At the crest of a wave the centrifugal force on the particles of water acts upwards against gravity, at the trough it acts downwards with gravity. The apparent or virtual weight of a body will therefore be less in the crest than the actual and more in the trough than the actual. This apparent weight may be 20 per cent. more or less than the real weight, according as it is in the trough or the crest.

This has been frequently verified by experience. If a spring balance is used, the indications on the dial for a given weight will be found less when the vessel is on the crest, and more when in the trough of a wave. This is also the explanation of the well-known phenomena of the tenderness of sailing boats on the crest of a long smooth wave. The virtual weight is considerably less than the actual, and consequently the righting moment is less than in still water. The wind moment is not affected in this way, and so on the crest of a wave, a boat, of sufficient stiffness in still water, is liable to be blown over to a large angle and possibly to capsize.

The effect of the centrifugal force at other portions of the wave is to alter the amount of the virtual gravity, and to cause it to

LENGTH

LENGTH

FIG. 195.-Wave profile.

act in a line of action perpendicular to the wave slope at any particular point. In considering rolling among waves we do not usually consider the variation of the virtual weight, but we must consider the variation of the line of action of the virtual buoyancy and gravity. This line of action will have its maximum inclination at about quarter the length of the wave from crest to crest or trough to trough. A small raft, as in Fig. 195, will always tend to keep normal to the wave surface; this normal is termed the virtual upright at any particular instant.

If now we take a ship floating broadside on to a series of waves (supposed long in comparison with the size of the ship), we shall have the set of forces as shown in Fig. 196. The inclination of the wave normal to the vertical is 0, and this wave normal is the virtual upright. If the ship is as shown the righting force is not due to the angle 0 + 0', but to the angle '.

If the ship has a very quick period compared with that of the wave, she will quickly come to the virtual upright, and so will take up the motion of the wave. This would be the case in a raft as in Fig. 195, and it is found that a ship of very short period does not roll very much or ship much water when rolling among waves, because she always keeps the deck parallel to the wave surface.

If a ship has a long period compared with that of the wave, the ship, at any particular instant, as in Fig. 196, does not come to

WAVE

VIRTUAL

UPRIGHT

SURFACE

the virtual upright with any suddenness, and the wave profile passes on and soon acts in the contrary direction. The ship therefore remains steady, never heeling to large angles. This quality of remaining nearly upright when among waves is termed steadiness, and is obtained in ships with a long period. We have seen above that a long period is mainly obtained by giving a small metacentric height. Such a ship is crank, i.e. is easily inclined by external forces, but in a seaway is most likely to be exceedingly steady.

FIG. 196.

If, however, a ship has her double period (port to port or vice versa) equal to the period of the waves (time the length of the wave is traversed), we have a serious state of things. This timing is termed synchronism. As each wave passes the ship, an impulse is given timing with the period of the ship herself, and the tendency of this is to produce larger and larger angles of oscillation. If the vessel was perfectly isochronous for large as well as small angles, and no resistances were acting, such a system of waves would inevitably capsize her. The actual conditions operating, however, are as follows:

1. As large angles are reached a ship departs from isochronous rolling, and the condition of synchronism with the wave is not fulfilled.

2. Resistances operate, and, especially in a vessel with bilge keels, the energy imparted by the wave is soon absorbed by the energy taken out of the ship by the various resistances. When

For a simple pendulum swinging 30° each side of the vertical 7 per cent. increase of period is noticed as compared with a small oscillation; for 45° the increase is 18 per cent.; for 60°, 37 per cent.

these are equal no further increment of rolling can take place.

3. A succession of waves of precisely the same period is a very unlikely occurrence.

It has often been noticed that ships with a great reputation for steadiness at sea occasionally roll heavily. This is doubtless caused by the fact that a succession of waves has been met with having a period approximately synchronizing with the double period of the ship.

In a ship thus rolling heavily a slight alteration of the course would be sufficient to destroy the synchronism, since what affects the ship is the apparent period of the waves, and if the ship's course be taken obliquely to the wave advance, the synchronism is at once destroyed.

The longer the period of a ship the less chance there is of meeting synchronizing waves. A series of waves of 16 seconds period is quite exceptional, so that the battle-ships of the British Navy having 16 seconds for their double period should be very steady, and this is borne out by actual experience. Atlantic storm waves have periods about 10 seconds, and it is only the smaller vessels of the Navy which have their double period as low as this; see above for periods of some typical ships.

Observations of Rolling.1-If a ship is rolling in still water, and a pendulum could be suspended at the centre of oscillation, then the point of suspension of such a pendulum would have no motion, and the pendulum would always remain vertical; the angles indicated would therefore give the angles of oscillation of the ship. If, however, the point of suspension is somewhere else, then as the ship rolls this point has motion and the pendulum does not give the true vertical. If one takes a fishing-rod, for instance, with a few feet of line, it is evident that, if the rod is swayed backwards and forwards, the line does not remain vertical. The same state of things obtains on board a ship; the pendulum does not hang vertically, and the angle it indicates will be in excess or defect of the true angle to which the ship rolls, unless it happens to be suspended at the centre of oscillation. If the point of suspension is above this centre the angle indicated will be in excess, if below, the angle will be in defect. (It is usually assumed that the centre of oscillation is near the C.G. of

1 In Sir W. H. White's "Manual of Naval Architecture" a whole chapter is devoted to this subject.

the ship.) In the above illustration of the fishing-rod, if the line is very long, the motion of the rod does not have any sensible effect on the line, so that for all practical purposes the line will remain vertical. This is the principle which has been effectively employed in instruments for measuring rolling, viz. that a pendulum of very long period is not appreciably affected by the motion of the ship, but will maintain itself practically vertical as the ship rolls.

When a ship is rolling at sea, an ordinary pendulum is still less likely to give correct angles, as in addition to rolling, the ship has a bodily movement among the waves.

Mr. Froude's apparatus for rolling records is most valuable when accurate observations are desired, as it shows automatically the rolling of the ship together with a time record. It is, however, too elaborate an instrument for ordinary use. It primarily depends on a heavy wheel, so weighted that its C.G. is very close to the axis of suspension (16 in. below), giving a very long period for a single oscillation, viz. 34 seconds.

An instrument, Fig. 197, which is very simple and which has given admirable results has been devised by Mr. Mallock (I.N.A.,

case is not communicated to the mass of the fluid, and in the interior of the box the fluid is practically at rest when the box is in motion. The paddle is adjusted so that its C.G. is just below the

1 This is interesting as analogous to the case of a ship with great moment of inertia and small metacentric height, both of which conduce to a long period.

axis. When free it has a complete period of 4 seconds, but when enclosed in the box the complete period is between 30 and 40 seconds. The paddle therefore remains practically vertical, and the box being attached to the ship, the pointer I will show on the paddle the angle of roll. This rolling indicator is now issued to ships of the Royal Navy.

The method of observing angles of roll by the use of battens is very simple. It can, however, only be used when the horizon or some fixed object (as a star) is visible. The battens are arranged so that they can be rigged up on the fore bridge as Fig. 198.

There are two battens on which a scale of angles is painted, the zero corresponding to the horizontal when the ship is upright. A bracket is placed at the middle line (attached to the chart house, say) having a horizontal slit. The horizon or distant object can be sighted through this slit at the extremity of each roll, and the angle can be noted. The time of each roll should also be noted by another observer. Forms are issued to ships of the Royal Navy giving detailed instructions.

1 "Mr. Mallock's instrument is exceedingly simple, it is always in place, it may be put anywhere, it is always measuring the angle of heel and is ready to be observed. I have invariably heard it spoken of very highly by those who have used it."-Mr. Philip Watts, F.R.S. (I.N.A., 1901).