Wave.

Wave. When the surface of a sheet of water is disturbed waves are invariably produced. These may vary in magnitude from the huge rollers of the Atlantic to the tiniest observable ripple. The broad characteristic of such wave-motion is that as the waves pass over the surface at a considerable speed the liquid itself simply rises and falls with a slight to-and-fro motion in a steady rhythmic manner. The wave, in short, is a particular state of motion handed on from one portion of the water to another. It is energy, not matter, that is transmitted. In the case of ordinary sea-waves it is not easy to see that gravity is the effective dynamic agent in their propagation. In accordance with hydrodynamic principles the tendency must be for the water at the crest of a wave to be pulled down to the level of the water in the neighbouring trough, and for the latter to be pushed up to the level of the former. Corresponding to this surface oscillation there must be at any point below the surface an oscillation in pressure. Also at any instant the pressure will vary from point to point along a horizontal line drawn in the fluid in the direction in which the wave-motion is being propagated. These conditions obviously imply a definite motion of the particles of the fluid. If we suppose this motion of the fluid itself to be so small that we may neglect the square of its value, we are able to determine mathematically the character of the motion in the simplest type of oscillatory waves sustained by the action of gravity. Two simple cases are usually distinguished according as the length of the wave is great or small compared with the depth of the liquid. In the former case we have the propagation of long waves in shallow water. Each particle of liquid describes an ellipse with longer axis horizontal. As the depth increases this ellipse diminishes in size, and becomes more elliptical. At the bottom the liquid simply moves to and fro in a straight horizontal line. The velocity of the wave is equal to the square root of the product of the total depth of the liquid into the acceleration due to gravity, in symbols $\backslash sqrt\{gh\}$. The shallower the water the more slowly will the waves pass.

In the other simple case we find a type of which the deep-sea wave may be taken as an example. Each particle of fluid describes a circle, which rapidly diminishes in size as the depth increases. The velocity of wave-propagation is given by the formula $v^2\; =\; gl/2\backslash pi$, where $l$ is the wave-length and $\backslash pi$ is the ratio 3.14159... Thus, with a wave-length 4$\backslash pi$ or about 12.6 feet, the velocity of propagation will be 8 feet per second or 5.6 miles per hour. Again, with a wave-length of 100$\backslash pi$—a length by no means uncommon with Atlantic rollers after a gale—the waves will travel at 40 feet per second or 27.3 miles per hour. Atlantic waves from 500 to 600 feet long and from 44 to 48 feet high have been observed to take from 10 to 11 seconds to pass. This gives a velocity of 50 or 55 feet per second. The theoretical formula gives 50.6 and 55.4 feet per second respectively. The Hydrographical Bureau of Washington records the observation of a wave half a mile long which took 23 seconds to pass. According to the formula it should have taken 22.7 seconds. These comparisons show that so far as the value of the velocity is concerned the simple theory is very satisfactory.

This theory, which is avowedly an approximation, gives a wave form whose crests are similar to the troughs. Now a glance is sufficient to show that the crests are sharper and the troughs flatter than this similarity would require. Stokes has, however, carried the approximation a step further, and finds that the steeper crests and flatter troughs are quite accounted for; also that the particles are, on the whole, carried forward in the direction of propagation of the wave.

In the case of oscillatory waves the disturbance, as already noted, rapidly diminishes as the depth increases. Thus at a depth equal to one wave-length the disturbance of the water is only $\backslash frac\{1\}\{5\backslash frac\{1\}\{2\}\}$th of that at the surface; and at a depth of two wave-lengths, only $\backslash frac\{1\}\{3\backslash sqrt\{5\}\backslash sqrt\{5\}\backslash sqrt\{5\}\}$th of that at the surface. With the largest ocean waves the agitation has an inappreciable effect at even moderate depths.

Near a shore-line or beach the to-and-fro motion of water along the bottom must have its effect on the material accumulated there. Signor Cornaglia has made some very valuable observations on the formation of beaches along the shores of the Mediterranean. He finds (as the theory also indicates) that the bottom water under a crest moves in the direction of the wave-motion, and that the water under a trough moves in the opposite direction. A stone or other object lying on the bottom will, as wave follows wave, be subjected to pressures acting in alternate directions. If the bottom is inclined, as in the case of a shelving beach, the shoreward push due to a passing crest is greater than the seaward push when the trough is passing. The seaward push will, however, be aided by the resolved part of the weight of the stone acting down the slope. At a certain depth, depending on the inclination of the slope and on the size of the waves overhead, the landward push will just be balanced by the seaward push together with the resolved part of the weight. In shallower depths, that is, nearer the shore, the landward push will have the advantage, and the effect of the waves will be to carry material up the beach. On the other hand, in greater depths, that is, farther from the shore, the other forces will have the advantage, and the materials will tend to be carried out to deeper water. Thus there exists near to every shore a neutral line or strip, above which sediment tends to move up, and below which it tends to be carried away to greater depths. In the Mediterranean this neutral line lies at a depth of from 26 to 33 feet. Signor Cornaglia shows the importance of this phenomenon in relation to estuaries and harbours, which become silted up if their mouths and openings are on the land side of the neutral line, but remain deep if their outlets are on the sea side of the neutral line.

As oscillatory waves flow in upon a shelving beach they gradually change character. The troughs become flatter, and the crests become sharper. At length the crest begins to curl over, and finally topples as a breaker upon the beach. If we regard these waves as oscillatory waves in shallow water, we see that their velocity should diminish with the depth. Hence there is a tendency for the advancing wave, which, so to speak, strives to keep its original momentum, to be retarded by friction on the bottom. Hence the crest tends to outstrip the lower parts of the wave, and the result is the breaker. It is certain, however, that as the depth becomes smaller and smaller the waves lose their true oscillatory character and become solitary waves or waves of translation.

The solitary wave was discovered and studied by Scott Russell, the eminent engineer. To this class belongs the long wave which accompanies a canal-boat, and which we see traversing the canal when the boat is stopped. The most favourable rate at which a canal-boat can be drawn is when its velocity is such that it rides on the crest of the solitary wave. When this condition is not fulfilled part of the work done by the horse is expended in producing fresh solitary waves. The speed of the wave is measured by $\backslash sqrt\{gh\text{'}\}$, where $h\text{'}$ is the height of the crest of the wave above the bottom of the canal. In a canal 8 feet deep the boat's most favourable speed would be a little greater than 16 feet per second, or about 11 miles per hour. Boussinesq and Rayleigh have worked out the theory of the solitary wave, and their conclusions agree very well with the observations made by Scott Russell. For example, the latter observed that when the height of the wave was equal to the depth of the undisturbed liquid the wave began to break; and the theory shows that with this relation of depth and height the water at the crest is moving horizontally with the speed of the wave. One characteristic of this solitary wave is that when it is a crest the water is displaced forward a definite amount and does not return. On the other hand, when it is a trough the water is displaced backward—i.e. opposite to the direction of propagation of the solitary trough. For this reason the solitary wave is also called the wave of translation. It is found that a solitary trough is very unstable; and it is doubtful if a solitary crest can remain permanently of the same form even on a frictionless fluid.

Stationary waves in running water, such as may be seen on any shallow brook with uneven bed, belong to another class of waves. One particular inequality, a large stone for example, will produce one conspicuous wave and a number of smaller ones accompanying it. The conditions of the problem are difficult to state; and very little has been done in the discussion of it. Lord Kelvin has shown that a certain relation between the speed of the water and the average depth determines whether there is a crest or trough formed over the inequality.

With water-waves whose wave-lengths are longer than 1 foot gravity is practically the efficient agent in propagating the wave-motion. But when the waves get small another agent comes into play—viz. the pressure due to the Surface-tension (q.v.) of the curved surface of the water. This pressure acts downwards over the convex crest and upwards over the concave trough. Its magnitude increases with the curvature, which becomes greater as the wave-length becomes smaller. Thus the influence of the surface-tension in accelerating the speed of the wave becomes more pronounced as the wave is taken shorter and shorter. When a wavelet becomes so small that the surface-tension is more effective than gravity, it is distinguished by the name of ripple. There is a certain particular wave-length for which gravity and surface-tension have equal effects as regards the speed. Shorter wave-lengths are ripples, longer wave-lengths are waves; and the speed of the wavelet having this critical wave-length is a minimum. All other ripples and waves, whatever be their wave-lengths, travel faster. For any speed greater than this minimum speed two wave-lengths correspond. In other words, for any given wave propagated mainly by gravity there corresponds a ripple propagated at the same speed mainly by surface-tension. These statements are all contained in the following formula which expresses the speed ($v$) of waves on the surface of fresh water in terms of the wave-length ($l$): $v^2\; =\; 61.4l\; +\; 31/l$. The units are the inch and second. The first term on the right is the part due to gravity, the second the part due to surface-tension. The minimum speed (9.34 inches per second) is given when these two terms equal each other, so that the critical wave-length which separates ripples from waves is 0.71 inch. When the wave-length is $2\backslash frac\{1\}\{2\}$ inches the speed is $12\backslash frac\{1\}\{2\}$ inches per second, instead of 12 inches per second as it would be under the influence of gravity alone. Now the same value of $v$ is obtained if we substitute 0.217 for $l$ in the formula. That is, the ripple of wave-length 0.217 inch travels with the same speed as the wavelet of wave-length $2\backslash frac\{1\}\{2\}$ inches. It is easy to see from the formula just given that the product of the wave-lengths (in inches) of the ripple and wavelet which move with the same speed must equal $\backslash frac\{31\}\{61\}l$ or .505, so that if the one wave-length is given the other can be at once calculated.

A very pretty experiment due to Scott Russell, but first discussed fully by Lord Kelvin, gives a practical demonstration of the fact that to every ripple there corresponds a wave having the same speed. Let a thin rod, dipping vertically in smooth water, be drawn through it at a speed of 10 inches per second or more. In front of the rod a group of ripples advancing with it will be formed, and behind will be seen the corresponding wavelets travelling also with the rod. These combine to produce a beautiful pattern on the surface of the water. If we increase the relative speed of the rod and the water the ripples in front get shorter, and the wavelets behind get longer. On the other hand, a diminution of speed causes the ripples to lengthen and the wavelets to shorten until for a particular speed they are equal in wave-length. For slower speeds wave patterns are not formed.

Any disturbance of the surface of water results in the formation of waves. A stone falling in, or a fish rising, produces a series of wavelets or ripples travelling outwards in widening circles. Of all agents that have to do with the generation of waves wind is, however, the most effective. An almost inappreciable puff of air will produce tiny ripples on an otherwise smooth water-surface. The first production of such ripples must depend upon the variations of vertical pressure accompanying the breeze; but once they are fairly started the momentum of the wind itself will have its direct influence on the wavelets. The power of wind in lashing up the surface of water is shown on a grand scale in every storm at sea.

Besides water-waves, which have been the chief object of discussion in this article, there are numerous other kinds of wave-motion which are essentially involved in many physical phenomena. For example, Sound (q.v.) has to do with waves travelling in elastic media, either as waves of compression and dilatation or as waves of distortion. A similar kind of wave-motion accompanies Earthquakes (q.v.). Then, again, the whole of physical optics is built upon an elaborate theory of wave-propagation, the exact nature of which is yet to be discovered (see LIGHT, POLARISATION, &c.). Maxwell's brilliant theory that the wave-motion which constitutes light is an electric rather than an elastic phenomenon has virtually carried the day; but it remains to find a dynamical explanation of electricity itself, or, at any rate, to conceive an ethereal medium whose fundamental property is a capacity to transmit wave-motions of the kind required. For an account of phenomena depending on the coexistence of wave-motions, see INTERFERENCE. See also HARBOUR.