Waves and ripples in water, air, and æther

by Sir J. A. Fleming

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]).

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.