 3 in Fig. 35; and _c_, _e_, and _f_ are the three corresponding sets of echeloned waves. For the sake of simplicity, the waves are shown on one side only. If, then, we imagine the ship to move uniformly forwards, its bows are always producing new inclined waves, which move with it, and it is always, so to speak, leaving the old ones behind. All these echelon waves produced by the bow of the ship are included within two sloping lines which each make with the direction of the ship’s line of movement, an angle of 19° 28´.[16] This angle can be thus set off: Draw a circle (see Fig. 36), and produce the diameter BC of this circle for a distance, CA, equal to its own length. From the end A of the produced diameter draw a pair of lines, AD, AD′, called tangents, to touch the circle. Then each of these lines will make an angle of 19° 28´ with the diameter. If we suppose a ship-to be placed at the point marked A in the diagram (see Fig. 36), all the echelon waves it makes will be included within these lines AD, AD′.

[Illustration: FIG. 35.]

Moreover, the angle of the lines will not alter, whether the ship goes fast or slow. This is easily seen in the case of a duck swimming on a lake. Throw bits of bread to a duck so as to induce it to swim faster or slower, and notice the system of inclined or echelon ripples made by the duck’s body as it swims. It will be seen that the angle at which the two lines, including both the trains of echelon ripples meet each other is not altered as the duck changes its speed.

[Illustration: FIG. 36.]

This echelon system of inclined waves is really only a part of a system of waves which is completed by a transverse group in the rear of the vessel. A drawing has been given by Lord Kelvin, in his lecture on “Ship Waves,” of the complete system of these waves, part of which is as represented by the firm lines in Fig. 37. This complete system is difficult to see in the case of a real ship moving over the water. The inclined rear system of waves can sometimes be well seen from the deck of a lake steamer, such as those on the large Swiss or Italian lakes, and may sometimes be photographed in a snap-shot taken of a boy’s yacht skimming along on a pond.

[Illustration: FIG. 37.]

In addition to the inclined bow waves, there is a similar system produced by the stern of a vessel, which is, however, much more difficult to detect. The other two wave-systems produced by a ship are generally called the transverse waves. There is a system of waves whose crest-lines are at right angles to the ship, and they may be seen in profile against the side of any ship or yacht as it moves along. These transverse waves are really due to the unequal pressures resulting from the distribution of the stream-lines delineating the movement of the water past the ship.

If we return again to the consideration of the flow of a perfect fluid round an ovoid body, it will be remembered that it was shown that, in consequence of the fact that the stream-lines are wider apart near the bow and stern than they are opposite the middle part of the body, the pressure in the fluid was greater near the bow and stern than at the middle. When a body is not wholly submerged, but floats on the surface as does a ship, these excess pressures at the bow and stern reveal themselves by forcing up the water-surface opposite the ends of the vessel and lowering it opposite the middle. This may be seen on looking at any yacht in profile as it sails. The yacht appears to rest on two cross-waves, one at the bow and one at the stern, and midships the water is depressed (see Fig. 38).

[Illustration: FIG. 38.]

These waves move with the yacht. If the ship is a long one, then each of these waves gives rise to a wave-train; and on looking at a long ship in motion, it will be seen that, in addition to the inclined bow wave-system, there is a series of waves which are seen in profile against the hull.

[Illustration: FIG. 39.]

When a ship goes at a very high speed, as in the case of torpedo-boat destroyers, the bow of the vessel is generally forced right up on to the top of the front transverse waves, and the boat moves along with its nose entirely out of water (see Fig. 39). In fact, the boat is, so to speak, always going uphill, with its bows resting on the side of a wave which advances with it, and its stern followed by another wave, whilst behind it is left a continually lengthening trail of waves, which are produced by those which move with the boat.

The best way to see all these different groups of ship-waves is to tow a rather large model ship without masts or sails—in fact, a mere hulk—over smooth water in a canal or lake. Let one person carry a rather long pole, to the end of which a string is tied; and by means of the string let the model ship be pulled through the water. Let this person run along the banks of the canal or lake, and tow the ship steadily through the water as far as possible at a constant speed. Let another person, provided with a hand camera, be rowed in a boat after the model, and keep a few yards behind. The second observer will be able to photograph the system of ship-waves made by the model, and secure various photographs when the model ship is towed at different rates. The echelon and transverse waves should then be clearly visible, and if the water is smooth and the light good, it is not difficult to secure many useful photographs.

By throwing bits of bread to ducks and swans disporting themselves on still water, they also may be induced to take active exercise in the right direction, and expose themselves and the waves or ripples that they make to the lens of a hand camera or pocket kodak. From a collection of snap-shot photographs of these objects the young investigator will learn much about the form of the waves made by ships, and will see that they are a necessary accompaniment of the movement of every floating object on water. By conducting experiments of the above kind under such conditions as will enable the exact speed of the model to be determined, and the resistance it experiences in moving through the water, information has been accumulated of the utmost value to shipbuilders.

Our scientific knowledge of the laws of ship-resistance we owe chiefly to the labours of two great engineers, Mr. Scott Russell and Mr. William Froude. Mr. Froude’s work was begun privately at Torquay about the year 1870, and was subsequently continued by him for the British Admiralty. Mr. Froude was the first to show the value and utility of experiments made with model ships dragged through the water. He constructed at Torquay an experiment tank about 200 feet in length, which was a sort of covered swimming-bath, and he employed for his experiments model ships made of wood or paraffin wax, the latter being chosen because the model could be so easily cut to the desired shape, and all the chips and the model itself could be melted up and used over again for subsequent experiments. Without detailing in historic order his discoveries, suffice it to say that, as the outcome of his work, Mr. Froude was able to state two very important laws which relate to the relative resistance experienced when two models of different sizes are dragged through the water at different speeds.

The first of these relates to what is called the “_corresponding speeds_.” Suppose we have a real ship 250 feet long, and we make an exact model of this ship 10 feet long, then the ship is twenty-five times longer than the model. Mr. Froude’s law of corresponding speeds is as follows:—

If the above model and the ship are both made to move over still water, the ship going five times as fast as the model, the system of waves made by the model will exactly reproduce on a smaller scale the system of waves made by the ship. In other words, if we were to take a couple of photographs, one of the ship going at 20 miles an hour, and one of the model one twenty-fifth of its size going at 4 miles an hour, and reduce the two photographs to the same size, they would be exactly alike in every detail.

Expressed in more precise language, the first law of Froude is as follows: When a ship and a model of it move through smooth water at such speeds that the speed of the ship is to the speed of the model as the square root of the length of the ship is to the square root of the length of the model, then these speeds are called “_corresponding speeds_.” At corresponding speeds the wave-making power of the model resembles that of the ship on a reduced scale. If we call _L_ and _l_ the lengths of the ship and the model, and _S_ and _s_ the speeds of the ship and the model, then we have—

_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_ = _L_^3/_l_^3

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.

It is in solving the above kind of practical problem that Mr. Froude’s methods of experimenting with models in a tank are of such immense value. The first thing that the naval architect does in designing a ship is to prepare a series of drawings, showing the form of the hull of the vessel. From these drawings a model is constructed exactly to scale. In England, following Mr. Froude’s practice, these models are usually made of paraffin wax, about 12 or 14 feet long and 1 inch in thickness. In the United States wood is used. These models are constructed with elaborate care and by the aid of special machinery, and are generally 10 or 12 feet in length, and some proper fraction of the length of the real vessel they represent. The models are then placed in a tank and experiments are made, the object of which is to ascertain the force or “pull” required to drag the model through the water at various speeds.

The tank belonging to the British Admiralty is at Haslar, Gosport, near Portsmouth, and the experiments are now conducted there by Mr. E. Edmund Froude, who continues the scientific work and investigations of his distinguished father, Mr. William Froude. This Admiralty tank at Haslar is 400 feet in length. The well-known firm of shipbuilders, Messrs. Denny Bros., of Dumbarton, Scotland, have also a private experimental tank of the same kind. The Government of the United States of America have a similar tank at Washington, the Italian Government have one at Spezzia, and the Russian Admiralty has also made one. These tanks resemble large swimming-baths, which are roofed over (see Fig. 40).

[Illustration: FIG. 40.—An experimental tank for testing ship models (Washington).[18]]

Over the water-surface is arranged a pair of rails, on which runs a light carriage or platform. This carriage is drawn along by a rope attached to a steam-engine, which moves at a very uniform rate, and its speed can be exactly ascertained and automatically recorded. This moving carriage has a rod or lever depending from it, to which the model ship is attached. The pull on this rod is exactly registered on a moving strip of paper by very delicate recording mechanism. The experiment is conducted by placing the model at one end of the tank, and taking a run at known and constant speed to the other end. The experimentalist is thus able to discover the total resistance which it is necessary to overcome in pushing the model ship at a certain known speed through the water. The immersed surface of the model being measured and the necessary calculations made, he can then deduct from the total resistance the resistance due to skin friction, and the residue gives the resistance due to wave-making. Suppose, then, that the experiment has been performed with a model of a ship yet to be built, the run being taken at a “corresponding speed.” The observations will give the wave-making resistance of the model, and from Mr. Froude’s second law the wave-making resistance of the real ship is predicted. Adding to this the calculated skin-friction resistance of the real ship, we have the predetermined actual total ship-resistance at the stated speed. For the sake of giving precision to these ideas, it may be well to give an outline of the calculations for a real ship, as given in a pamphlet by Mr. Archibald Denny.[19]

The tank at the Leven shipyard, constructed by Messrs. Denny Bros. for their own experiments, is 300 feet long, 22 feet wide, and 10 feet deep, and contains 1500 tons of fresh water. At each end are two shallower parts which serve as docks for ballasting and trimming models. As an example of the use of the tank in predicting the power required to drive a ship of certain design through the water, Mr. A. Denny gives the following figures: The ship to be built was 240 feet in length, and from the drawings a model was constructed 12 feet in length, or one-twentieth the size.

It was then required to predetermine the power required to drive the ship through the water at a speed of 13¹⁄₂ knots. A knot, be it remarked, is a speed or velocity of 1 nautical mile an hour, or 6080 feet per hour. It will be seen that this is not far from 100 feet per minute.

By Froude’s first law, the corresponding speed for the 12-foot model is therefore—

13¹⁄₂ × 6080/60 × √(12/240) = 306 feet per minute

The model was accordingly dragged through the tank at a speed of nearly 5 feet per second, and, after deducting from the total observed pull the resistance due to the calculated skin friction of the model, it was found that the resistance to the motion of the model at this speed due to wave-making was 1·08 lb. Hence, by Froude’s second law, the wave-making resistance of the ship was predetermined to be—

1·08 × (240/12)^3 × 40/39 = 8850 lbs.

The last fraction 40/39 is a correcting factor in passing from fresh water to salt water.

The surface of the proposed ship was 10,280 square feet, and the skin friction was known to be 1·01 lb. per square foot at a speed of 13·5 knots. Hence the total skin resistance of the ship would be—

10,280 × 1·01 × 40/39 = 10,620 lbs.

Adding to this, the 8850 lbs. for wave-making resistance, we have a total resistance of 19,470 lbs. predetermined as the total pressure required to be overcome in moving the ship at a speed of 13·5 knots. Hence, since 1 horse-power is defined to be a power which overcomes a resistance of 33,000 lbs. moved 1 foot per minute, it is easy to see that 19,470 lbs. overcome at a rate of 13·5 knots represents a power of—

(19,470 × 13·5 × 6080)/(33,000 × 60) = 810 horse-power

But now, in the case of a screw-driven steamer, a part of the power is lost in merely churning up the water, and a part in internal frictional losses in the engine and screw-shaft.

It is not far from the truth to say that 50 per cent. of the applied engine-power is lost in useless water-churning. Hence, for the above steamer, an actual power of at least 1600 H.P. would have to be applied to the screw-shaft. To allow, however, for the loss of power in friction, and to allow a margin for emergencies, it would be usual to provide for such a steamer engines of at least 3000 _indicated_ horse-power.

Each shipbuilder has, however, at call a mass of data which enable him, from actual measured mile trials, to determine the rates between the calculated driving horse-power and the indicated horse-power of the engines, and so enable him, in the light of experience, to provide in any new ship the exact amount of steam-power necessary to produce the required speed. As an instance of how accurately this can be done by the aid of the tank experiments, Mr. A. Denny gives an example drawn from experience in building the well-known paddle-steamers _Princess Josephine_ and _Princess Henriette_ for the Belgian Government Dover to Ostend fast mail-steamer service.

The speed guaranteed before the boats were built was 20¹⁄₂ knots. The estimate was made for 21 knots, and the actual results of trials on the measured mile, when the ships were built, showed that each did 21·1 knots on prolonged and severe test.

The reader, therefore, cannot fail to see how important are these methods, laws, and researches of Mr. Froude.

The above-described process for testing models is being continually conducted in the case of all new battleships and cruisers for the British Navy, and also is pursued by the naval constructors of other nations. In connection with the extensive programme of battleship construction which has been carried out of late years, Sir William White, the late eminent Chief Director of Naval Construction, states that it is not too much to say that these methods of investigation and experiment have placed in the hands of the naval architect an instrument of immense power for guiding him safely and preventing costly mistakes. Sir William White has declared that it would have been impossible to proceed with the same certainty in battleship design, were it not for the aid afforded by these methods.

Mr. Froude was not content, however, with experiments made with models. He ascertained by actual trials the total force required to drive an actual ship through the water at various speeds, and obtained from other experiments valuable data which showed the proportion in which the total resistance offered to the ship was divided between the skin friction and the wave-making resistance.

Then he made experiments on a ship of 1157 tons, viz. H.M.S. _Greyhound_. This vessel was towed by another vessel of 3078 tons, viz. H.M.S. _Active_, by means of a tow-rope and a dynamometer, which enabled the exact “pull” on this hawser to be ascertained when the _Greyhound_ was towed at certain speeds. The following are some of the results obtained:—

Speed in knots Strain in tons of H.M.S. _Greyhound_. on towing-rope. 4 knots 0·6 tons 6 ” 1·4 ” 8 ” 2·5 ” 10 ” 4·7 ” 12 ” 9·0 ”

It will be seen that the total resistance increases very rapidly with the speed, varying in a higher ratio than the square of the speed.

In addition, the indicated horse-power of the engine of the _Greyhound_ was taken when being self-driven at the above speeds, and it was found that only 45 per cent. of the indicated horse-power of the engines was used in propelling the ship, the remaining 55 per cent. being wasted in engine and shaft friction and in useless churning of the water by the screw.

It is an important thing to know how this total resistance is divided between skin friction and wave-making resistance.

Mr. R. E. Froude has kindly furnished the author, through the intermediation of Sir William White, with some figures obtained from experiments at Haslar, showing the proportion of the whole ship-resistance which is due to skin friction for various classes of ships going at certain speeds.

At full speed. At 10 knots. Battleships 55 per cent. 79 per cent. Cruisers 55 ” 84 ” Torpedo-boat destroyers 43 ” 80 ”

The above table gives the percentage which the skin friction forms of the total resistance, and the remainder is, of course, wave-making and eddy-resistance.

The curves shown in Fig. 41 (taken, by kind permission of the editor, from an article by Mr. E. H. Tennyson-D’Eyncourt, in _Cassier’s Magazine_ for November, 1901) give, in a diagrammatic form, an idea of the manner in which the two principal sources of ship-resistance vary with the speed.

[Illustration: FIG. 41.—(Reproduced, by permission, from _Cassier’s Magazine_.)]

It will be seen that when a ship is going at a relatively slow speed, the greater portion of the whole resistance is due to skin friction, but when going at a high speed, the greater portion of the resistance is due to wave-making. Hence the moral is that ships and boats intended to move at a high speed must be so fashioned as to reduce to a minimum the wave-making power. In general, the naval architect has to consider many other matters besides speed. In battleship design he has to consider stability, power of carrying guns and armour, and various other qualities. In passenger-steamers he has to take into consideration capacity for passengers and freight, also steadiness and sea-going qualities; and all these things limit and control the design. There is one class of vessel, however, in which everything is sacrificed to speed, and that is in racing-yachts. Hence, in the design of a racing-yacht, the architect has most scope for considerations which bear chiefly upon the removal of all limitations to speed. A little examination, therefore, of the evolution of the modern racing-yacht shows how the principles we have endeavoured to explain have had full sway in determining the present form of such boats.

Attention has chiefly been directed to this matter in connection with the international yacht race for the possession of the America Cup.

In 1851 a yacht named the _America_ crossed the Atlantic and made her appearance at Cowes to compete for a cup given by the Royal Yacht Squadron. Up to that time British yachts had been designed with full bluff bows and a tapering run aft. These boats were good sea-boats, but their wave and eddy making powers were considerable. The _America_ was constructed with very fine lines and a sharp bow, and was a great advance on existing types of yacht. In the race which ensued the _America_ won the cup, and carried it off to the United States.

Since that date there has been an intermittent but steady effort on the part of British yachtsmen to recover the trophy, so far, however, without success.

[Illustration:

AMERICA, 1851.

VIGILANT, 1893.

PURITAN, 1885.

DEFENDER, 1895.

VOLUNTEER, 1887.

COLUMBIA, 1899.

United States yachts entered for the America Cup race, 1851–1899. (FIG. 42.)]

[Illustration:

GENESTA, 1885.

VALKYRIE III., 1895.

THISTLE, 1887.

SHAMROCK, 1899.

VALKYRIE II., 1893.

SHAMROCK II., 1901.

FIG. 42.—British yachts entered for the America Cup race, 1885–1901.]

In a very interesting article in _Harmsworth’s Magazine_, in 1901, Mr. E. Goodwin has traced the gradual evolution of the modern yacht, such as _Shamrock II._ or the _Columbia_, from the _America_.

No doubt the methods of “measurement” in force at the time, or the dimensions which determine whether the boat can enter for the Cup race or not, have had some influence in settling the shape. The reader, however, will see, on comparing the outlines of some of the competing yachts as shown in Fig. 42,[20] that there has been a gradual tendency to reduce the underwater surface as much as possible, and also to remove the wave-making tendency by overhanging the bows. The only rule now in force restricting the yacht size for the Cup race is that it must not be more than 90 feet in length when measured on the water-line. In order that the yacht may have stability, and be able to carry a large sail-surface, it must have a certain depth of immersed hull. This is essential also to prevent the boat from making leeway when sailing with the wind abeam. But consistently with this object, the two great aims of the yacht-builder are, _first_, to reduce as much as possible the skin friction by making the yacht-surface smooth and highly polished. Thus modern racing-yachts are not always built of wood, but very often of some metal, such as bronze, steel, or aluminium alloys, which admit of a very high polish. This hull-surface is burnished as much as possible before the race, to reduce to a minimum the skin friction. Then in the _second_ place, the designer aims at fashioning the form of the bow of the yacht so as to reduce as much as possible its wave-making qualities. A fine type of modern yacht glides through the water with hardly any perceptible bow wave at moderate speeds.

Thus the following extract from the _Chicago Recorder_ of September 4, 1901, respecting Sir Thomas Lipton’s yacht, _Shamrock II._, during her trials for the Cup race, shows how marked a feature this is in the case of a yacht of the best modern type:—

“With her owner, designer, builder, manager, and sailmaker on board, the yacht _Shamrock II._ sailed her seventh trial race to-day off Sandy Hook. Although at times there was not more than a three-knot air, at no time did the yacht act sluggishly.

“She slipped through the water at an amazingly good rate under the influence of her great mainsail and light sails. The water was smooth, but even when pressed to a speed of 9 knots _the yacht made a very small wave at the bow_, and left an absolutely clean wake.”

We may say, therefore, that the ideal form of yacht is one which would travel through the water without making any wave at all at bow or stern. This condition can, however, only be reached approximately, but the clear recognition of the principle has enabled yachts to be designed with vastly greater speed powers than in the old days of bluff bows and tapering bodies.

Before passing away from the subject of waves made by ships, it is desirable to refer a little more in detail to the complicated wave-system made by a ship in motion. This has been most carefully elucidated by Lord Kelvin, who, in this as in so many other matters, is our great teacher. Lord Kelvin has shown that if a small floating body is towed through the water at a uniform speed, it originates a system of waves, each one of which is of the form shown in Fig. 43. The whole system of waves formed is represented in Fig. 37, where the position of the ship or moving object is at the point marked A.

The key to a correct comprehension of this ship wave-system is to be found in the fact explained in