WAVES AND RIPPLES IN WATER,
AIR, AND ÆTHER.
—⋄—
WATER WAVES AND WATER RIPPLES.
WE have all stood many times by the seashore, watching the waves, crested with white foam, roll in and break upon the rocks or beach. Every one has more than once cast a stone upon still water in a lake or pond, and noticed the expanding rings of ripples; and some have voyaged over stormy seas, whereon great ships are tossed by mighty billows with no more seeming effort than the rocking of a cradle. In all these things we have been spectators of a wave-motion, as it is called, taking place upon a water surface. Perhaps it did not occur to us at the time that the sound of the splash or thunder of these breaking waves was conveyed to our ears as a wave-motion of another sort in the air we breathe, nay, even that the light by which we see these beautiful objects is also a wave-motion of a more recondite description, produced in a medium called the æther, which fills all space.
A progressive study of Nature has shown us that we are surrounded on all sides by wave-motions of various descriptions—waves in water, waves in air, and waves in æther—and that our most precious senses, our eyes and ears, are really wave-detectors of a very special form. The examination of these waves and their properties and powers has led us to see that waves in water, air, and æther, though differing greatly in detail, have much in common; and many things about them that are difficult to understand become more intelligible when we compare these various wave-motions together. In these lectures, therefore, I shall make use of your familiar experiences concerning sea and water waves to assist you to understand some of the properties of air waves to which we owe our sensations of sound and music; and, as far as possible, attempt an explanation of the nature of æther waves, created in the all-pervading æther, to which are due not only light and sight, but also many electrical effects, including such modern wonders as wireless telegraphy. In all departments of natural science we find ourselves confronted by the phenomena of wave-motion. In the study of earthquakes and tides, telegraphs and telephones, as well as terrestrial temperature, no less than in the examination of water waves and ripples, sound, music, or light and heat, we are bound to consider waves of some particular kind.
Fastening our attention for the moment on surface water waves, the first question we shall ask ourselves is—What is a wave? If we take our station on a high cliff looking down on the sea, on some clear day, when the wind is fresh, we see the waves on its surface like green rounded ridges racing forward, and it appears at first sight as if these elevations were themselves moving masses of water. If, however, we look instead at some patch of seaweed, or floating cork, or seagull, as each wave passes over it, we shall notice that this object is merely lifted up and let down again, or, at most, has a small movement to and fro. We are led, therefore, to infer that, even when agitated by waves, each particle of water never moves far from its position when at rest, and that the real movement of the water is something very different from its apparent motion. If we place on the surface of water a number of corks or pieces of paper, and then watch them as a wave passes over them, we shall notice that the corks or bits of paper rise and fall successively, that is, one after the other, and not all together. A little more careful scrutiny will show us that, in the case of sea waves in deep water, the motion of the floating object as the wave passes over it is a circular one, that is to say, it is first lifted up, then pushed forward, next let down, and, lastly, pulled back; and so it repeats a round-and-round motion, with the plane of the circle in the direction in which the wave is progressing. This may be illustrated by the diagram in Fig. 1, where the circular dotted lines represent the paths described by corks floating on the sea-surface when waves are travelling over it.
Accordingly, we conclude that we have to distinguish clearly between the actual individual motion of each water particle and that general motion called the wave-motion. We may define the latter by saying that to produce a wave-motion, each separate particle of a medium, be it water, or air, or any other fluid, must execute a movement which is repeated again and again, and the several particles along any line must perform this same motion one after the other, that is, lagging behind each other, and not simultaneously. We might illustrate this performance by supposing a row of fifty boys to stand in a line in a play-ground, and each boy in turn to lift up his arm and let it down again, and to continue to perform this action. If all the boys lifted up their arms together, that would not produce a wave-motion; but if each boy did it one after the other in order, along the rank, it would constitute a wave-motion travelling along the line of boys. In more learned language, we may define a wave-motion by saying that a wave-motion exists in any medium when the separate portions of it along any line execute in order any kind of cyclical or repeated motion, the particles along this line performing the movement one after the other, and with a certain assigned delay between each adjacent particle as regards their stage in the movement.
It will be evident, therefore, that there can be many different kinds of waves, depending upon the sort of repeated motion the several parts perform.
Some of the numerous forms of wave-motion can be illustrated by mechanical models as follows:—
A board has fastened to it a series of wooden wheels, and on the edge of each wheel is fixed a white knob. The wheels are connected together by endless bands, so that on turning one wheel round they all revolve in the same direction. If the knobs are so arranged to begin with, that each one is a little in advance of its neighbour on the way round the wheel, then when the wheels are standing still the knobs will be arranged along a wavy line (see Fig. 2). On turning round the first wheel, each knob will move in a circle, but every knob will be lagging a little behind its neighbour on one side, and a little in advance of its neighbour on the other side. The result will be to produce a wave-motion, and, looking at the general effect of the moving knobs, we shall see that it resembles a hump moving along, just as in the case of a water wave.
The motion of the particles of the water in a deep-sea wave resembles that of the white knobs in the model described. Those who swim will recall to mind their sensations as a sea wave surges over them. The wave lifts up the swimmer, then pushes him a little forward, then lets him down, and, lastly, drags him back. It is this dragging-back action which is so dangerous to persons who cannot swim, when they are bathing on a steep coast where strong waves are rolling in towards the shore.
Two other kinds of wave-motion may be illustrated by the model shown in Fig. 3. In this appliance there are a number of eccentric wheels fixed to a shaft. Each wheel is embraced by a band carrying a long rod which ends in a white ball. The wheels are so placed on the shaft that, when at rest, the balls are arranged in a wavy line. Then, on turning round the shaft, each ball rises and falls in a vertical line, and executes a periodic motion, lagging behind that of its neighbour on one side. The result is to produce a wave-motion along the line of balls. By slightly altering the model, each ball can be made to describe a circle in a direction at right angles to the line of the balls, and then we have a sort of corkscrew wave-motion propagated along the line of balls.
Again, another form of wave-motion may be illustrated by the model shown in Fig. 4. In this case a number of golf-balls are hung up by strings, and spiral brass springs are interposed between each ball. On giving a slight tap to the end ball, we notice that its to-and-fro motion is handed on from ball to ball, and we have a wave-motion in which the individual movement of the balls is in the direction of the wave-movement, and not across it.
The kind of wave illustrated by the model in Fig. 3 is called a transverse wave, and that shown in Fig. 4 is called a longitudinal wave.
At this stage it may be well to define the meaning of some other expressions which will be much used in these lectures. We have seen that in a wave-motion each part of the medium makes some kind of movement over and over again; and of its neighbours on either side, one is a little ahead of it in its performance, and the other a little in arrear. If we look along the line, we shall see that we can select portions of it which are exactly in the same stage of movement—that is, are moving in the same way at the same time. The distance between these portions is called one wave-length. Thus, in the case of sea waves, the distance between two adjacent crests, or humps, is one wave-length.
When we use the expression, a long wave, we do not mean a wave which is of great length in the direction of the ridge, but waves in which the crests, or humps, are separated far apart, measuring from crest to crest across the ridges.
Strictly speaking, the wave-length may be defined as the shortest distance from crest to crest, or hollow to hollow, or from one particle to the next one which is in the same stage of its movement at the same time.
Another way of illustrating the same thing would be to pleat or pucker a sheet of paper into parallel ridges. If we make these pleats very narrow, they would represent what we call short waves; but if we make these pleats very far apart, they would represent long waves.
Another phrase much used is the term wave-velocity. Suppose that a seagull were to fly along over a set of sea waves so as to keep always above one particular hump, or wave-crest; the speed of the gull, reckoned in miles per hour or feet per minute, would be called the speed of the waves. This is something very different from the actual speed of each particle of water.
A third and constantly used expression is the term wave-frequency. If we watch a cork floating on a wave-tossed sea, we observe that it bobs up and down so many times in a minute. The number of times per second or per minute that each particle of the medium performs its cycle of motion is called the wave-frequency, or simply the frequency.
Again, we employ the term amplitude to denote the extreme distance that each individual particle of the medium moves from its mean position, or position of rest. In speaking of sea waves, we generally call the vertical distance between the crest and the hollow the height of the wave, and this is twice the amplitude. With regard to the height of sea waves, there is generally much exaggeration. Voyagers are in the habit of speaking of “waves running mountains high,” yet a sea wave which exceeds 40 feet in height is a rare sight. Waves have been measured on the Southern Indian Ocean, between the Cape of Good Hope and the Island of St. Paul, and of thirty waves observed the average height was found to be just under 30 feet. The highest was only 37¹⁄₂ feet in height. On the other hand, waves of 16 to 20 feet are not uncommon. Travellers who have crossed the Atlantic Ocean in stormy weather will often recount experiences of waves said to be 100 feet high; but these are exceedingly rare, if even ever met with, and unless wave-heights are obtained by some accurate method of measurement, the eye of the inexperienced voyager is apt to be deceived.
In all cases of wave-motion there is a very close connection between the wave-velocity, or speed, the wave-length, and the wave-frequency. This connection is expressed by the numerical law that the velocity is equal to the product of the length and the frequency.
Thus, supposing we consider the case of Atlantic waves 300 feet from crest to crest, which are travelling at the rate of 27 miles an hour, it is required to calculate the frequency or number of times per minute or per second that any floating object, say a boat, will be lifted up as these waves pass over it.
We must first transform a speed of 27 miles per hour into its equivalent in feet per second. Since one mile is 5280 feet, 27 miles per hour is equal to 2376 feet per minute. Accordingly, it is easy to see that the wave-frequency must be 7·92, or nearly 8, because 7·92 times 300 is 2376. The answer to the question is, then, that the floating object will rise and fall eight times a minute. This rule may be embodied in a compact form, which it is desirable to hold firmly in the memory, viz.—
Wave-velocity = wave-length × wave-frequency.
This relation, which we shall have frequent occasion to recall, may be stated in another manner. We call the period of a wave the time taken to make one complete movement. The periodic time is therefore inversely proportional to the frequency. Hence we can say that the wave-length, divided by the periodic time, gives us the wave-velocity.
In the case of water waves and ripples, the wave-velocity is determined by the wave-length. This is not the case, as we shall see, with waves in air or waves in æther. In these latter cases, as far as we know, waves of all wave-lengths travel at the same rate. Long sea waves, however, on deep water travel faster than short ones.
A formal and exact proof of the law connecting speed and wave-length for deep-sea waves requires mathematical reasoning of an advanced character; but its results may be expressed in a very simple statement, by saying that, in the case of waves on deep water, the speed with which the waves travel, reckoned in miles per hour, is equal to the square root of 2¹⁄₄ times the wave-length measured in feet. Thus, for instance, if we notice waves on a deep sea which are 100 feet from crest to crest, then the speed with which those waves are travelling, reckoned in miles per hour, is a number obtained by taking the square root of 2¹⁄₄ times 100, viz. 225. Since 15 is the square root of 225 (because 15 times 15 is 225), the speed of these waves is therefore 15 miles an hour.
In the same way it can be found that Atlantic waves 300 feet long would travel at the rate of 26 miles an hour, or as fast as a slow railway train, and much faster than any ordinary ship.[1]
The above rule for the speed of deep-sea waves, viz. wave-velocity = square root of 2¹⁄₄ times the wave-length, combined with the general rule, wave-velocity = wave-length multiplied by frequency, provides us with a useful practical method of finding the speed of deep-sea waves which are passing any fixed point. Suppose that a good way out at sea there is a fixed buoy or rock, and we notice waves racing past it, and desire to know their speed, we may do it as follows: Count the number of waves which pass the fixed point per minute, and divide the number into 198; the quotient is the speed of the waves in miles per hour. Thus, if ten waves per minute race past a fixed buoy, their velocity is very nearly 20 miles an hour.[2]
Waves have been observed by the Challenger 420 to 480 feet long, with a period of 9 seconds. These waves were 18 to 22 feet high. Their speed was therefore 50 feet per second, or nearly 30 knots. Atlantic storm waves are very often 500 to 600 feet long, and have a period of 10 to 11 seconds. Waves have been observed by officers in the French Navy half a mile in length, and with a period of 23 seconds.
It has already been explained that in the case of deep-sea waves the individual particles of water move in circular paths. It can be shown that the diameter of these circular paths decreases very rapidly with the depth of the particle below the surface, so that at a distance below the surface equal only to one wave-length, the diameter of the circle which is described by each water-particle is only ¹⁄₅₃₅ of that at the surface.[3] Hence storm waves on the sea are a purely surface effect. At a few hundred feet down—a distance small compared with the depth of the ocean—the water is quite still, even when the surface is tossed by fearful storms, except in so far as there may be a steady movement due to ocean currents.
By a more elaborate examination of the propagation of wave-motion on a fluid, Sir George Stokes showed, many years ago, that in addition to the circular motion of the water-particles constituting the wave, there is also a transfer of water in the direction in which the wave is moving, the speed of this transfer depending on the depth, and decreasing rapidly as the depth increases. This effect, which is known to sailors as the “heave of the sea,” can clearly be seen on watching waves on not very deep water. For the crest of the wave will be seen to advance more rapidly than the hollow until the wave falls over and breaks; and then a fresh wave is formed behind it, and the process is repeated. Hence waves break if the depth of water under them diminishes; and we know by the presence of breakers at any place that some shallow or sandbank is located there.
It is necessary, in the next place, to point out the difference between a mere wave-motion and a true wave. It has been explained that in a wave-motion each one of a series of contiguous objects executes some identical movement in turn. We have all seen the wind blowing on a breezy day across a cornfield, and producing a sort of dark shadow which sweeps along the field. This is clearly caused by the wind bending down, in turn, each row of cornstalks, and as row after row bows itself and springs up again, we are presented with the appearance of a wave-motion in the form of a rift rushing across the field.
A very similar effect can be produced, and another illustration given of a wave-motion, as follows: Coil a piece of brass wire into an open spiral like a corkscrew, and affix to it a small fragment of sealing-wax (see Fig. 5). Hold this in the sun, and let the shadow of it fall upon paper. Then turn it round like a screw. We shall see that the shadow of the spiral is a wavy line, and that, as it is turned round, the humps appear to move along just as do the crests of sea waves, but that the shadow of the little bit of sealing-wax simply moves up and down.
Another wave-motion model may be made as follows: Procure a painter’s comb. This is a thin steel plate, cut into long narrow teeth. Provide also a slip of glass about 3 inches wide and 12 inches long. Paint one side of this glass with black enamel varnish, and when it is quite dry scratch a wavy line upon it (see Fig. 6). Place the glass slip close in front of the comb before the light, and, holding the comb still, move the glass slip to and fro, lengthways. The observer will see a row of dots of light lying in a wavy line, and these, as the glass moves, will rise and fall. If the movement is rapid enough, the appearance of a wave moving along will be seen.[4] In all these exhibitions of wave-motion the movement of the particles is due to a common cause, but the moving particles do not control each other’s motion. There is no connection or tie between them. Suppose, however, that we suspend a series of heavy balls like pendulums, and interconnect them by elastic threads (see Fig. 7), then we have an arrangement along which we can propagate a true wave. Draw the end ball to one side, and notice what takes place when it is released. The first ball, being displaced, pulls the second one through a less distance, and that the third one, and the third the fourth, and so on. This happens because the balls are tied together by elastic threads, which resist stretching. When the first ball is released, it is pulled back by the tension of the thread connecting it to its neighbours, and it begins to return to its old position. The ball possesses, however, a quality called inertia, and accordingly, when once set in motion, its motion persists until an opposing force brings it to rest. Hence the returning ball overshoots the mark, and passes to the opposite side of its original position of rest. Then, again, this displacement stretches the elastic threads connecting it to its fellows, and a controlling or retarding force is thus created, which brings it to rest, and forces it again to return on its steps. We see, therefore, that each ball must oscillate, or swing to and fro, and that its movement is gradually communicated to its neighbours. A wave-motion is thus started, and a true wave is propagated along the line of balls, in consequence of the presence of elasticity and inertia. The necessary conditions for the production of a true wave in a medium of any kind are therefore: (1) that the medium must elastically resist some sort of deformation; and (2) when it is deformed at any place, and returns to its original state, it must overshoot the mark or persist in movement, in consequence of inertia, or something equivalent to it.
Briefly speaking, any material or medium in or on which a true self-propagating wave-motion can be made must resist and persist. It must have an elastic resistance to some change or deformation, and it must have an inertia which causes it to persist in movement when once set in motion. These two qualities, or others equivalent to them, must invariably be present if we are to have a true wave produced in a medium.
These things may be best understood by considering, for example, the production of surface waves on water. Let us ask ourselves, in the first place, what alteration or change it is that a water-surface resists. The answer is, that, for one thing, it resists being made unlevel. A still water surface is everywhere a level surface. If we attempt to make it unlevel by pouring water on to it at one point, or by heaping it up, the water surface would resist this process. We can dig a hole in sand, or heap up sand to form a hillock, but we know full well we cannot do the same thing with water. If, for instance, some water is placed in a glass tube shaped like the letter ⋃, then it stands at the same level in both limbs. Again, if water is set in motion, being a heavy substance, it cannot be brought to rest instantly. Like every other body, it possesses inertia. Accordingly, if we do succeed by any means in making a depression in a water-surface for an instant, the water would immediately press in to fill up the hole; but more, it would, so to speak, overshoot the mark, and, in consequence of its inertia, it would create a momentary hump, or elevation, in the place on the surface where an instant ago there was a depression.
This elevation would again subside into a hollow, and the process would be continued until the water-motion was brought to rest by friction, or by the gradual dispersion of the original energy. The process by which a wave is started on the surface of water, as a consequence of these two qualities of resistance to being made unlevel and persistence in motion, is beautifully shown by the study of waves made by throwing stones into a pond. The events which give rise to the expanding wave are, however, over so quickly that they can only be studied by the aid of instantaneous photography. The most interesting work on this subject is that of Professor A. M. Worthington, who has photographed, by the exceedingly brief light of an electric spark, the various stages of the events which happen when a drop of water or a stone falls into water.[5] These photographs show us all that happens when the falling object touches the water, and the manner in which it gives rise to the wave or ripple which results. Some of Professor Worthington’s results for a drop of water falling into milk are reproduced in the appended diagrams. In the first place (Fig. 8) the drop is seen just entering the water. As it plunges down, it leaves behind it a cavity, or, as it may be called, a hole in the water (see Fig. 9).
This hole, at a certain stage, begins to fill up. The water rushes in on all sides, and the impetus carries up the inrushing water so that it builds up a tall pillar of water in the place where an instant ago there was a hole (see Fig. 10). No one could anticipate such an extraordinary effect; but the instantaneous photographs, taken by the light of an electric spark, which reveal it, cannot but be truthful.
The next stage is that this pillar of water breaks up, and falls back again on the surface. Hence the water, at the place where the drop plunges into it, is subjected to two violent impulses—a downward, succeeded by an up-lifting, force. The effect of this is exactly analogous to that of giving a blow to the interconnected string of balls in the model shown in Fig. 7—it propagates a wave. In Fig. 10 is illustrated the next stage, in which this outward-moving initial wave-crest is shown.
So much for the events revealed by the flash-light of an electric spark; but succeeding these there is a long train of interesting wave-making performance which can be watched with the eye, or the stages photographed with a hand camera. This wave production is best seen when a large stone is thrown into calm water in a lake or pond.
A story is recorded of the great artist Turner, that he once spent a morning throwing stones into a pond. A friend reproved him for his idleness. “No,” said the painter, “I have not been idle; I have learnt how to paint a ripple.” If the artist’s eye has to be carefully trained to notice all that there is to see when a stone is hurled into a pond, it is not strange that a careless observer cannot grasp at once what really happens to the water in this ordinary occurrence.
The photograph in Fig. 11 will, however, show one stage in the event. As soon as the first wave-crest, the origin of which we have already explained, is formed, it begins to move outwards in a circular form, and as it moves it gives rise to a wave-train, that is, it multiplies itself into a series of concentric ripples, or waves, which move outwards, multiplying in number, but getting smaller as they move.
Thus if a large stone is thrown far out into a deep, still lake, after the first splash we shall see a circular wave spreading out from the place where the commotion was made in the water. As we look at this wave we shall see it growing in size and multiplying itself. At first there is but a single wave, then two, four, seven, ten, or more concentric ripples are seen, each circular wave expanding and getting feebler, but seeming to give birth to others as it moves. Moreover, a very careful examination will show us that the whole group of waves, or the wave-train, has an outward motion with a less speed than any individual wave. This observation will serve to initiate the conceptions of a wave-train and of a wave-group velocity. At first it is difficult to understand that a group of waves may move more slowly than the individual waves which compose it. If, however, we cast a stone into a pond, and look very carefully at what takes place, we shall see that the circular expanding band of ripples has an ill-defined but visible inner and outer edge, and that wavelets or ripples which compose it are being continually brought into existence on its inner edge, and dying away on its outer edge. Waves, so to speak, pass through the ripple band with a greater speed than that at which the whole band of waves moves forward. This rather difficult, but important, idea of the distinction between the velocity of a group of waves and that of an individual wave was first suggested by Sir George Stokes, who set a question in a Cambridge Examination on the subject in 1876, and subsequently it was elucidated by Professor Osborne Reynolds[6] and Lord Rayleigh.
It can be further explained as follows: Let us consider a wave-motion model such as that represented in Fig. 7, in which a number of suspended heavy balls are connected to one another by elastic threads. Let one ball in the centre be drawn on one side and then released. It will swing to and fro, and will start a wave outwards in both directions. If the row of balls is sufficiently long, it will be seen that the ball by which the wave was started soon comes to rest, and that the wave-motion is confined to a certain group of balls on either side. As time goes on, the wave-motion in each group dies away on the side nearest the origin, and extends on the side furthest away. Hence the particular group of balls which are the seat of the visible wave-motion is continually being shifted along. The rate at which the centre of this active group of vibrating balls is displaced may be called the velocity of the wave-train. The velocity of the wave is, however, something greater, since the waves are all the time moving through the group. This wave-velocity is numerically estimated by taking the product of the wave-length and frequency of the motion.
At this stage it is necessary to explain that waves are not merely a mode of motion; they are a means of conveying energy. It is difficult to give in a compact form any simple definition of what is meant in modern scientific writings by the word Energy.
Briefly speaking, we may say that there are two fundamental agencies or things in Nature with which we are in contact, manifesting themselves in many different forms, but of which the total quantity is unchangeable by human operations. One of these is called Matter. This term is the collective name given to all the substance or stuff we can see or touch, and which can be weighed or has weight. All known solids, liquids, or gases, such things as ice, water, steam, iron, oil, or air, are called material substances, and they have in common the two qualities of occupying space or taking up room, and of having weight. Experiment has shown that there are some eighty different kinds of simple matter which cannot be transformed into each other, and these forms are called the Elements. Any other material substance is made up of mixtures or combinations of these elements. The elementary substances are therefore like the letters of the alphabet, which, taken in groups, make up words, these last corresponding to compound chemical bodies. Exact research has shown that no chemical changes taking place in a closed space can alter the total weight or amount of gravitating matter in it. If a chemist and numerous chemicals were enclosed in a great glass ball, and the ball balanced on a gigantic but very sensitive pair of scales, no operations which the chemist could conduct in the interior of his crystal laboratory would alter, by the ten-thousandth part of a grain, the total weight of it all. He might analyze or combine his chemicals, burn or mix them as he pleased, but as long as nothing entered or escaped from the ball, the total gravitating mass would remain precisely the same. This great fact is called the Law of Conservation of Matter, and it teaches us that although a scuttle of coal may seem to disappear when burnt, yet the weight of the ashes and of all the gaseous products of combustion are together equal to the weight of the original coal and the air required to burn it.
In addition to various material substances we find that we have to recognize different forms of something called Energy, associated with Matter. Thus an iron ball may be more or less hot, more or less electrified or magnetized, or moving with more or less speed. The production of these states of heat, electrification, magnetization, or movement, involves the transfer to the iron of Energy, and they are themselves forms of Energy. This Energy in all its various forms can be evaluated or measured in terms of Energy of movement. Thus the Energy required to heat a ball of iron weighing one imperial pound from a temperature of the melting point of ice to that of boiling water, is nearly equal to the Energy required to impart to it a speed of 1000 feet a second.
In the same way, every definite state of electrification or magnetization can be expressed in its mechanical equivalent, as it is called. Moreover, it is found that we can never create any amount of heat or mechanical motion or other form of energy without putting out of existence an equivalent of energy in some other form. We are therefore compelled to consider that Energy stands on the same footing as Matter in regard to our inability to create or destroy it, and its constancy in total amount, as far as we can ascertain, gives it the same character of permanence. The difference, however, is that we cannot, so to speak, ear-mark any given quantity of energy and follow it through all its transformations in the same manner in which we can mark and identify a certain portion of Matter. The moment, however, that we pass beyond these merely quantitative ideas and proceed to ask further questions about the nature of Energy and Matter, we find ourselves in the presence of inscrutable mysteries. We are not able as yet to analyze into anything simpler this “something” we call Energy which presents itself in the guise of heat or light, electricity or magnetism, movement or chemical action. It is protean in form, intangible, yet measurable in magnitude, and all its changes are by definite equivalent amount and value. There is a most rigid system of book-keeping in the transactions of the physical universe. You may have anything you like in the way of Energy served out to you, but the amount of it is debited to your account immediately, and the bill has to be discharged by paying an equivalent in some other form of Energy before you can remove the goods from the counter.
Matter in its various forms serves as the vehicle of Energy. We have no experience of Energy apart from Matter of some kind, nor of Matter altogether devoid of Energy. We do not even know whether these two things can exist separately, and we can give no definition of the one which does not in some way presuppose the existence of the other. Returning, then, to the subject of waves, we may say that a true wave can only exist when Energy is capable of being associated with a medium in two forms, and the wave is a means by which that Energy is transferred from place to place.
It has already been explained that a true wave can only be created in a medium which elastically resists some kind of deformation, and persists in motion in virtue of inertia. When any material possesses such a quality of resistance to some kind of strain or deformation of such a character that the deformation disappears when the force creating it is withdrawn, it is called an elastic material. This elasticity may arise from various causes. Thus air resists being compressed, and if the compressing force is removed the air expands again. It possesses so-called elasticity of bulk. In the case of water having a free surface there is, as we have seen, a resistance to any change of level in the surface. This may be called an elasticity of surface form. Whenever an elastic material is strained or deformed, energy has to be expended on it to create the deformation. Thus to wind up a watch-spring, stretch a piece of indiarubber, compress some air, or bend a bow, requires an energy expenditure.
As long as the material is kept strained, it is said to have potential energy associated with it. This term is not a very expressive one, and it would be better to call it Energy of strain, or deformation. If, however, we relax the bent bow or release the compressed air, the Energy of Strain disappears, and we have it replaced by Energy of Motion. The arrow which flies from a bow carries with it, as energy of motion, some part of the energy of strain associated with the bent bow.
A little examination of wave-motion shows us, therefore, that we always have at any instant associated with the material in which the wave is being propagated, both Energy of Strain and Energy of Motion. It can be shown that in a true wave of permanent type, the whole energy at any one moment is half energy of strain and half energy of motion, or, as it is called, half potential and half kinetic.
Thus if we consider a wave being propagated along a line of balls elastically connected, at any one moment some of the balls are moving with their greatest velocity, and some are at the extremity of their swing. The former have energy of motion, and the latter energy of strain.
Or, look at a train of sea waves. Some parts of the water are at any moment lifted high above the average level of the sea, or are much below it, but are otherwise nearly at rest. These portions possess what is called potential energy, or energy of position. Other parts of the water are at the average level of the sea, but are moving with considerable velocity, and these portions possess energy of motion. Every other part of the wave has in some degree both energy of motion and energy of position, and it can be shown that the energy of the whole wave is half of one kind and half of the other.
As a wave progresses over the surface, wave-energy is continually being imparted to portions of the water in front, and it is transferred away from others in the rear. In the very act of setting a fresh particle of water in oscillation, the portions already vibrating must diminish their own motion. They may hand on the whole of their energy or only a part of it to their neighbours. This distinction is a very important one, and it determines whether a single act of disturbance shall create a solitary wave or wave-train in a medium.
The difference may be illustrated as follows: Consider a row of glass or steel balls suspended by threads so hung as to be quite close to each other (see Fig. 12). Withdraw the first ball, and let it fall against the second one. The result is that the last ball of the row flies off with a jerk. In this case the whole energy imparted to the first ball is transmitted along the row of balls. The first ball, on falling against the second one, exerts on it a pressure which slightly squeezes both out of shape. This pressure is just sufficient to bring the first ball to rest. The second ball, in turn, expands after the blow and squeezes the third, and so on. Hence, in virtue of Newton’s Third Law of Motion, that “action and reaction are equal and opposite,” it follows that the pressure produced by the blow of the first ball is handed on from ball to ball, and finally causes the last ball to fly off.
In this case, owing to the rigid connection between the elastic balls, each one hands on to its neighbour the whole of the energy it receives. Supposing, however, that we separate the balls slightly, and give the first ball a transverse, or side-to-side swing. Then, owing to the fact that there is no connection between the balls, the energy imparted to the first ball would not be handed on at all, and no wave would be propagated.
Between these two extremes of the whole energy transferred and a solitary wave produced, and no energy transferred and no wave produced, we have a condition in which an initial disturbance of one ball gives rise to a wave-train and part of the energy is transferred.
For if we interconnect the balls by loose elastic threads, and then give, as before, a transverse or sideways impulse to the first ball, this will pull the second one and set it swinging, but it will be pulled back itself, and will be to some extent deprived of its motion. The same sharing or division of energy will take place between the second and third, and third and fourth balls, and so on. Hence the initial solitary vibration of the first ball draws out into a wave-train, and the originally imparted energy is spread out over a number of balls, and not concentrated in one of them. Accordingly, as time goes on, the wave-train is ever extending in length and the oscillatory motion of each ball is dying away, and the original energy gets spread over a wider and wider area or number of balls, but is propagated with less speed than the wave-velocity for that medium.
There need be no difficulty in distinguishing between the notion of a wave-velocity and a wave-train velocity, if we remember that the wave travels a distance equal to a wave-length in the time taken by one oscillation. Hence the wave-velocity is measured by taking the quotient of the wave-length by the time of one complete vibration.
If, for example, the wave-length of a water wave is 4 inches, and we observe that twelve waves pass any given point in 3 seconds, we can at once infer that the wave-velocity is 16 inches per second. The transference of energy may, however, take place so that the whole group of waves moves forward much more slowly. They move forward because the waves are dying out in the rear of the group and being created in the front, and the rate of movement of the group is, in the case of deep-water waves, equal to half that of the single-wave velocity.
A very rough illustration of this difference between a group velocity and an individual velocity may be given by supposing a barge to be slowly towed along a river. Let a group of boys run along the barge, dive over the bows, and reappear at the stern and climb in again. Then the velocity of the group of boys on the barge is the same as the speed of the barge, but the speed of each individual boy in space is equal to the speed of the barge added to the speed of each boy relatively to the barge. If the barge is being towed at 3 miles an hour, and the boys run along the boat also at 3 miles an hour, then the velocity of the group of boys is only half that of the individual boy, because the former is 3 miles an hour and the latter is 6 miles an hour.
Before leaving the subject of sea waves there are two or three interesting matters which must be considered. In the first place, the breaking of a wave on the shore or on shallow water calls for an explanation. If we watch a sea wave rolling in towards the beach, we shall notice that, as it nears the shore, it gets steeper on the shore side, and gradually curls over until it falls and breaks into spray. The reason is because, as the wave gets into the shallow water, the top part of the wave advances more rapidly than the bottom portion. It has already been explained that the path of the water-particle is a circle, with its plane vertical and perpendicular to the wave-front or line.
Accordingly, if the wave is moving in shallow water, the friction of the water against the bottom retards the backward movement at the lowest position of the water, but no such obstacle exists to the forward movement of the water at its highest position. An additional reason for the deformation of the wave on a gently sloping shore may be found in the fact that the front part of the wave is then in shallower water, and hence moves more slowly than the rearward portion in deeper water. From both causes, however, the wave continually gets steeper and steeper on its landward side until it curls over and tumbles down like a house which leans too much on one side. The act of curling over in a breaking wave is a beautiful thing to watch, and one which attracts the eye of every artist who paints seascapes and storm waves, or of any lover of Nature who lingers by the shore.
Another matter of interest is the origin of sea waves. Undoubtedly they are due originally to the action of the wind upon the water. Whenever two layers of fluid lie in contact with each other, and one moves faster than the other, the faster-moving layer will throw the other into waves. This is seen, not only in the action of moving air or wind upon water, but even in the action of air upon air or water upon water. From the tops of high mountains we may sometimes look down upon a flat surface of cloud beneath. On one occasion the author enjoyed a curious spectacle from the summit of an Alpine peak. The climb up had been through damp and misty air, but on reaching the summit the clouds were left behind, and a canopy of blue sky and glorious sunshine were found overhead. Beneath the clouds lay closely packed like a sea of white vapour, and through this ocean of cloud the peaks of many high mountains projected and stood up like islands. The surface of this sea of white cloud, brilliantly illuminated by the sunshine, was not, however, perfectly smooth. It was tossed into cloud waves and billows by the action of currents of air blowing over its upper surface, and it had a striking resemblance to the surface of a rough sea. When such a cloud layer is not too thick, the ruffling of its upper or under surface into cloud waves may thin it away into regular cloud rolls, and these cloud rollers may then be cut up again by cross air-currents into patches, and we have the appearance known as a “mackerel sky.”
Another familiar phenomenon is that known as the “ripple-mark” on wet sand. As the tide ebbs out over a smooth bank of sea-sand, it leaves the surface ploughed into regular rounded ridges and furrows, which are stationary waves on the sand. This is called the ripple-mark. It is due to the fact that the sand, when covered by the water, forms a surface which in a certain sense is fluid, being saturated and filled with water, but the movement of this bottom sand-logged water is hindered by the sand, and hence the layer of overlying water moves over it at a different speed in ebbing out, and carves it into what are virtually sand waves.
Even a dry sand or snow surface may in this manner be moulded into a wave-form by the wind, and very curious effects of this kind have been noticed and described by Dr. Vaughan Cornish, who has made a great study of the science of waves.[7]
The production of waves on water by means of a current of air blowing over it is easily exhibited on a small scale by blowing through an indiarubber pipe, the end of which is held near the surface of the water in a tub or tank. The exact manner in which the moving air gets a grip of the water is not quite plain, but it is clear that, if once an inequality of level is set up, the moving air has then an oblique surface against which it can press, and so increase the inequality by heaping up the water in some places, and hollowing it out in others.
Hence oscillations of the water-surface are set up, which go on accumulating. These waves then travel away with a speed depending upon their wave-length, and we may have great disturbances of the sea-surface at places where there is no actual storm-wind. These “echoes of a far-off storm” are known as a “ground swell.” In some localities the inhabitants are able to apprise themselves of the coming of a storm by noticing movements of the sea which indicate the arrival of waves which have travelled more quickly than the storm-centre itself.
Every visitor to the seaside will have noticed occasions on which the sea is violently disturbed by waves, and yet the air in the locality is tolerably calm. In this case the waves have been propagated from some point of disturbance at a distance.
A study of breaking waves shows us that the cause of their great power to effect damage to coast structures, such as piers, harbour works, and shipping in harbours, is really due to the forward motion of the water as the wave is breaking. Every cubic foot of water weighs 63¹⁄₂ lbs., so that a cubic yard of water weighs about three-quarters of a ton. If this water is moving with a speed of many feet per second in a forward direction, the energy of motion stored up in it is tremendous, and fully sufficient to account for the destructive power of storm waves on a coast.
The total volume of water which is comprised in the space occupied by even one sea-storm wave of moderate dimensions may have a mass of many hundreds of tons, and its energy of motion may easily amount to that of an express train in motion. Hence when, in the last stage of its career, this mass of water is hurled forward on the shore, its destructive effects are not a matter for surprise.
We must now leave the subject of waves in the open sea on a large level surface, and consider that of waves in narrow channels, such as canals or rivers. The laws which govern water-wave production in a canal can best be studied by placing some water in a long tank with glass sides. If at one end we insert a flat piece of wood and give it a push forward, we shall start what is called a long wave in the tank. The characteristic of this kind of wave is that the oscillatory motion is chiefly to-and-fro, and not up-and-down. This may be very easily seen by placing some bran in the water, or floating in it some glass balls which have been adjusted so as to just float anywhere in the water. When this is done, and a wave started in the tank, it runs up and down, being reflected at each end (see Fig. 13).