its range, the former will flow away with the tidal current, while the latter will be thrown on to the beach.

Although the wave immediately in the rear of the one that breaks retains its undulating character, any floating substance within its range simply rising and falling and getting no nearer to the shore, while a substance contained in the shoreward wave is thrown forward beyond the water-line, yet the particles of water below the rear wave have a certain amount of translatory movement due to the effect which the shoaling of the bed has on the water below the wave's orbit.

This horizontal motion of the particles below the wave in shallow water is proved to be sensible by the broken water that occurs on the surface over a reef of rocks or on the edge of a shoal lying below the orbit of the wave.

When a wave meets a barrier projecting from the bed of the sea the water is thrown upward, and the wave which comes in contact with this obstruction attains an increased height.

The translatory movement of the particles of water below the wave accounts for substances being thrown, during heavy gales, on to the beach from greater depths than the height of the wave breaking on shore would appear to warrant.

This translatory movement, however, only extends for a short distance from the shore-line. Beyond this the waves, being mere undulations, are incapable of conveying material towards the land.

In order to find out the depth at which waves act on the seabed, an experiment was made in Lake Ontario, where in storms the waves are of considerable dimensions, by anchoring four empty boxes on the sloping sand-bed of the lake at equal distances over a length of 650 yards, in depths of 6 feet, 12 feet, 18 feet, and 20 feet. After storms it was found that the first box in the shallow water became filled with sand; the box in 12 feet of water half full; in the one at 18 feet there was very little sand; and at 20 feet there was no sand in the box.

In storms or heavy ground swells waves will break in depths that are great in comparison with their height, where the bottom is abrupt or uneven, or where there is sudden shoaling.

Ground-swell waves which are of great length, although of small height, break in depths where ordinary waves would remain undulations.

This translatory movement of the water below the wave itself

was termed by Colonel Emy in his treatise on waves1 Flot de Fond, and he considered that the destructive effect of the sea on maritime works was due in a great measure to this action. Although to a certain extent this is the case, it is generally considered that Colonel Emy attached greater importance to this movement than actually prevails.

The theory as set out in his treatise is, that as waves approach the shore and begin to feel the effect of the shoaling, the undulatory movement of the lower molecules of the water is changed into a horizontal one, independent of the undulatory movement of the upper part of the wave; and that where the bed of the sea rises abruptly this horizontal or translatory movement accumulates with each successive wave, the wave itself being thus raised above the level which it would otherwise attain. By this means the depth and volume of the water that finally breaks on the shore is increased, and consequently the force and destructive effect with which the wave strikes a cliff or sea-wall against which it is projected.

The wave-stroke is much heavier on a steep shore than when the beach consists of a long flat slope. In the latter case the waves are much broken up, the whole space between the shore and deep water becoming a mass of broken water. Tennyson gives a true description of these waves on the flat sands of the Lincolnshire coast :

"As the crest of some slow arching wave,

Heard in dead night along that table shore,
Drops flat; and after the great waters break
Whitening for half a league, and thin themselves,
Far over sands marbled with moon and cloud,
From less and less to nothing."

Ground Swell or Rollers.-These are the product of wind waves generated at some distant part of the ocean.

When wave-motion is once set up in the ocean, it continues for a considerable interval of time, and extends over a wider space than that covered by the original cause of disturbance, the effect being transmitted beyond the sphere of the gale. Thus frequently there is a heavy ground-swell on the coast without any corresponding gale. Or the wave-motion may travel at a greater rate than that of the movement of the gale which causes 1 "Du Mouvement des Ondes et des Travaux Hydraulique Maritime." Paris, 1831.

the disturbance, in which case a ground-swell precedes and becomes the harbinger of the coming gale.

As waves due to distant gales travel across the ocean shorewards, they coalesce and form long low undulations.

The effect of a ground-swell extends to a greater depth than that of ordinary wind waves; and it exerts a greater power of transmission near the bottom than shorter waves in the same depth.

As the long waves due to a ground-swell approach shallow water, where the depth is constantly diminishing and the space for their volume is contracted, the momentum contained in the moving water sensibly raises the height of the wave and increases its velocity. These waves are therefore always more powerful and exert a greater percussive effect on a sea-wall or cliff, and the back wash is more destructive to a beach than ordinary wind

waves.

Breaking Waves.-Waves are said to break when, owing to the water becoming shallow, they are no longer able to complete their undulation; and the water of the wave is thrown forward on the beach with a violence proportionate to the momentum it has acquired.

The depth of water in which a wave ceases to be an undulation and breaks varies with circumstances.

The least depth in which a wave can complete its undulation is when it reaches water the depth of which in repose is only equal to half the height of the wave from trough to crest.

The general result of observation of coast waves shows that as a rule, and under ordinary conditions, a wave breaks when it enters water the depth of which is equal to, or little exceeds, its height from trough to crest.

Waves are, however, known to break during very heavy gales, and when the fetch is very extended, in depths considerably greater than their height.

As illustrating the effect of the shoaling of the water in changing oscillating to breaking waves, the instance given by Mr. Shield in his book on Harbours1 may be quoted. At Peterhead there is a quay wall resting on a rock base about 2 feet above low water. At a short distance from the wall the rock bed dips, the depth of the water increasing to about 30 feet; there is then a ledge of rocks 160 feet from the wall, on which the "Harbour Constructions," W. Shield. London, 1895.

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water shoals to 22 feet, after which it deepens again to over 50 feet. With waves of 4 to 5 feet in height, and when the tide is from 7 or 8 feet up the wall, the undulations only rise and fall against the face, and are reflected without delivering any perceptible stroke. When, owing to the state of the tide, the water becomes more shallow, the waves are tripped up by the rock immediately in front of the wall, and, breaking, deliver such a heavy stroke on it that the broken water is thrown upwards to a height of 100 feet. When a wave breaks and the water assumes a progressive accelerated horizontal motion, it becomes capable of carrying forward with it any movable material with which it comes in

contact.

The water falling over from the crest of a breaking wave has a very mordant effect on the surface of the beach, and that receding down the slope has both a strong erosive and transporting power.

The mean level of the sea at the place where a wave breaks on the shore is raised by the action of an onshore wind, and this is further increased by the impetus of the waves. Hence the surface of the water forms a slope upwards towards the shore, causing an under-current towards the sea, which has considerable effect in pulling beaches down and conveying the material seaward. This action, also, is the main factor in maintaining the depth of water in bays. Captain Calver describes this as "the scavenging process caused by waves due to on-shore gales. An on-shore wind always produces an off-shore current, and the more violent the gale the greater the erosive effect.

The height of a wave, its length, and the velocity with which it moves are all governed by the depth of the water. The force of a breaking wave and its percussive effect on a cliff or sea-wall are therefore in proportion to the cube of the depth of the water in which it breaks.

The percussive force of the blow is diminished in proportion to the angle at which the wave strikes the object with which it comes in contact, either horizontally or vertically.

The force of waves on a beach, therefore, varies with the slope. The flatter the beach and the shallower the water, the less the eroding and transporting effect of the breaking wave. For this reason, shingle beaches lying at steeper slopes than sand beaches are more affected by gales, and there is a much greater disturbance of material.

As a rule, the depth of water in which wave-action takes place where sea-walls and other coast-protection works are situated, is that due to the rise of the tide, the beach generally being uncovered at low water. Waves of the magnitude of those where breakwaters are placed have not, therefore, to be encountered by the class of works here dealt with.

The length of a wave is the distance measured from the top of the crest of one wave to that of the next.

For waves of 5 feet in height or over, the velocity of motion varies as the depth of the water in repose, and is the same that a heavy body would acquire when falling freely from a height equal to that measured from the centre of gravity of the wave to its lowest part; or a height equal to that from the crest of the wave to the level of the water in repose. Approximately T V when h is the total height of the wave in feet, and 2'

= 8

V the velocity in feet per second.

The height is the vertical distance from the bottom of the trough to the top of the crest.

Small shore waves, or the wavelets due to the rising and falling tides, vary considerably in their dimensions according to circumstances.

Approximately, it may be taken that breaking waves having a height of 3 to 5 feet will have a length of 18 feet and a velocity of 10 feet a second, with 10 waves in a minute; waves 1 foot in height, a length of 13 feet, and a velocity of 6 feet a second, with 14 in a minute; and wavelets 6 inches high, a length of 6 feet, and a velocity of 23 feet a second, with 15 to 20 in a minute.

The period of the wave and the velocity do not always accord in these waves as with those in deeper water, as there is a slack interval between two waves.

Shore waves occur in series, the waves of each series varying in height, one of the series attaining a maximum height, and the others declining until a minimum is reached, when they begin again to increase in size.

It was stated by Mr. Palmer, in his paper on the "Motion of Shingle Beaches" (Phil. Trans. Royal Society, 1834), and the statement has frequently been repeated, that with eight breakers a minute the effect on the beach is accumulative, but that with ten breakers a destructive action sets in.

This rule, however, cannot be held to apply. The number of