SECTION I.
THE STRUCTURE OF SEA-WAVES.
It is essential to the just understanding of the best methods for opposing the violence of waves, that the phenomena which constitute wave-motion be understood. It can be said with some degree of confidence that there is no instance in nature of a perfectly quiescent surface of water. Air and water are both mediums of extreme mobility, and the individual molecules of both, and of all other substances, are continually in a state of motion, with different velocities, in paths different in direction and length. There is thus a continual interlacing of particles. When air covers water, some of the particles of air, in their excursions, strike the surface of the water, producing unequal pressures upon it, and giving rise to ripples which the vision is not acute enough to detect. If the original surface of the water were perfectly smooth, and if all parts of it continued equally exposed to an equal wind, waves could not be produced. But, with the minute corrugations which are always present upon the smoothest water, it is to be observed that it does not occur that water is all equally exposed to equal winds. The pressures of the moving air upon the crests and posterior portions of the minute corrugations are greater than those on the hollows and anterior portions. There is thus a tendency to heap up the water at the places of greatest pressure, which is augmented by the rotational or vortex motion produced by the viscosity of the air. These actions produce new forms and inequalities, which, exposed to the wind, generate new modifications of its force and give rise to further deviations from the primitive condition of the fluid. Imagine an isolated example in which the water has been suddenly heaped up by a gust of wind. The action of gravity tends to force this heap to the level of the original surface, which causes the particles of water in the heap to push forward the particles immediately in front of them out of their former place to another place further on, and they repose in their new place at rest as before the original heaping up. Thus in succession volume after volume continues to carry on a process of displacement which only ends witli the exhaustion of the displacing force originally impressed and communicated from one to another successive mass of water. As the particles of water crowd upon one another in the act of going out of their old places into the new, the crowd forms a temporary heap visible on the surface of the water, and as each successive mass is displacing its successor, there is always one such heap, and this heap travels apparently at that point where the process of displacement is going on; and although there may be only one crowd, yet it consists of always another and another set of migrating particles. This moving crowd constitutes a true wave. The velocity of the wave is the velocity with which the heap is seen to move. Its form is the form of the heap. Its length is the distance from crest to crest, and its height is the distance from the crest to the surface of the water before the disturbance.
The motions of the individual particles of water are different from the motion of translation which the wave has. Consider a particle in a mass of water about to be traversed by a wave-form. The action of gravity on the heap behind it tends to press it forward where it is confronted by a solid wall of water. Under the action of these two opposite forces the particle is driven upward and forward until the particles which have displaced it have made room for themselves, then it sinks, and finally comes to rest a little in advance of the place from which it started. The motion of migration of each individual particle is thus in a closed orbit. The propagation of the wave is the advancement of a mere form. The actual translation of water in the propagation of unbroken waves is small. The motion of each particle takes place in a vertical plane parallel to the direction of propagation of the wave. The path or orbit described by each particle is approximately elliptic, and, in water of nearly uniform depth, the longer axis of the elliptic orbit is horizontal and the shorter, vertical. When at the top of its path, the particle moves forward as regards
the direction of propagation; when at the bottom, backwards, as shown by the curved arrows in the accompanying figure. The straight arrow denotes the direction of propagation.
The particles at the surface describe the largest orbits. The extent of the motion horizontally and vertically diminishes with the depth below the surface. A particle in contact with the bottom of water of moderate depth moves backward and forward in a horizontal straight line as at D. On the ocean, where the water is deep as compared with the length of a wave, the paths of the particles are nearly circular, and the motion is insensible at great depths.
When waves are first raised at sea their crests are smooth and rounded, as represented in Fig. 2. As the wind freshens the crests rise higher and become more acuminate. Rankine has investigated the limiting forms which waves assume before breaking, and has concluded that in the steepest possible oscillatory waves of the irrotational kind, the crests become at the vertex infinitely curved in such a manner that a section of the crest by the plane of motion presents two branches of a curve which meet at an angle of π/2. Some years later Stokes concluded that the two branches of the crest are inclinedat an angle of ± π/3 to the vertical, and at an angle of 2/3π to each other, not π/2 as supposed by Rankine.
After the prolonged action of the wind, when the crests of the waves rise to a considerable height and become sharper and sharper, the passage of the air over the crests with high velocity tends to impart its velocity to them. Owing to the inertia of the lower masses of water, the imparting of this velocity is resisted. The paths of the particles become distorted, as shown in Fig. 3, the front of each wave gradually becomes steeper than the back, and the crests seem to advance faster than the troughs, until at length the front of the wave curls over and breaks, as shown in Fig. 4.
Large sea-waves seem to be the result of a building-up process carried on by the joint action of large and small waves. If, for any cause, there be one wave larger than those surrounding it, its size will be continually increased at the expense of the smaller ones. For these smaller waves, in passing over the top of the larger, offer increased obstruction to the wind and cause the formation of cusps when the waves coincide. The delicate equilibrium incident to a cusped form is easily destroyed by the action of the wind, and the crest of the wave breaks into fragments which go to increase the volume of the large wave, leaving the small ones yet smaller. Therefore, whatever influence prevents the breaking of waves acts also as an agency to prevent their increase in size. No fact of observation and no method of sound reasoning has yet led to the conclusion that the spreading of oil on the surface of water agitated by waves can exercise any sensible effect in lessening the size or velocity of the waves themselves. It is in the breaking of the waves that the oil finds its field of action.
SECTION II.
WHY OIL SPREADS OVER THE SURFACE OF WATER.
There is an attraction of one particle of water for another, and there is an attraction of one particle of oil for another, but there is a repulsion between a particle of water and a particle of oil. If we attempt to mix oil and water, the two liquids separate from each other of themselves, and, in the act of separation, sufficient force is brought into play to set in motion considerable masses of the fluids.
Imagine an individual particle of water within a mass of water. The particles on every side of the individual particle attract it, and the attractions of opposite particles on every side tend to neutralize each other, so that the individual particle has almost perfect mobility. The surface particles, however, inasmuch as all the rest of the fluid is below them, are drawn inward toward the mass of the fluid, and a certain tension is produced. This tension is potential energy, and is inherent in the surface particles in virtue of their position. If we consider an oily film spread over the surface of a body of water, it will appear that the particles near the surfaces which separate the oil from the water and from the air must have greater energy than those in the interior of the film. The excess of energy due to this cause will be proportional to the area of the surface of separation.
In a unit of area separating any two fluids, let the energy which the particles have in virtue of the tensions due to their positions, be defined as the surface energy per unit of area.
When the area of the surface is increased in any way, work must be done; and when the surface is allowed to contract, it does work upon other bodies. Hence it acts like a stretched sheet of india rubber, and exerts a tension of the same kind.
Let the equation to the curve BCA be = f(x). Take any ordinate, as CD, whose length is y, and let the whole tension exerted across this line be represented by ψ, then the superficial tension is measured by the tension across a unit of length of y, or since ψ is the tension across the whole ordinate y, if T, which is constant, is the superficial tension per unit of length, ψ = Ty = T.f(x). Suppose that the variable ordinate y is originally in contact with the axis OB, and that the surface included between the curve and the two axes is produced by drawing the ordinatey away from the axis OB toward the right by the action of the force p. If we consider OB and DC, which is equal toy, to be two rods wet with oil and placed between the curve and the axis of X, and then drawn asunder, the oily film BCADO will be formed. Let E represent the superficial energy per unit of area. Then the work done in forming the film will be
E ∫ f(x).dx
But if ψis the variable force required to draw the ordinate y from the axis OB, the same work may be written
∫ ψ.dx.
Therefore
Work = ∫ ψ.dx=E ∫ f(x).dx, (I)
substituting the value of ψ in (I),
T ∫ f(x).dx = E ∫ f(x).dx,
or
T=E,
or the numerical value of the superficial energy per unit of area is equal to the superficial tension per unit of length.
Let Tao represent the superficial tension of the surface separating air from oil.
Let Taw represent the superficial tension of the surface separating air from water.
Let Tow represent the superficial tension of the surface separating oil from water.
Let the figure represent an exaggerated picture of a layer of oil on the surface of a body of water. Let P be a point of the line forming the common intersection of the surfaces separating the air, oil, and water. For equilibrium of these three media, the three tensions Tao, Taw, and Tow must be in equilibrium along the line of A common intersection, and since these tensions are known, the angles their directions make with one another can easily be determined. For, by constructing a triangle, ABC, having lines proportional to these tensions for its sides, the exterior angles of this triangle will be equal to the angles formed by the three surfaces of separation which meet in a line.
But it is not always possible to construct a triangle with three given lines as its sides. If one of the lines is greater than the sum of the other two, the triangle is impossible. For the same reason, if any one of the superficial tensions is greater than the sum of the other two, the three fluids cannot be in equilibrium in contact.
If, therefore, the tension of the surface separating air from water is greater than the sum of the tensions of the surfaces separating air from oil and oil from water, then a drop of oil cannot be in equilibrium on the surface of water. The edge of the drop where the air meets the oil and the water becomes thinner and thinner till it covers a vast expanse of water.
M. Quinke has determined the superficial tensions of different liquids in contact with each other and with air, and the following is an extract from his table of results. The tension is measured in grams per linear centimeter at 20° centigrade.
Liquid | Specific gravity. | Tension of surface separating liquid from air. | Tension of surface separating liquid from water. |
Water, | 1.0000 | .08235 | .00000 |
Olive oil, | 0.9136 | .03760 | .02096 |
Although olive oil is here taken as the representative of oils, it is not considered so well adapted for use at sea as some of the others.
Whale oil seems to be the best adapted for this use, but the surface tensions of this oil do not seem to be determined. It may be presumed that they do not differ greatly from the values given for olive oil.
SECTION III.
HOW BREAKERS LOSE THEIR FORCE THROUGH THE OPERATION OF SURFACE TENSIONS.
Let us imagine a “break” to occur after the surface of the water is covered by the oily film. For every square centimeter of film torn asunder there will be destroyed .05856 centigrammeter of potential energy, being the sum of .03760 and .02096, the potential or surface energies, in centigram meters per square centimeter, of the surfaces separating air from oil and oil from water; and there will be generated for every square centimeter of free surface of water formed .08235 centigrammeter of potential energy. Thus the mere act of breaking the film of oil causes an expenditure of energy, because it lays bare a surface having a tension greater than the sum of the tensions of the surfaces separating air from oil and oil from water. But there is a further loss of energy. Suppose, after a “break ” has occurred, a layer of water glides over a layer of oil. The superficial energy in the surface separating the oil from the air, amounting to .03760 centigrammeter per square centimeter, is replaced by .10331 centigrammeter per square centimeter, being the sum of .08235 and .02096, the superficial energies per square centimeter of the surfaces separating air from water and water from oil respectively. Therefore, when water breaks over an oily film, there is required for the formation of each square centimeter of a layer of water on the oily film, .10331 minus .03760, or .06571 centigrammeter of work.
SECTION IV.
HOW THE FILM OF OIL ACTS AS A SHIELD TO PREVENT THE DERANGEMENT OF THE WAVE MECHANISM.
It has been pointed out that when waves are propagated across any body of liquid, the individual particles of the liquid describe closed orbits. At the highest points of these orbits, or in the crests of the waves, the particles are moving in the direction of the propagation of the wave. The centrifugal and centripetal forces acting upon each particle are in equilibrium, and, for a unit of mass, are each equal to v2/r, in which r = the radius of the orbit, and v = the velocity of the particle along the orbit.
When the wind is blowing over the waves with a velocity greater than the velocity of propagation, and in the same direction with it, the moving air tends to impart to the particles of water a velocity additional to the normal velocity of revolution in their orbits, causing the distortion of the orbits and the disintegration of the crests of the waves. The force which the moving air exerts to draw the water along with it is due to the viscosity of air. When wind blows over water, all the air does not pass over the surface of the water. On account of the high degree of adhesion between air and water, a thin stratum of air remains in contact with the water, and it is the action of the internal friction or viscosity of air tending to draw this stratum along which causes the tractive effect of wind on water.
In the figure, A represents the crest of a wave, and the dotted area represents the air above it. Let BC represent the stratum of air which remains in contact with the water, and let c, the distance between the horizontal planes BC and DE, be the thickness of the layer of air which undergoes shearing strain when the velocity of the wind relatively to the velocity of propagation of the waves is V The air at the height of the plane DE will then be moving relatively to the water with a velocity V, while the velocity of any intermediate stratum will be proportional to its height above BC. The rate at which shearing strain is increasing in the area between the planes DE and BC is measured by the velocity of the upper
plane divided by the distance between the planes, or by V/c.
Let the direction of the arrow denote the direction in which the wind and waves are moving. Let F denote the shearing stress, which is measured by the horizontal force exerted by the air on a unit of area of the plane B C, and acting from C toward B. The ratio of the force to the rate of increase of the shearing stress is called the coefficient of viscosity, and is generally denoted by the symbol μ. We may therefore write:
F = μ V/c (1)
Maxwell has determined μ for air at θ° centigrade to be
μ = .0001878 (1 + .00366 θ°) (2)
the centimeter, gram, and second being units. So that
F = .0001878 (1 + .00366 θ°) V/c. (3)
If R is the amount of this force on a rectangular area of length a and breadth b,
R = abF = .0001878 (1 + .00366 θ°) abV/c.
Suppose the velocity of propagation of the waves to be 15 miles per hour, and the velocity of the wind to be 40 miles per hour, and that the thickness of the stratum undergoing shearing strain is 5 feet, and that the temperature is 20°. Then V= 40 — 15 = 25 miles per hour = 42 feet or 1280.15 centimeters per second, c = 5 feet = 152.4 centimeters, and θ°= 20°. The force exerted on each square centimeter of the crest of the wave would be .0001878(1 +.00366 X 20°) 42/5= .0017 gram.
In the above case, if the height of the wave be 10 feet and its length 300 feet, we have, from the following proportion which obtains in oscillatory wave-motion,
mean speed of particle/speed of wave = circumference of particle's orbit/length of wave,
mean speed of particle = 2.6 feet per second = 79.25 centimeters per second.
There is thus, according to these moderate suppositions, a force of .0017 gram acting upon each square centimeter of a surface directly connected with a system of particles moving in the direction in which the force is exerted with a velocity of 79.25 centimeters per second. When a film of oil is spread over the surface of the water, this tractive force is not brought to bear on the surface of the water as long as the oily film remains unbroken, but acts upon the surface of the film, whose particles, being entirely separate from the particles of water, do not share their motion. The surface of the water is thus shielded from the action of the wind in the same manner as if a skin of india-rubber were spread over it, and the only action of the wind in this case is to move the film over the surface of the water with a force equal to μ V/c upon each square centimeter of surface.
SECTION V.
THE ACTION OF A FILM OF OIL TO PREVENT THE GROWTH OF WAVES AND THE FORMATION OF SHARP CRESTS.
The following passage, illustrating the mode of the formation of sea-waves, is taken from the Report on Waves, made to the British Association for the Advancement of Science, in 1842 and 1843, by John Scott Russell, M. A., F. R. S.:
“An observer of natural phenomena who will study the surface of a sea or large lake during the successive stages of an increasing wind, from a calm to a storm, will find in the whole motions of the surface of the fluid, appearances which illustrate the nature of the various classes of waves . . . . and which exhibit the laws to which these waves are subject. Let him begin his observations in a perfect calm, when the surface of the water is smooth and reflects like a mirror the images of surrounding objects. This appearance will not be affected by even a slight motion of the air, and a velocity of less than half a mile an hour (8$ inches per second) does not sensibly disturb the smoothness of the reflecting surface. A gentle zephyr flitting along the surface from point to point may be observed to destroy the perfection of the mirror for a moment, and on departing, the surface remains polished as before; if the air have a velocity of about a mile an hour, the surface of the water becomes less capable of distinct reflection, and on observing it in such a condition, it is to be noticed that the diminution of this reflecting power is owing to the presence of those minute corrugations of the superficial film which form waves of the third order. These corrugations produce on the surface of the water an effect very similar to the effect of those panes of glass which we see corrugated for the purpose of destroying their transparency, and these corrugations at once prevent the eye from distinguishing forms at a considerable depth, and diminish the perfection of forms reflected in the water. To fly-fishers this appearance is well known as diminishing the facility with which the fish see their captors. This first stage of disturbance has this distinguishing circumstance, that the phenomena on the surface cease almost simultaneously with the intermission of the disturbing cause, so that a spot which is sheltered from the direct action of the wind remains smooth, the waves of the third order being incapable of travelling spontaneously to any considerable distance, except under the continued action of the original disturbing force. This condition is the indication of present force, not that which is past. While it remains it gives that deep blackness to the water which the sailor is accustomed to regard as an index of the presence of wind, and often as the forerunner of more.
“The second condition of wave-motion is to be observed when the velocity of the wind acting on the smooth water has increased to two miles an hour. Small waves then begin to rise uniformly over the whole surface of the water; these are waves of the second order, and cover the surface with considerable regularity. Capillary waves disappear from the ridges of these waves, but are to be found sheltered in the hollows between them, and on the anterior slopes of these waves. The regularity of the distribution of these secondary waves over the surface is remarkable; they begin with about an inch of amplitude, and a couple of inches long; they enlarge as the velocity or duration of the wave increases; by and by conterminal waves unite; the ridges increase, and if the wind increase, the waves become cusped, and are regular waves of the second order. They continue enlarging their dimensions, and, the depth to which they produce the agitation increasing simultaneously with their magnitude, the surface becomes extensively covered with waves of nearly uniform magnitude.”
Observation has thus shown that, in the generation of oscillatory waves, ripples or capillary waves are first formed, and that it is to the union of conterminal ripples and to their more abundant formation with the increased force of the wind that the growth of waves is due. The existence of a certain definite tension, equal to .08235 gram per linear centimeter, at the common surface of air and water has been pointed out. The water surface under this tension is in perfect equilibrium.
When wind blows over the surface of a body of water, the tangential force which the air, in virtue of its viscosity, exerts on the surface of the water is of different degrees of intensity at different places, owing to the minute corrugations which are always present on the surface of a body of water, and to the eddying motion of the air. At the places where the tangential force is greatest, the surface film of water is drawn along and heaped upon the portions of the surface immediately in front of them, destroying their surface tension or energy of position, and, by laying bare new surface in the places from which they are moved, generating a like amount of surface tension. Through this action heaps or ripples are formed, and surface tension is being constantly generated and destroyed. The formation of ripples takes place on waves already in existence in the same manner as upon a surface of water originally at rest, and by continually uniting with the larger waves, they impart those dangerous qualities to the wave which result from high and acuminate crests.
When a film of oil is spread over the surface of the water, this heaping-up action, which, in the case of the water film, results in the formation of ripples, cannot take place. This has been demonstrated by the experiments of Mr. John Aitken, described in the Proceedings of the Royal Society of Edinburgh, Vol. XII, 1882-83, No. 113.
Let A represent the crest of a wave covered by the film of oil BC, and let P be a point of greatest action of the tangential force of the wind, which is supposed to move in the direction of the arrow. The tendency of this action is to drive the film into a heap immediately in front of P. By this action, a greater tension is generated in the film at b and a lesser tension at a. The greater tension at b tends to draw the portion at b' ahead, and the lesser tension at a allows the tension at a' to draw the portion at a ahead. So that, instead of a tendency toward heaping up, there is a tendency to move the entire surface film along at a uniform rate. The formation of ripples is therefore stopped, and the growth of the waves and the formation of “breaking” crests, as far as they result from this cause, are prevented.