From the motion of the bran we can see that the water swings backwards and forwards in a horizontal line with a pendulum-like motion, but its up-and-down or vertical motion is much more restricted. A wave of this kind travels along a canal with a speed which depends upon the depth of the canal. If waves of this kind are started in a very long trough, the wave-length being large compared with the depth of the trough,[8] it can be shown that the speed of the wave is equal to the velocity which would be gained by a stone or other heavy body in falling through half the depth of the canal. Hence, the deeper the water, the quicker the wave travels. This can be shown as an experimental fact as follows: Let two galvanized iron tanks be provided, each about 6 feet long and 1 foot wide and deep.

At one end of each tank a hollow cylinder, such as a coffee-canister or ball made water-tight, is floated, and it may be prevented from moving from its place by being attached to a hinged rod like the ball-cock of a cistern. The two tanks are placed side by side, and one is filled to a depth of 6 inches, and the other to a depth of 3 inches, with water. Two pieces of wood are then provided and joined together as in Fig. 14, so as to form a double paddle. By pushing this through the water simultaneously in both tanks at the end opposite to that at which the floating cylinders are placed, it is possible to start two solitary waves, one in each tank, at the same instant. These waves rush up to the other end and cause the floats to bob up. It will easily be seen that the float on the deeper water bobs up first, thus showing that the wave on the deeper water has travelled along the tank more quickly than the wave on the shallower water.

In order to calculate the speed of the waves, we must call to mind the law governing the speed of falling bodies. If a stone falls from a height its speed increases as it falls. It can be shown that the speed in feet per second after falling from any height is obtained by multiplying together the number 8 and a number which is the square root of the height in feet.

Thus, for instance, if we desire to know the speed attained by falling from a height of 25 feet above the earth’s surface, we multiply 8 by 5, this last number being the square root of 25. Accordingly, we find the velocity to be 40 feet per second, or about 26 miles an hour.

The force of the blow which a body administers and suffers on striking the ground depends on the energy of motion it has acquired during the fall, and as this varies as the square of the speed, it varies also as the height fallen through.

Let us apply these rules to calculate the speed of a long wave in a canal having water 8 feet deep in it. The half-depth of the canal is therefore 4 feet. The square root of 4 is 2; hence the speed of the wave is that of a body which has fallen from a height of 4 feet, and is therefore 16 feet per second, or nearly 11 miles an hour. When we come to consider the question of waves made by ships, in the next chapter, a story will be related of a scientific discovery made by a horse employed in dragging canal-boats, which depended on the fact that the speed of long waves in this canal was nearly the same as the trotting speed of the horse.

It may be well, as a little digression, to point out how the law connecting height fallen through and velocity acquired by the falling body may be experimentally illustrated for teaching purposes.

The apparatus is shown in Fig. 15. It consists of a long board placed in a horizontal position and held with the face vertical. This board is about 16 feet long. Attached to this board is a grooved railway, part of which is on a slope and part is horizontal. A smooth iron ball, A, about 2 inches in diameter, can run down this railway, and is stopped by a movable buffer or bell, B, which can be clamped at various positions on the horizontal rail. At the bottom of the inclined plane is a light lever, T, which is touched by the ball on reaching the bottom of the hill. The trigger releases a pendulum, P, which is held engaged on one side, and, when released, it takes one swing and strikes a bell, G. The pendulum occupies half a second in making its swing. An experiment is then performed in the following manner: The iron ball is placed at a distance, say, of 1 foot up the hill and released. It rolls down, detaches the pendulum at the moment it arrives at the bottom of the hill, and then expends its momentum in running along the flat part of the railway. The buffer must be so placed by trial that the iron ball hits it at the instant when the pendulum strikes the bell. The distance which the buffer has to be placed from the bottom of the hill is a measure of the velocity acquired by the iron ball in falling down the set distance along the hill. The experiment is then repeated with the iron ball placed respectively four times and nine times higher up the hill, and it will be found that the distances which the ball runs along the flat part in one half-second are in the ratio of 1, 2, and 3, when the heights fallen through down the hill are in the ratio of 1, 4, and 9.

The inference we make from this experiment is that the velocity acquired by a body in falling through any distance is proportional to the square root of the height. The same law holds good, no matter how steep the hill, and therefore it holds good when the body, such as a stone or ball, falls freely through the air.

The experiment with the ball rolling down a slope is an instructive one to make, because it brings clearly before the mind what is meant by saying, in scientific language, that one thing “varies as the square root” of another. We meet with so many instances of this mode of variation in the study of physics, that the reader, especially the young reader, should not be content until the idea conveyed by these words has become quite clear to him or her.

Thus, for instance, the time of vibration of a simple clock pendulum “varies as the square root of the length;” the velocity of a canal wave “varies as the square root of the depth of the canal;” and the velocity or speed acquired by a falling ball “varies as the square root of the distance fallen through.” These phrases mean that if we have pendulums whose lengths are in the ratio of 1 to 4 to 9, then the respective times of their vibration are in the ratio of 1 to 2 to 3. Also a similar relation connects the canal-depth and wave-velocity, or the ball-velocity and height of fall.

Returning again to canal waves, it should be pointed out that the real path of a particle of water in the canal, when long waves are passing along it, is a very flat oval curve called an ellipse. In the extreme cases, when the canal is very wide and deep, this ellipse will become nearly a circle; and, on the other hand, when narrow and shallow, it will be nearly a straight line. Hence, if long waves are created in a canal which is shallow compared with the length of the wave, the water-particles simply oscillate to and fro in a horizontal line. There is, however, one important fact connected with wave-propagation in a canal, which has a great bearing on the mode of formation of what is called a “bore.”

As a wave travels along a canal, it can be shown, both experimentally and theoretically, that the crest of the wave travels faster than the hollow, and as a consequence the wave tends to become steeper on its front side, and its shape then resembles a saw-tooth.

A very well known and striking natural phenomenon is the so-called “bore” in certain tidal rivers or estuaries. It is well seen on the Severn in certain states of the tide and wind. The tidal wave returning along the Severn channel, which narrows rapidly as it leaves the coast, becomes converted into a “canal wave,” and travels with great rapidity up the channel. The front side of this great wave takes an almost vertical position, resembling an advancing wall of water, and works great havoc with boats and shipping which have had the misfortune to be left in its path. To understand more completely how a “bore” is formed, the reader must be reminded of the cause of all tidal phenomena. Any one who lives by the sea or an estuary knows well that the sea-level rises and falls twice every 24 hours, and that the average interval of time between high water and high water is nearly 12¹⁄₂ hours. The cause of this change of level in the water-surface is the attraction exerted by the sun and moon upon the ocean. The earth is, so to speak, clothed with a flexible garment of water, and this garment is pulled out of shape by the attractive force of our luminaries; very roughly speaking, we may say that the ocean-surface is distorted into a shape called an ellipsoid, and that there are therefore two elevations of water which march across the sea-covered regions of the earth as it revolves on its axis. These elevations are called the tidal waves. The effects, however, are much complicated by the fact that the ocean does not cover all parts of the earth. There is no difficulty in showing that, as the tidal wave progresses round the earth across each great ocean, it produces an elevation of the sea-surface which is not simultaneous at all places. The time when the crest of the tidal wave reaches any place is called the “time of high tide.” Thus if we consider an estuary, such as that of the Thames, there is a marked difference between the time of high tide as we ascend the estuary.

Taking three places, Margate, Gravesend, and London Bridge, we find that if the time of high tide at Margate is at noon on any day, then it is high tide at Gravesend at 2.15 p.m., and at London Bridge a little before three o’clock. This difference is due to the time required for the tidal wave to travel up the estuary of the Thames.

When an estuary contracts considerably as it proceeds, as is the case with the Bristol Channel, then the range of the tide or the height of the tidal wave becomes greatly increased as it travels up the gradually narrowing channel, because the wave is squeezed into a smaller space. For example, the range of spring tides at the entrance of the Bristol Channel is about 18 feet, but at Chepstow it is about 50 feet.[9] At oceanic ports in open sea the range of the tide is generally only 2 or 3 feet.

If we look at the map of England, we shall see how rapidly the Bristol Channel contracts, and hence, as the tidal wave advances from the Atlantic Ocean, it gets jambed up in this rapidly contracting channel, and as the depth of the channel in which it moves rapidly shallows, the rear portion of this tidal wave, being in deeper water, travels faster than the front part and overtakes it, producing thus a flat or straight-fronted wave which goes forward with tremendous speed.[10]

We must, in the next place, turn our attention to the study of water ripples. The term “ripple” is generally used to signify a very small and short wave, and in ordinary language it is not distinguished from what might be called a wavelet, or little wave. There is, however, a scientific distinction between a wave and a ripple, of a very fundamental character.

It has already been stated that a wave can only exist, or be created, in or on a medium which resists in an elastic manner some displacement. The ordinary water-surface wave is termed a gravitation wave, and it exists because the water-surface resists being made unlevel. There is, however, another thing which a water-surface resists. It offers an opposition to small stretching, in virtue of what is called its surface tension. In a popular manner the matter may thus be stated: The surface of every liquid is covered with a sort of skin which, like a sheet of indiarubber, resists stretching, and in fact contracts under existing conditions so as to become as small as possible. We can see an illustration of this in the case of a soap-bubble. If a bubble is blown on a rather wide glass tube, on removing the mouth the bubble rapidly shrinks up, and the contained air is squeezed out of the tube with sufficient force to blow out a candle held near the end of the tube.

Again, if a dry steel sewing-needle is laid gently in a horizontal position on clean water, it will float, although the metal itself is heavier than water. It floats because the weight of the needle is not sufficient to break through the surface film. It is for this reason that very small and light insects can run freely over the surface of water in a pond.

This surface tension is, however, destroyed or diminished by placing various substances on the water. Thus if a small disc of writing-paper the size of a wafer is placed on the surface of clean water in a saucer, it will rest in the middle. The surface film of the water on which it rests is, however, strained or pulled equally in different directions. If a wire is dipped in strong spirits of wine or whisky, and one side of the wafer touched with the drop of spirit, the paper shoots away with great speed in the opposite direction. The surface tension on one side has been diminished by the spirit, and the equality of tension destroyed.

These experiments and many others show us that we must regard the surface of a liquid as covered with an invisible film, which is in a state of stretch, or which resists stretching. If we imagine a jam-pot closed with a cover of thin sheet indiarubber pulled tightly over it, it is clear that any attempt to make puckers, pleats, or wrinkles in it would involve stretching the indiarubber. It is exactly the same with water. If very small wrinkles or pleats, as waves, are made on its surface, the resistance which is brought into play is that due to the surface tension, and not merely the resistance of the surface to being made unlevel. Wavelets so made, or due to the above cause, are called ripples.

It can be shown by mathematical reasoning[11] that on the free surface of a liquid, like water, what are called capillary ripples can be made by agitations or movements of a certain kind, and the characteristic of these surface-tension waves or capillary ripples, as compared with gravitation waves, is that the velocity of propagation of the capillary ripple is less the greater the wave-length, whereas the velocity of gravitation on ordinary surface waves is greater the greater the wave-length.

It follows from this that for any liquid, such as water, there is a certain length of wave which travels most slowly. This slowest wave is the dividing line between what are properly called ripples, and those that are properly called waves. In the case of water this slowest wave has a wave-length of about two-thirds of an inch (0·68 inch), and a speed of travel approximately of 9 inches (0·78 foot) per second.

More strictly speaking, the matter should be explained as follows: Sir George Stokes showed, as far back as 1848, that the surface tension of a liquid should be taken into account in finding the pressure at the free surface of a liquid. It was not, however, until 1871 that Lord Kelvin discussed the bearing of this fact on the formation of waves, and gave a mathematical expression for the velocity of a wave of oscillatory type on a liquid surface, in which the wave-length, surface tension, density, and the acceleration of gravity were taken into account. The result was to show that when waves are very short, viz. a small fraction of an inch, they are principally due to surface tension, and when long are entirely due to gravity.

It can easily be seen that ripples run faster the smaller their wave-length. If we take a thin wire and hold it perpendicularly in water, and then move it quickly parallel to itself, we shall see a stationary pattern of ripples round the wire which moves with it. These ripples are smaller and closer together the faster the wire is moved.

Ripples on water are formed in circular expanding rings when rain-drops fall upon the still surface of a lake or pond, or when drops of water formed in any other way fall in the same manner. On the other hand, a stone flung into quiet and deep water will, in general, create waves of wave-length greater than two-thirds of an inch, so that they are no longer within the limits entitling them to be called ripples. Hence we have a perfectly scientific distinction between a ripple and a wave, and a simple measurement of the wave-length will decide whether disturbances of oscillatory type on a liquid surface should be called ripples or waves in the proper sense of the words.

The production of water ripples and their properties, and a beautiful illustration of wave properties in general, can be made by allowing a steady stream of water from a very small jet to fall on the surface of still water in a tank. In order to see the ripples so formed, it is necessary to illuminate them in a particular manner.

The following is a description of an apparatus, designed by the author for exhibiting all these effects to a large audience:⁠—

The instrument consists essentially of an electric lantern. A hand-regulated or self-regulating arc lamp is employed to produce a powerful beam of light. This is collected by a suitable condensing-lens, and it then falls upon a mirror placed at an angle of 45°, which throws it vertically upwards. The light is then concentrated by a plain convex lens placed horizontally, and passes through a trough of metal having a plane glass bottom. This trough is filled to a depth of half an inch with water, and it has an overflow pipe to remove waste water. Above the tank, at the proper distance, is placed a focussing-lens, and another mirror at an angle of 45° to throw an image of the water-surface upon a screen. The last lens is so arranged that ripples on the surface of the water appear like dark lines flitting across the bright disc of light which appears upon the screen. Two small brass jets are also arranged to drop water into the tank, and these jets must be supplied with water from a cistern elevated about 4 feet above the trough. The jets must be controlled by screw-taps which permit of very accurate adjustment. These jets should work on swivels, so that they may be turned about to drop the water at any point in the tank.

The capillary ripples which are produced on the water-surface by allowing water to drop on it from a jet, flit across the surface so rapidly that they cannot be followed by the eye. They may, however, be rendered visible as follows: A zinc disc, having holes in it, is arranged in front of the focussing-lens, and turned by hand or by means of a small electric motor. This disc is called a stroboscopic disc. When turned round it eclipses the light at intervals, so that the image on the screen is intermittent. If, now, one of the water-jets is adjusted so as to originate at the centre of the tank a set of diverging circular ripples, they can be projected as shadows upon the screen. These ripples move at the rate of 1 or 2 feet per second, and their shadows move so rapidly across the field of view that we cannot well observe their behaviour. If, however, the metal disc with holes in it is made to revolve and to intermittently obscure the view, it is possible to adjust its speed so that the interval of time between two eclipses is just equal to that required by the ripples to move forward through one wave-length. When this exact speed is obtained, the image of the ripples on the screen becomes stationary, and we see a series of concentric dark circles with intermediate bright spaces (see Fig. 16), which are the shadows of the ripples. In this manner we can study many of their effects. If, for instance, the jet of water is made to fall, not in the centre of the trough, but nearer one side, we shall notice that there are two sets of ripples which intersect—one of these is the direct or original set, and the other is a set produced by the reflection of the original ripples from the side of the trough. These direct and reflected ripple-shadows intersect and produce a cross-hatched pattern. If a slip of metal or glass is inserted into the trough, it is very easy to show that when a circular ripple meets a plane hard surface it is reflected, and that the reflected ripple is also a circular one which proceeds as if it came from a point, Q, on the opposite side of the boundary, just as far behind that boundary as the real centre of disturbance or origin of the ripple P is in front of it (see Fig. 17). In the diagram the dotted curves represent the reflected ripple-crests.

If we make two sets of ripples from origins P and Q (see Fig. 18), at different distances from a flat reflecting boundary, it is not difficult to trace out that each set of ripples is reflected independently, and according to the above-mentioned rule. We here obtain a glimpse of a principle which will come before us again in speaking of æther waves, and furnishes an explanation of the familiar optical fact that when we view our own reflection in a looking-glass, the image appears to be as far behind the glass as we are in front of it.

A very pretty experiment can be shown by fitting into the trough an oval band of metal bent into the form of an ellipse. If two pins are stuck into a sheet of card, and a loop of thread fitted loosely round them, and a pencil employed to trace out a curve by using it to strain the loop of thread tight and moving it round the pin, we obtain a closed curve called an ellipse (see Fig. 19). The positions of the two pins A and B are called the foci. It is a property of the ellipse that the two lines AP and BP, called radii vectores, drawn from the foci to any point P on the curve, make equal angles with a line TT′ called a tangent, drawn to touch the selected point on the ellipse. If we draw the tangent TT′ to the ellipse at P, then it needs only a small knowledge of geometry to see that the line PB is in the same position and direction as if it were drawn through P from a false focus A′, which is as far behind the tangent TT′ as the real focus A is in front of it. Accordingly, it follows that circular ripples diverging from one focus A of an ellipse must, after reflection at the elliptical boundary, be converged to the other focus B. This can be shown by the use of the above described apparatus in a pretty manner.

A strip of thin metal is bent into an elliptical band and placed in the lantern trough. The band is so wide that the water in the trough is about halfway up it. At a point corresponding to one focus of the ellipse, drops of water are then allowed to fall on the water-surface and start a series of divergent ripples. When the stroboscopic disc is set in revolution and its speed properly adjusted, we see that the divergent ripples proceeding from one focus of the ellipse are all converged or concentrated to the other focus. In fact, the ripples seem to set out from one focus, and to be, as it were, swallowed up at the other. When, in a later chapter, we are discussing the production and reflection of sound waves in the air, you will be able to bring this statement to mind, and it will be clear to you that if, instead of dealing with waves on water, we were to create waves in air in the interior of a similar elliptically shaped room, the waves being created at one focus, they would all be collected at the other focus, and the tick of a watch or a whisper would be heard at the point corresponding to the other focus, though it might not be heard elsewhere in the room.

With the appliances here described many beautiful effects can be shown, illustrating the independence of different wave-trains and their interference. If we hurl two stones into a lake a little way apart, and thus create two sets of circular ripples (see Fig. 20), we shall notice that these two ripple-trains pass freely through each other, and each behave as if the other did not exist. A careful examination will, however, show that at some places the water-surface is not elevated or disturbed at all, and at others that the disturbance is increased.

If two sets of waves set out from different origins and arrive simultaneously at the same spot, then it is clear that if the crests or hollows of both waves reach that point at the same instant, the agitation of the water will be increased. If, however, the crest of a wave from one source reaches it at the same time as the hollow of another equal wave from the other origin, then it is not difficult to see that the two waves will obliterate each other. This mutual destruction of wave by wave is called interference, and it is a very important fact in connection with wave-motion. It is not too much to say that whenever we can prove the existence of interference, that alone is an almost crucial proof that we are dealing with wave-motion. The conditions under which interference can take place must be examined a little more closely. Let us suppose that two wave-trains, having equal velocity, equal wave-length, and equal amplitude or wave-height, are started from two points, A and B (see Fig. 21). Consider any point, P. What is the condition that the waves from the two sources shall destroy each other at that point? Obviously it is that the difference of the distances AP and BP shall be an odd number of half wave-lengths. For if in the length AP there are 100 waves, and in the distance BP there are 100¹⁄₂ waves, or 101¹⁄₂ or 103¹⁄₂, etc., waves, then the crest of a wave from A will reach P at the same time as the hollow of a wave from B, and there will be no wave at all at the point P. This is true for all such positions of P that the difference of its distances from A and B are constant.

But again, we may choose a point, Q, such that the difference of its distances from A and B is equal to an even number of half wave-lengths, so that whilst in the length AQ there are, say, 100 waves, in the distance BQ there are 101, 102, 103, etc., waves. When this is the case, the wave-effects will conspire or assist each other at Q, and the wave-height will be doubled. If, then, we have any two points, A and B, which are origins of equal waves, we can mark out curved lines such that the difference of the distances of all points on these lines from these origins is constant. These curves are called hyperbolas (see Fig. 22).

All along each hyperbola the disturbance due to the combined effect of the waves is either doubled or annulled when compared with that due to each wave-train separately. With the apparatus described, we can arrange to create and adjust two sets of similar water ripples from origins not far apart, and on looking at the complicated shadow-pattern due to the interference of the waves, we shall be able to trace out certain white lines along which the waves are annulled, these lines being hyperbolic curves (see Fig. 23). With the same appliances another characteristic of wave-motion, which is equally important, can be well shown.

We make one half of the circular tank in which the ripples are generated much more shallow than the other half, by placing in it a thick semicircular plate of glass. It has already been explained that the speed with which long waves travel in a canal increases with the depth of the water in the canal. The same is true, with certain restrictions, of ripples produced in a confined space or tank, one part of which is much shallower than the rest. If waves are made by dropping water on to the water-surface in the deeper part of the tank, they will travel more quickly in this deeper part than in the shallower portion. We can then adjust the water-dropping jet in such a position that it creates circular ripples which originate in deep water, but at certain places pass over a boundary into a region of shallower water (see Fig. 24). The left-hand side of the circular tank represented in the diagram is more shallow than the right-hand side.

When this is done, we notice two interesting facts, viz. that the wave-lines are bent, or refracted, where they pass over the boundary, and that the waves are shorter or nearer together in the shallower region. This bending, or refraction, of a wave-front in passing the boundary line between two districts in which the wave has different velocities is an exceedingly important characteristic of wave-motion, and we shall have brought before us the analogous facts in speaking of waves in air and waves in æther.

It is necessary to explain a little more in detail how it comes to pass that the wave-line is thus bent. Imagine a row of soldiers, ab, marching over smooth grass, but going towards a very rough field, the line of separation SS between the smooth and the rough field being oblique to the line of the soldiers (see Fig. 25). Furthermore, suppose the soldiers can march 4 miles an hour over the smooth grass, but only 3 miles an hour over the rough field. Then let the man on the extreme left of the line be the first to step over the boundary. Immediately he passes into a region where his speed of marching is diminished, but his comrade on the extreme right of the row is still going easily on smooth grass. It is accordingly clear that the direction of the line of soldiers will be swung round because, whilst the soldier on the extreme left marches, say, 300 feet, the one on the extreme right will have gone 400 feet forward; and hence by the time all the men have stepped over the boundary, the row of soldiers will no longer be going in the same direction as before—it will have become bent, or refracted.

This same action takes place with waves. If a wave meets obliquely a boundary separating two regions, in one of which it moves slower than in the other, then, for the same reason that the direction of the row of soldiers in the above illustration is bent by reason of the retardation of velocity experienced by each man in turn as he steps over the dividing line, so the wave-line or wave-front is bent by passing from a place where it moves quickly to a place where it moves more slowly. The ratio of the velocities or speeds of the wave in the two regions is called the index of refraction.

We can, by arranging suitably curved reflecting surfaces or properly shaped shallow places in a tank of water, illustrate all the facts connected with the change in wave-fronts produced by reflection and refraction.

We can generate circular waves or ripples diverging from a point, and convert them, by reflection from a parabolic reflector, into plane waves; and again, by means of refraction at a curved or lens-shaped shallow, converge these waves to a focus.

Interesting experiments of this kind have been made by means of capillary ripples on a mercury surface by Mr. J. H. Vincent, and he has photographed the ripples so formed, and given examples of their reflection and refraction, which are well worth study.[12]

We do not need, however, elaborate apparatus to see these effects when we know what to look for.

A stone thrown into a lake will create a ripple or wave-train, which moves outwards at the rate of a few feet a second. If it should happen that the pond or lake has an immersed wall as part of its boundary, this may form an effective reflecting surface, and as each circular wave meets the wall it will be turned back upon itself as a reflected wave. At the edge of an absolutely calm sea, at low tide, the author once observed little parallel plane waves advancing obliquely to the coast; the edge of the water was by chance just against a rather steep ledge of hard sand, and each wavelet, as it met this reflecting surface, was turned back and reflected at an angle of reflection equal to that of incidence.

It is well to notice that a plane wave, or one in which the wave front or line is a straight line, may be considered as made up out of a number of circular waves diverging from points arranged closely together along a straight line. Thus, if we suppose that a, b, c, d, etc. (see Fig. 26), are source-points, or origins, of independent sets of circular waves, represented by the firm semicircular lines, if they send out simultaneous waves equal in all directions, the effect will be nearly equivalent to a plane wave, represented by the straight thick black line, provided that the source-points are very numerous and close together.

Supposing, then, we have a boundary against which this plane wave impinges obliquely, it will be reflected and its subsequent course will be exactly as if it had proceeded from a series of closely adjacent source-points, a′, b′, c′, d′, etc., lying behind the boundary, each of which is the image of the corresponding real source-points, and lies as far behind the boundary as the real point lies in front of it.

An immediate consequence of this is that the plane reflected wave-front makes the same angle with the plane reflecting surface as does the incident or arriving wave, and we thus establish the law, so familiar in optics, that the angle of incidence is equal to the angle of reflection when a plane wave meets a plane reflecting surface.

At the seaside, when the tide is low and the sea calm or ruffled only by wavelets due to a slight wind, one may often notice trains of small waves, which are reflected at sharp edges of sand, or refracted on passing into sudden shallows, or interfering after passing round the two sides of a rock. A careful observer can in this school of Nature instruct himself in all the laws of wave-motion, and gather a fund of knowledge on this subject during an hour’s dalliance at low tide on some sandy coast, or in the quiet study of seaside pools, the surface of which is corrugated with trains of ripples by the breeze.