Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/445

WAVE and that part at least of the flash is clue to the heat developed by practically instantaneous and very great compression of each layer of air to which this violent motion extends. (5) Gravitation and Surface-Tension Waves in Liquids. Leaving out of consideration, as already sufficiently treated in a special article, the whole subject of TIDES, whether in oceans or in tidal rivers, there remain many different forms of water-waves all alike interesting and important. The most usual division of the free waves is into long waves, oscillatory waves, and ripples. The first two classes run by gravity, the third mainly by surface- tension (see CAPILLARY ACTION). But, while the long waves agitate the water to nearly the same amount at all depths, the chief disturbance due to oscillatory waves or to ripples is confined to the upper layers of the water, from which it dies away with great rapidity in successive layers below. We will treat of these three forms in the order named. (6) Long Waves. The first careful study of these waves was made by Scott RUSSELL (q.v.) in the course of an inquiry into traffic on canals. He arrived at the remarkable result that there is a definite speed, depending on the depth of the water, at which a horse can draw a canal-boat more easily than at any other speed, whether less or greater. And he pointed out that, when the boat moves at this speed, it agitates the water less, and there- fora damages the banks less, than at any lower. This particular speed is thus, in fact, that of free propagation of the wave raised by the boat ; and, when the boat rides, as it were, on this wave, its speed is maintained with but little exertion on the part of the horse. If the boat be made to move slower, it leaves behind it an ever- lengthening procession of waves, of course at the expense of additional labour on the part of the horse. The theory of the motion of such, a wave is based on the hypothesis that all particles in a transverse section of the canal have, at the same instant, the same horizontal speed. However great this horizontal motion may be, the vertical motion of the water may be very small, for it depends on the change of horizontal speed from section to section only. In the investigation which follows, the energy of this vertical motion will be neglected (even at the surface, where it is greatest) in comparison with that of the horizontal motion. The hypothesis is proved to be well grounded by the actual observation of the motion of the water when a long wave of slight elevation or depression passes. A long box, with parallel sides of glass, partly filled with water, represents the canal ; and the wave is produced by slowly and slightly tilting the box, and at once restoring it to the horizontal position. The nature of the motion of the water is shown by particles of bran suspended in it. Such an apparatus may be usefully employed in verifying the theoretical result below, as to the connexion between the speed of the wave and the depth of the water, observations of the passage of the crest being made with great exactness by means of a ray of light reflected from the surface of the water in a vertical plane parallel to the length of the canal. It may also be employed, by tilting it about an inclined position, for the study of the changes which take place in the wave as it passes from deeper to shallower water, or the reverse. The statement of (1) above is immediately applicable to this question. For, if h be the (undisturbed) depth of the water, p its density, y and y the elevations in two succes sive transverse sections at unit distance from one another, the difference of pressures (at the same level) in the two sections is gp(y - //). The acceleration of a horizontal cylinder of unit section is the difference of pressures divided by p. But the whole depth is increased at each point in proportion as the thickness of a transverse slice is diminished. Hence, by the reasoning in (2) above, ffp(y -y). p p -p h and the speed of propagation of the wave is that which a stone would acquire by falling through half the depth of the water. That the speed ought to be independent of the density of the liquid is clear from the fact that it is the weight of the disturbed portion which causes the motion, and that this (for equal waves in different liquids) changes proportionally to the mass to be moved. Since we have made no hypothesis as to the form of the wave, our only assumptions being that the vertical motion is not only small in comparison with the depth, but incon siderable in comparison with the horizontal motion, while the latter is the same at all depths in any one transverse section, it is clear that, under the same limitations, a wave of depression will run at the same speed as does a wave of elevation. A solitary wave of elevation obviously carries across any fixed transverse plane a quantity of water equal to that which lies above the undisturbed level. If H be the mean height of this raised water, b the breadth of the canal (supposed rectangular), and X the length of the wave, the volume of this water is bXH. But all particles in the transverse section behave alike ; and, when the wave has passed, the particles in all transverse sections have been treated alike. Hence the final result of the passage of the wave is that the whole of the water of the canal has been translated, in the direction of the wave s motion, through the space bXH/bk, or AH/A. If the wave had been one of depression, the translation of the water would have been in the opposite direction to that of the wave s motion. Hence, when the wave consists of an elevation followed by a depression of equal volume, it leaves the water as it found it. Thus any permanent displacement of the water is due to inequality of troughs and crests. A hint, though a very imperfect one, as to the formation of breakers on a gently sloping beach, is given by con sidering that in shallow water the front and rear of an ordinary surface-wave must move at different rates, the front being in shallower water than the rear and therefore allowing the rear to gain upon it. (7) Oscillatory Waves. The typical example of these waves is found in what is called a "swell," or the regular rolling waves which continue to run in deep water after a storm. Their character is essentially periodic, and this feature at once enables us to select from the general integrals of the equations of nou- rotatory fluid-motion the special forms which we require. The investigation may, without sensible loss of completeness for application, be still further simplified by the assumption that the disturb ance is two-dimensional, i.e., that the motion is precisely the same in any two vertical planes drawn parallel to the direction in which the waves are travelling. The investi gation is, unfortunately, very much more simple in an analytical than in a geometrical form. If the axis of x be taken in the surface of the undisturbed water, in the direction in which the waves are travelling, and that of y vertically downwards, the equation for the velocity-potential (see HYDUOMECHANICS) is simply This is merely the "equation of continuity," the condition that no liquid is generated, and none annihilated, during the motion. The type of solution ve seek, as above, is represented by <j> = Ycos(w- nx) ,

where Y depends on >j alone. If this can be made to satisfy the