Foreword.—They made master mariners and real sailors in the old days by a process of long training and direct observation of the elements. But, learning of the results, they evidently cared little about how those effects were produced. We find some of the most commonplace things were often the least understood in these older vocations—passed along from master to apprentice, generation after generation.
Since our modern pace precludes that old, slow thoroughness, training for skilled crafts must now use a different approach. The necessary grasp of the results is developed by carefully analyzing and explaining the motions and forces until clear mental pictures of these actions are acquired. Thus the student gains some ability for coping with them before he is subjected to their sudden hazards.
It appears that the accepted standards of seamanship have been touched quite gently, thus far, by the investigations of this scientific age. Yet adequate knowledge of the elemental forces with which our ships contend is vital to the success and safety of every voyage. We will therefore try to apply here some of that modern method of instruction, so as to develop a better understanding of ocean waves.
Several decades ago the famous engineer and mathematician, Dr. Rankine,1 proved conclusively that the theory of rotation of the water met every possible requirement and condition for the familiar propagation of ocean waves, except for one point of confusion we can clear up here by more recent knowledge. Somewhat later, Admiral A. W. Stahl, U. S. Navy,2 developed a similar beautiful treatise of wave action, in full agreement with Dr. Rankine, and used it for an investigation of the power that might be obtained from ocean waves by various devices. Their trochoidal form of ocean waves has been recognized by all leading authorities, but the practical significance of the wave motion has somehow escaped. Their analyses in the higher realms of mathematics are far too mysterious, so we will try here to carry them through to more tangible applications in the mariner’s field.
Wave motion.—They show us that each particle of water in a series of waves in deep water travels in a circle. Figure 1 shows this motion in a wave traveling from left to right while every particle of water at the surface of the ocean goes around and around its circular path, clockwise, at uniform speed as indicated by the arrows. Keep in mind one certain particle of water, such as a particular cubic inch, and we will follow its motion while we develop the familiar form of the wave.
At the top of the crest, A, this unit of water moves to the right in the same direction as the wave. After the crest of the wave has passed, the same particle of water, at B, moves downward. In the trough of the wave our bit of water is moving to the left, as shown at C, in the opposite direction to the travel of the waves. As the next crest approaches, the water is rising, at D, and when the crest reaches this spot, the same particle of water again moves to the right as it did during the passage of the previous crest. Wave after wave passes this point rapidly, but the water stays in its same locality. The height and diameter of this circular path is, of course, the height of the wave from crest to trough.
It is easy enough to see the vertical movement of the water, at B and D, but very few seem to have realized fully that the water has an equal horizontal motion at the top and bottom of its travel. You may have noticed how the crest of a wave flips some floating object along in its path, without realizing the horizontal movement of the water supporting these objects. There is a really dangerous example of this horizontal motion of the water in a wave. When a boat is near some piling, or near a ship large enough to span several of these wave forms and therefore practically fixed, the powerful horizontal thrust of the water at the top of its orbit, A, may throw the boat violently against the ship or structure, thereby damaging the boat or crushing a person trying to fend off or struggling with lines. Then the trough of the wave, C, pulls the small craft away for another smash on the next crest. It floats in, and is moved by the same few barrels of water all through this pitched battle of men with the elements.
Each particle of water revolving in this path nudges or crowds its neighbor along and transmits this same motion to the water ahead of it as the wave progresses. In Fig. 2 we have taken the length of a wave as the distance from point A to M. In order to construct the resulting curve of this wave we have here chosen to divide this distance into twelve equal parts for convenience in plotting the positions of the small particles of water through one length of this wave form. While the wave travels a distance equal to its length between successive crests, each particle of water travels one complete circle, 360°. This means it turns through 30° during each of the twelve divisions of its wave length shown, A to M.
As the wave is assumed to travel from left to right and the particles of water at each subdivision are rotating clockwise, we see that the crest of the wave still has one- twelfth of its length to go before it reaches the circular path of particle B, so this particle B still has one-twelfth of its circle, 30°, to travel before it reaches its top point. Therefore, when the crest is at A, our particle in circle B will be in the position shown by the heavy dot. We may similarly plot the location of all other particles in this wave curve. Particle D in the orbit one-quarter wave length ahead of crest at A still has one-quarter of the circle to travel before it reaches its top or crest position, and particle G half way between crests is just at the bottom of its circle. Drawing a smooth curve through all these plotted points gives us the surface form of a wave on deep water with a length from A to M and a height equal to the diameter of the circle of rotation. This is the form of a curve known to mathematicians as the trochoid.
The direction in which the water is then traveling is also marked on the lower wave outline of Fig. 3 on each of the twelve particles we selected for plotting this wave form. All are moving with the same velocity because all have the same uniform rotation at the surface, so when we use the length of such arrows to represent the velocity of each particle, this diagram now shows equal velocity though in different directions, all along the surface. Arrows will be used in this way to show certain water velocities and directions on many later diagrams.
Some of the very interesting features of this wave will be shown more clearly by drawing the form of a higher wave in the same manner, Fig. 4. The wave outline drawn in Fig. 2 has a height equal to one- twelfth of its length, and the height of the Fig. 4 wave is one-seventh of its length. This higher wave emphasizes the characteristics noticeable in Fig. 2. The crests are sharper and the troughs broader than in the regular “sine” curve with equal crests and troughs. Turn this diagram upside down for just one look, and the sharp crests will be more noticeable. As the waves become higher in proportion to their length, the crests become very much sharper and the troughs much broader. This agrees so definitely with common observations that we can accept this trochoidal theory as entirely reasonable.
They tell us there are “theoretical reasons for believing that, in nature, trochoidal waves never reach a sharper angle than 30° of slope at the breaking cusp.” We can easily go along with that from our observations of waves “in nature” except, of course, for breakers on beaches and the shallow water waves described later. This angle of 30° is also shown by a dotted line on the right-hand crest of Fig. 4.
Wave impedance.—We know how a vessel is slowed down when it is heading into such waves. Compared to a calm sea, the humps of water come very much higher on the blunt part of the bow, well above the normal water line. And, of course, this offers greater resistance to the ship’s progress. Yet very few realize that at the same time the wave comes up so much higher on the blunt lines above the forward water line, all that high water around the bow is really traveling horizontally at considerable speed, against the desired motion of the vessel as shown by the arrows. Add to all that the pitching of a vessel in a rough sea, frequently burying its nose deeper in the crests, and it is no wonder such waves just about make good on their threat of “Thou shalt not pass.” Figure 5 illustrates the position here discussed.
You will notice that the water in the trough, traveling with the vessel, contacts only the lower part of the hull, and at places where there is little, if any, chance of pushing even a slow boat ahead appreciably. Therefore, we might as well say that after feeling the impact of the horizontal motion of the wave approaching high on its bow, the ship regains practically none of its lost speed by reason of the equal forward motion of the water in the trough of the wave.
This same action would add considerably to the drift or leeway of a ship that is blown off her course by a storm. It also washes floating wreckage along with the waves when its exposure to the wind is a mere fraction of its underwater body, and finally brings the flotsam up on the beach as the waves turn toward the shore line, long after the wind has ceased. Wave action does not propel things through the water when they are completely submerged.
It is now time to call attention to a fact well known to experienced mariners. A shorter choppy sea is much rougher on a small boat than on the ship with length enough to span several waves. The longer, higher ocean waves are more severe for the larger ships, and not particularly dangerous for the smaller lifeboats, if properly loaded, unless a secondary set of smaller, shorter waves is imposed on the long wave forms by weather conditions. The small boat, drawn to scale on Fig. 5, spans only a short section of the long wave, so the surface that floats it is not far from a straight line and there is no great difference in the horizontal velocity of the water throughout the boat’s length.
Wave swing.—Pitching may be worse in some cases in waves about the same length as the vessel. The pitching depends considerably on the inertia or distribution of weights fore and aft, the same as the period of any pendulum depends on the length from the weight to its support. However, waves up to roughly double the length of the ship or boat generally present the greatest reduction in speed and the most serious maneuvering difficulty and hazard. In such waves the bow and stern are in water that is traveling in opposite directions. If the ship is not steered accurately into high waves, any angle of approach will result in having the bow thrown farther off its course by the crest of the wave while water in the trough of the wave at the stern, traveling in the opposite direction, will carry the stern farther around and still further increase the angle of the vessel with the wave, as indicated in Fig. 7. Then, when out of control, the vessel might fall off to the trough of the sea and be in real danger.
The extra swing and strain on insecure weights, or tending to shift cargo, may be better appreciated now, adding this strain to the customary ideas of the rolling forces from wallowing in the trough. Remember that while the wave is swinging the vessel sharply from its course, the effect of the rudder is diminished and may be entirely lost. The water around the stern, in the trough of the wave, Figs. 5 and 7, is traveling in the same direction as the desired course of the ship, and so the ordinary flow of water past the rudder is greatly reduced and might even be reversed if the ship has lost much of its speed—shuddering after a high wave has slapped it down. It should be quite evident that the helmsman or automatic pilot would be practically helpless in a storm to offset the violent swing of the ship into the trough, crosswise of the seas, if the ship approached the waves at a bad angle.
In the case of steering into the waves, after the ship comes up from its dive, Fig. 6, so that the bow is in the trough where water is moving in the same direction as the ship, the resistance and bow wave will then be less than average, so the ship can try to pick up speed momentarily. At the same time, the water of the crest high on the stern is adding its rotational velocity to the speed of the ship’s wake and the rudder then has deep, fast water for the best steering conditions. Notice the arrows showing the movement of the water at bow and stern. In this situation, also, the horizontal motion of the wave action tends somewhat to straighten the course of the ship into the next wave, because the water of the trough at the bow is moving forward and the crest around the afterbody is pulling the stern aft and into line.
However, the faster flow of water past the stern means that the propeller does not have its usual thrust to take advantage of this favorable moment for picking up speed, unless the engine revolutions are increased proportionally. But alert engineers stand by to prevent the engine from speeding up in a storm. They rightfully take every precaution to avoid racing of the engine when the propeller gets out of water in a heavy sea. And we suspect that more than a few of them think the propeller is at least part way out of water when the engine begins to speed up so as to take hold of this faster water under the stern, while the propeller is still well submerged under the crest of a wave.
Wave velocities.—We had better get a clear idea of the real velocity of that water in the waves this ship is bucking. Some very complicated calculations in higher mathematics, investigating the requirements of continuity, dynamical and boundary equilibrium, finally work out to a rather simple relation between the length and speed of our trochoidal wave. To avoid possible confusion, we will speak of the velocity with which waves pass a fixed point as the speed of the crest, to distinguish it from the velocity of the water itself around its circular path. The wave form travels along the ocean surface rapidly but the water does not.
The speed of the crest is not affected by the height of the wave—only by the wave length. This speed of the crest varies as the square root of the wave length. Waves 25 feet long travel at 6.7 knots, the speed of waves 100 feet long is 13.4 knots, and the speed of larger waves 400 feet long between crests is 26.8 knots, showing twice the speed for four times the length. This shows that with waves 25 feet long, 27.2 crests would pass a fixed point in one minute, 13.6 waves 100 feet long would pass in a minute, or 6.8 crests of waves 400 feet long. As the speed is only doubled when waves become four times as long, half as many pass in a given time, for it then takes each crest twice as long to travel one wave length.
TABLE I |
|||||||||
Length of wave, feet |
25 |
50 |
75 |
100 |
150 |
200 |
300 |
400 |
600 |
Speed of crest, knots |
6.7 |
9.5 |
11.6 |
13.4 |
16.4 |
19.0 |
23.2 |
26.8 |
32.8 |
Number waves per minutes |
27.2 |
19.2 |
15.7 |
13.6 |
11.1 |
9.6 |
7.8 |
6.8 |
5.5 |
A couple of the simplest formulas may be interesting to some, but the figures and speeds given in the following tables may help the average man size up these facts and relations more clearly. The established formulas are
V = √5.123 L and N = √5.123/L
where
L =Length of wave in feet,
V =Velocity of wave crest, feet per second,
N = number of waves passing a point per minute.
These formulas give the values in Table 1, which check closely with timed and measured waves.
It is not easy to measure or even closely guess the length of waves when you are in the middle of them, but with this little table you can count the number passing a spot in one minute and get a good estimate of their length. We will show later that the form changes and the speed of waves is reduced when they approach shallow water, but it is some fun to count the waves as they break on the beach, find from that their length out on deep water, and so tell something of the severity of the storm that started them. It is interesting that the longer waves travel at more than 30 knots. We can show that a powerful speed boat would have to make 35 to 40 knots in rough water to stay in the trough of long, high ocean waves, although the Kanaka’s surfboard can ride the front of the crests in to the beach with nothing but a skillful start and good balance.
TABLE II |
|||||||||
Velocity of Water in Knots |
|||||||||
Height of Wave, Feet |
Length of Wave in Feet |
||||||||
25 |
50 |
75 |
100 |
150 |
200 |
300 |
400 |
600 |
|
1 |
.85 |
.60 |
.49 |
.42 |
.35 |
.30 |
.24 |
.21 |
.17 |
2 |
1.7 |
1.2 |
.98 |
.85 |
.69 |
.60 |
.49 |
.42 |
.35 |
3 |
2.5 |
1.8 |
1.5 |
1.3 |
1.0 |
.90 |
.73 |
.63 |
.52 |
4 |
3.4 |
2.4 |
2.0 |
1.7 |
1.4 |
1.2 |
.98 |
.84 |
.69 |
5 |
|
3.0 |
2.4 |
2.1 |
1.7 |
1.5 |
1.2 |
1.1 |
.86 |
6 |
|
3.6 |
2.9 |
2.5 |
2.1 |
1.8 |
1.5 |
1.3 |
1.04 |
8 |
|
|
3.9 |
3.4 |
2.8 |
2.4 |
2.0 |
1.7 |
1.4 |
10 |
|
|
|
4.2 |
3.5 |
3.0 |
2.4 |
2.1 |
1.7 |
12 |
|
|
|
|
4.1 |
3.6 |
2.9 |
2.5 |
2.1 |
15 |
|
|
|
|
5.8 |
4.5 |
3.7 |
3.2 |
2.6 |
20 |
|
|
|
|
|
6.0 |
4.9 |
4.2 |
3.5 |
25 |
|
|
|
|
|
|
6.1 |
5.3 |
4.3 |
30 |
|
|
|
|
|
|
7.3 |
6.3 |
5.2 |
35 |
|
|
|
|
|
|
|
7.4 |
6.0 |
40 |
|
|
|
|
|
|
|
8.4 |
6.9 |
The water makes one complete revolution in the time it takes one wave length to pass any point, and the diameter of its circle is the height of the wave, so we can easily tabulate the velocity of the water in all these waves.
Table II lists the velocity in knots of water in waves whose heights for the longer waves run up to one-tenth of their length. This is the wave form shown in Fig. 5, which may not look too ominous until we appreciate its effect on the ship. We have many reports of higher waves and can hardly lay them all to hallucinations or exaggeration by unreliable men. Studying the largest velocities at the botton of these columns, you may feel inclined to draw a long breath of relief. Why worry about those velocities? Your ship or yacht certainly has plenty of speed to take care of that much wave motion. But no ship makes her speed in those seas, whether moderate or high powered. Furthermore, the listed figures give only half the story of Figs. 5 or 7. Take a 35-foot cruiser in waves 75 feet long and assume its nearly buried bow is bucking a 3 ½-knot current while water under the stern has an equal forward current around your rudder. Seven knots difference in velocity of the water at bow and stern of your boat makes the figures in the table much more significant. A ship in comparable seas may have 12 to 15 knots difference in water velocities from bow to stern when in the higher waves here listed, for the velocities in the table must be doubled to give the whole difference between crest and trough. It’s no honeymoon trip for the fast runabout or the proud Queen of the Seas, and the full bodied work boat or cargo vessel may be out of control and in peril with lesser waves.
Storm waves are not all as severe as those at the bottom of columns in Table II, yet some are reported greater. Bowditch claims that 50 feet height is the ultimate limit for long distances of ocean and gives 600 to 1,000 feet as the longest waves in the Southern Pacific and 2,600 feet for the longest wave ever recorded. These figures extend well beyond the values of our table. While we describe the motions and effects with the higher waves, any reader can adjust these values for different heights of waves. For waves of the same length on deep water, the water velocities will be in direct proportion to the height of the waves.
The diagram in Fig. 5 shows only the bare body of the wave, the fundamental form, or nudist of the species. We will see later that the same natural forces which produce these waves also dress them up in furbelows—at the expense of the ship. With white headdress and ruffles on the basic waves, a battered vessel is increasingly exposed to the lash of the elements, for the rolling broken water just in front of the wind-whipped crests travels with the full speed of the waves.
Towing.—Storms with high waves call for exceptional skill when it becomes necessary to tow any vessel, and the knowledge of wave motion has a special application to such service. By sag and stretch a very long hawser can take up only a limited amount of the constant differences in spacing between the towing vessel and the vessel being towed. Everyone probably knows that more progress can be made by a fairly steady pull, and that a moderate sized towline is sufficient for this, yet many a stout, oversize line has parted in a series of violent jerks.
In Fig. 8 we illustrate a situation which produces these periodic jerks, where the hawser leaves one vessel in the trough of the wave while the other is retarded on the crest of a wave. When the towing ship is bucking the crest, Fig. 8A, the hawser may be quite slack. But when the towing vessel is pulling in the next trough, as in Fig. 8B, where water is traveling with it, at the same time the crippled ship is held or flung back by its next crest, the line may get more of a yank than it can stand.
The steadiest pull possible under the circumstances, with the least severe jerks, will be maintained when both ships are on the crests at the same time and in the troughs at the same time. But even then the pull cannot be uniform with ships of different sizes, because the horizontal velocities of the water in crests and troughs will be more concentrated on the smaller ship than on the larger one spanning more of these changes in water velocity.
And steering calls for special attention while towing in a seaway. The effect of water velocities on the rudder, previously described, becomes more apparent on these slower moving vessels, as the water that supports them moves with considerable speed in one direction in the troughs and in the opposite direction when the rudder is in or near the crests. This steering difficulty will be more or less offset by the pull of the hawser from the stern of one to the bow of the other, so long as there is a good strain on it, tending to keep both ships in line. A slack hawser, of course, leaves both vessels to their own resources, at low speed, and subject to all the swing described for Fig. 7. The towed vessel, remember, has no propeller slip stream acting on its rudder.
Let’s look into another angle. We must not expect the crests of all these storm waves to line up straight for a thousand yards or more. While we have described here the motion of one wave in a series or line of travel, other series of similar waves at a fair distance on either side may sometimes be considerably out of step with the single set discussed. The rotating water in one path tends to keep the water on each side revolving with it in the same circle, only by rubbing against it— friction, turbulence, fluid viscosity or any thing you like to call it. This usually results in establishing a clear length of wave crest, or breadth of wave front, where all that water is moving in rhythm. The length of crests—at right angles to their direction of travel—is naturally shorter in the early stages of a storm before they have had much time to get into step, and tend to become longer and more regular as the wind continues and this friction on the water on each side brings a wider path into rotational agreement.
Now assume for the moment that the hawser has been adjusted for the usually successful length and pull in one series of waves. If some swing or yawing on a slack line, with the rudder inactive when most needed, should then permit the helpless towed vessel to get out of line into another series of waves that is more or less out of step with those around the towing vessel, all the violent conditions shown in Fig. 8 may then be encountered. It appears that the best seamanship calls for the steadiest pull and best spacing that conditions will permit, to keep the towed vessel fairly well in line with the ship ahead. This all refers to the more common practice of towing from bitts. A highly efficient towing machine, properly operated, may often succeed in maintaining a fairly steady pull without severe jerks, but an intelligent operator should relieve it from excessive work by selecting a reasonable average spacing of the ships.
A real job of towing is shown in the picture, Fig. 9, taken by Captain Beaton in 1912. We are told two towboats handled this sailing ship at a river bar, though one of the towboats was out of the picture to the left. Notice that both vessels are in the trough of the waves at this moment.
Depth of waves.—The depth to which the surface disturbance is felt may be of no particular concern to the mariner, but it is interesting. These waves do not penetrate to much depth. The water at each level below the wave on top revolves in its circle, the same as at the surface, but the circles decrease in size rapidly as we go down. The relative size of each lower circle depends on the relation of its distance below the surface to the wave length. Take, for example, waves 25 feet high and 300 feet long. At a depth of 150 feet, one half the wave length, the water particles revolve in a circle only a scant 13 inches in diameter. At a depth of 300 feet, equal to the wave length, the circle of movement is hardly more than a half inch, and at 600 feet depth the particles travel a theoretical circle that you could find only with a micrometer—one thousandth of an inch. These subsurface wave forms are the wrinkles occurring on what would otherwise be smooth fiat planes at the stated distances below calm water on the surface. Also, the length and velocity of the subsurface wrinkles are exactly the same as the surface waves above.
Taking another example of very choppy shorter waves, say 4 feet high and 40 feet long, on a lake or bay, we find about a 2-inch circular orbit at 20 feet depth—half the wave length—and only one-tenth of an inch at the depth of one wave length, 40 feet. Those who like to figure these relations can use Table III for the proportional subsurface movement, but Fig. 10 shows the amount of motion in all these wave forms for a depth equal to about three-quarters of the wave length. This same diagram is true for all waves with length ten times their height because the relative sizes of the lower orbits for proportional depths are certain fractions of the surface height.
This diagram now shows how the shape of one cubic foot of water changes as the wave progresses. Looking back at Fig. 2, which shows the circular paths of certain particles of water along the surface wave, one must not get confused by any impression that these represent large quantities of water rubbing on each other like wheels. Figure 10 shows, instead, that the only possible reduction of energy by internal friction in the liquid comes from the gradual change of form of each cubic foot as the wave passes. Suppose we imagine a very thin rubber box is placed around one cubic foot of water at the surface of a wave, made of rubber so thin and weak and completely relaxed that it serves only as a flexible division from the other water. Figure 11 shows how this would change its shape as one complete wave passes.
TABLE III |
|||
Subsurface Radii |
|||
D/L |
R |
D/L |
R |
0.00 |
1.00 |
.50 |
.0432 |
.10 |
.5335 |
.70 |
.0123 |
.20 |
.2846 |
1.00 |
.0019 |
.30 |
.1518 |
2.00 |
.0000035 |
.40 |
.0810 |
|
|
Here D = Depth of water below surface
L = Length of wave
R = Radius of circular orbit.
These forms were simply traced at the proper locations in Fig. 10 after drawing in the curves which tell how the vertical lines in still water will bend back and forth when the waves pass above them. This slow change of form in any unit of water explains how easily a series of waves can travel great distances from the original disturbance as a gradually reducing ground swell. Water has very little viscosity or fluid friction as the successive layers slide over each other the small amount necessary to let our cubic-foot box of water change shape as shown by Fig. 11, once for each wave that passes and only a few times a minute. There is enormous power in these high waves. Think of the weight of moving water. Every square mile of water one foot deep weighs a million tons. Then look at the velocity of all that water, and it is no wonder that such energy, or momentum, with small loss, can carry this wave motion long distances.
If you have good perception and can keep in mind the idea of flexible partitions between the vertical slices of water while looking at Figs. 10 and 11, it will be easier to understand how each chunk of water crowds the next one along to generate and transmit these trochoidal waves. And if it were only possible here to include a moving picture of successive Fig. 10’s, it could clear up forever the former indefinite ideas of wave action.
Shallow waves.—Up to this point we have discussed waves on relatively deep water. Let’s find out next why the waves on shallow water are even more dangerous. Yachtsmen know they take an extra beating in a choppy sea over shoal water, and few know better than the Coast Guard rescue crews the special hazard of the river bar or surf. The experienced mariner keeps his ship well out in deep water during a storm, realizing that the wave action nearer shore is more likely to throw the vessel out of control and bring her to grief. Yet the commercial fishermen, with their fine heritage of experience in such waters, probably know how to cope with these peculiar threats when Nature decides to enforce her stern law—survival of the fittest.
Waves formed by the wind on relatively shallow water, or rolling in toward the shoals from deep water, travel slower with even wider troughs and sharper crests. Their free circular path when in deep water is here restricted by the closer bottom into an ellipse with less height than length in its path of rotation. The calculated relations show that this effect of the bottom is negligible until the depth of water becomes less than half the wave length.
TABLE IV |
||
Proportions of Shallow Waves |
||
D/L |
H/B |
Relative Velocity of Shall Water Waves |
.10 |
.557 |
.746 |
.15 |
.736 |
.858 |
.20 |
.847 |
.920 |
.25 |
.917 |
.958 |
.30 |
.955 |
.977 |
.50 |
.996 |
.998 |
D/L gives the ratio of depth of water to length of wave,
H/B gives the ratio of wave height to the breadth of its orbit.
The third column gives the relative velocity of a shallow water wave compared to a deep-water wave of the same length.
These values are all interdependent to meet the various balancing conditions previously mentioned; when any number in column 3 is squared, we find it gives the corresponding number in column 2 for the same relative depth.
Waves formed on shallow lakes, bays, and sounds seldom develop beyond medium length because the limited areas do not afford the time and distance required to make long waves. The longest and highest waves for shallow water are likely to be those approaching shoals or river mouths from the vast ocean spaces.
Let’s examine the shallow water wave form, taking as an illustration a wave 80 feet long on water that is only 8 feet deep when quiet. Still using a height of one-tenth its length for direct comparison with the Fig. 6 diagram of a deep water wave, and the figures in Table IV where the depth is 10 per cent of the wave length, we find that the elliptical orbit of this shallow water wave 8 feet high will be about 14.4 feet horizontally. Constructing the diagram to show its characteristics, Fig. 12 gives the basic outline of this wave, without the white caps or breakers, spume and spindrift that high winds would add. Notice the very broad trough and the sharper crest, compared to the deep water wave of the same proportions. The water is still nearly 8 feet deep, for only the sharp peaks have been taken out of the original calm-weather depth. As on Fig. 3 the directions and velocities of the water are shown here by arrows, but they are not all equal on this shallow wave—the horizontal velocity at crests and troughs is nearly double the vertical velocity. The same outline form will be true for all other lengths of waves with the same proportions of height and length and the same relative depth of water.
It is particularly interesting to follow the changes when long deep-water waves approach shallow water. Data already given in above tables permit us to determine all these. Starting with the same storm waves 400 feet long and 40 feet high which we described earlier, we find that the length and height are reduced as the water shoals, to the values shown in Table V. We must remember that all the waves that roll in from the sea will pass any fixed point, so the number of waves in a minute, in such a series, is just the same as for deep water, and the speed of the crest is reduced in shallow water in the same proportion that their length is reduced by the changing depths.
TABLE V |
|||||
Depth |
Length |
Height |
Speed of Crest |
No. per Minute |
Horizontal Velocity of Water |
400 ft. |
400 ft. |
40 ft. |
26.8 kn. |
6.8 |
8.4 kn. |
200 ft. |
398 ft. |
39.8 ft. |
26.7 kn. |
6.8 |
8.4 kn. |
100 ft. |
373 ft. |
37.3 ft. |
25.0 kn. |
6.8 |
8.4 kn. |
80 ft. |
354 ft. |
35.4 ft. |
23.7 kn. |
6.8 |
8.4 kn. |
60 ft. |
326 ft. |
32.6 ft. |
21.8 kn. |
6.8 |
8.4 kn. |
40 ft. |
283 ft. |
28.3 ft. |
19.0 kn. |
6.8 |
8.4 kn. |
30 ft. |
255 ft. |
25.5 ft. |
17.1 kn. |
6.8 |
8.4 kn. |
20 ft. |
213 ft. |
21.3 ft. |
14.3 kn. |
6.8 |
8.4 kn. |
10 ft. |
155 ft. |
15.5 ft. |
10.4 kn. |
6.8 |
8.4 kn. |
Notice that the height is also reduced in exactly the same proportion as the length changes, so that, when conditions allow them to roll in to these restricted depths without wasting their energy, the relation of height to length in these shallow waves stays just the same as it was in deep water. The full wave height and force shown at each successive depth in Table V would occur when the series of deep-water waves come to a bank that shelves off rather rapidly. In the other extreme, if the depths of water listed in the table were spread over several miles of very gradual change from the beach, the waves would comb and break and so use up some part of their energy much farther out, and only the remnants of the original high waves would ever reach the regions with the shallower water shown by the lower numbers in this table. Then the heights and horizontal velocities of the “solid” water would be smaller, to match the smaller elliptical orbits of the remaining part of the original deep-water waves, but the lengths and speed of crest would still be the same as shown by the table. The form of a wave 220 feet long in the table, in water 22 feet deep, will be the same as drawn in Fig. 12.
We need to emphasize one departure from the Table V values for horizontal velocity of the “solid” water—another action which increases the effect of waves on a ship and adds a special hazard for smaller craft. Wherever the crests of these waves break into white water, the rolling tumbling broken water just in front of the crests travels with the full speed of the crests given in the fourth column of Table V or in Table I. The quantity of this faster water is negligible in the small whitecaps of moderate winds on deep water, but it may be a very considerable weight to be reckoned with on high waves in a storm. This velocity of broken water is most noticeable and especially severe where the waves comb and break over these shoals.
On the ordinary sloping beach, large storm waves break a considerable distance from shore, moderate waves break at a medium distance, and even the smallest waves or slight ground swells show the same typical sharpening crest, curl and collapse close to the sand line. Standing on the beach you can see the curved path at the end of these ellipses, looking straight into solid green water as the crests push up and curl forward approaching the shore. It looks like the same water standing on edge and moving in, but that is not true. New sheets of water are constantly being pushed up in front of what you saw a moment before. The ends of the ellipses get sharper as the shoaling water makes the orbits flatter, so the crests bend forward more and more. Finally, one of the new sheets of water pushed around these sharpening curves, up and forward, fails to reach the top in time to support all of the previous layer, the tip begins to break over a little, and this collapse of the combers increases progressively until a rolling froth of white water dashes up the beach.
Following seas.—Wherever the shoaling water causes sharp or breaking crests, while the horizontal motion at the crests and troughs is still near its deep-water storm velocity, that place is dangerous for craft of all sizes. A typical hazard of this class occurs when entering a river mouth from the sea. The difference in velocity of the water near bow and stern of a vessel, explained in Figs. 5 and 6 for deep water, is here exaggerated and reversed. The following seas lift the stern of the vessel and carry it along on the crests, while the bow is pushing down into water that has no such forward velocity, and will really have an equally strong current against the craft’s progress if the bow is near the trough when the stern is up. The rudder of the ordinary boat is then useless.
Too many boatsmen believe they can meet this situation by reversing the engine. They know, of course, that a propeller is not particularly effective in reverse, also that the propeller then throws its wash against the afterbody of the boat. But they have failed to grasp from earlier vague instruction on wave motion the very important fact that water in the crest of such waves has a forward velocity of its own. If the propeller were thrown in reverse, the engine would have to turn up at a good rate before it could begin to take hold in that forward stream of water. Furthermore, when the wave under the stern has swung the vessel, a reversed propeller at full power could only pull back at that angle, with an inactive or even contrary rudder, and will not offset the swing from her safe course.
We learn some of the best practices from the commercial fishermen. In one locality, at least, where these dangers arise, we find that many fishermen who get home safely at such times tow a suitable drag, as shown in Fig. 13. We also have first-hand reports that boats near them, without such resourcefulness, were seen to capsize and no trace ever found. But we find no evidence to indicate that these able seafaring men ever cared about the length of towline on the drag. (Some old salts still prefer to call it a drogue.)
From Fig. 13 it will be seen that, with the towline adjusted so the drag will be in the trough of the following wave at the same time the crest of the wave is lifting and swinging the stern, it will have the maximum desired effect to keep the vessel in line with the series of waves. Then, later, when the stern is in the trough of a wave, the drag will be in the following crest with the least possible pull on the stern, and the line might even be slack at this time when no pull is needed. It is at this moment that the line can best be shortened, by smart handling, if it appears desirable in order to keep the drag at its most useful distance astern—exactly half the length between crests. Don’t begrudge the loss of speed when the drag keeps your craft from capsizing. With such a drag, towed from the extreme stern, keep the engine kicking a fair stream past the rudder. We would estimate from previous tables that the minimum for a properly spaced drag shown in Fig. 13, when towed with full power in calm weather, should check the craft to about 3 knots less than normal full speed. All this gear must be stout enough to haul that drag at 15 knots or more through the water of the following trough during a storm.
Instead of searching for a jury-rigged collection of spars and dunnage when needed, we urge maintaining a suitable drag always at hand for coping with these following seas without “crossing the bar’’ of destiny. A stout hatch cover can be provided with attachments, or a removable cockpit seat, right over the lines you will use, can be fitted with a substantial eye-bolt in the middle of its lower side. With a line attached to its exact center it will stream perpendicular to the taut towline. Be sure to try out any rig you prepare, for a remarkable variety of theories about drags have failed, and new cordage unwinds under a strain more than well-used lines. A good sweep in a transom notch or oarlock, such as used in awkward situations for sculling, will be a lot better than the rudder and reversed engine when following combers try to throw your boat out of control. The Coast Guard never relies on the rudder in a surf; they use a steering oar for following seas.
Wave propagation.—The early investigators of wave motion could not account for the generation and growth of the trochoidal wave form, for their theoretical consideration of the perfect, frictionless fluid found no reason for the molecular rotation they believed necessary to set up the initial surface disturbance. Aviation has taught us a lot about winds and the inequalities of pressure and velocity. We have all seen a puff of wind ruffle the surface of calm water. As soon as such roughness occurs on the surface of the water, the wind velocity over these irregularities goes to work to increase them rapidly. The first waves developed are steep and short. As the wind continues, the waves become higher and longer, as long as the wind travels faster than the wave crests. The increase in size and length is rapid at first, for there is a great difference between the force of the wind and the speed of the crests. The rate of increase fades out as the speed of the crests approaches the wind velocity.
Figure 14 illustrates the path of the wind over the waves, with arrows just above the water showing the forces exerted by the wind on the basic wave form, tending to pull the water higher on the crests and push down on the troughs. A sailor knows that no sailing vessel could travel in any direction except straight down wind, if it were driven only by the impact of the wind, but it can travel in many other directions by making use of the sidewise thrust as the sails cut the wind and change the direction of large quantities. Air has considerable weight. It’s heavier than most of us realize, for a whole ton of it will go in a cubic box about 30 feet on each side. It requires a great force to change the direction of any heavy mass weighing many tons when traveling at high speed. This is the force of the sails deflecting the wind, with an opposite force of the wind against the sails, pushing on the windward side of the sails and pulling on the other side. These same forces, so familiar to the airplane pilot, act in the same way on the curved surface of the waves as the flow of air is deflected first one way, and then the other, just above the water.
As on the front of a sail or the top of an airplane wing, the wind curving over and down at the crest of a wave pulls upward on the water. Flowing down the front side of the crest, along the trough and up the other side, it pushes downward on the water between crests.
The wind also pushes and pulls the crests ahead. Look closely at the arrows above the wave in Fig. 14 indicating the push or pull of the wind on the water, and the arrows just below the surface showing the direction and velocity of the water. Notice that the wind, when traveling faster than the crests, will push downward and forward on the back of a wave in almost the same direction the water is moving, and in front of the wave crest the wind will pull the water upward and forward toward a lower pressure area tending to some turbulence—again closely in line with the velocity of the water at that point. So, in addition to its vertical forces at crests and troughs, the wind pressure crowds the water near the crests for the faster rotational velocity of higher waves and faster travel of longer waves.
Thus the wind will steadily increase the height and speed of the crests, faster when the wind whips around these curves at great speed, slower when there is not so much relative velocity. When the speed of the crests reaches the wind velocity, the wind merely goes along with the wave form and no longer turns these curves, so it has no more effect to increase their height or length.
To build up a series of long and high storm waves takes considerable time and requires long stretches of open sea, called “fetch” by the ancient mariners. When they have been built up to such size, they will carry great distances, and after the wind has ceased, they will proceed as smooth, unruffled swells of about the same length but losing height gradually, because there is very little friction in this slight movement between the layers of water. You can see that the “friction” of wind against the water surface which was formerly supposed to drive these waves is still at its maximum amount when the waves have reached their full height and length, but it does not act to increase them further.
Compound waves.—Up to this point we have discussed only the single basic wave form in a regular series. Looking back at Fig. 14 now, it will readily be seen that wind can kick up other disturbances as it passes along the smooth curves of the large primary wave, just as well as it can when the first waves began to be generated on smooth water. The larger fundamental waves will continue of their own momentum, and these secondary waves are then a fair indication of the present wind velocities. A table was prepared by some unknown authority years ago for estimating the surface wind by the condition of the sea. Our U. S. Navy thought enough of it to print it as a general guide for naval aviators, with the remark that it may not hold entirely true for all areas, especially in restricted waters where the wind does not have sufficient sweep for the full surface condition here shown reprinted with permission of the Navy as Table VI. These conditions are familiar to seafaring men.
In other cases of compound waves, without some special training on the mechanics of fluids it may not be so easy to grasp the fact that two regular series of waves from different directions can pass through the same water without interfering with each other. Refer back for a moment to Figs. 10 and 11 to keep in mind the manner in which each vertical section through a wave pushes the next slice along. One series of waves will push the water in the direction of its travel. If another set of ground swells is coming in another direction at the same time from a distant storm, this second series will move the same water back and forth to suit its own rotation. As a result, the water will move the combined distance of both orbits, each in its own direction, giving a disagreeable cross chop and a very irregular motion. Where the crests of the two sets of waves cross, water will be thrown up in higher peaks than either set of swells would produce alone. This situation is all too familiar to vessels on patrol in the Gulf of Alaska, where considerate skippers often can find no course that will head into the seas while the crews are at mess.
TABLE VI |
|||
Wind Force Prediction Table |
|||
Velocity in Knots |
Surface Condition |
Velocity in Knots |
Surface Condition |
0 |
Smooth, slick sea. |
20-22 |
Streaks are long and straight; white- |
2 |
Small, occasional ripples. |
|
caps on every crest; wind picks up |
3- 4 |
Small ripples all over—no calm areas. |
|
and carries mist along; large waves. |
5- 6 |
Well defined waves—smooth with no breaking. |
23-26 |
Large seas with waves forming on them; wind picks up and carries |
7- 9 |
Occasional whitecaps. |
|
occasional wave crest. |
10-11 |
Pronounced waves, frequent white- caps which carry a short distance. |
27-30 |
Heavy seas; pronounced white streaks; wind picks up frequent wave crests |
12-13 |
Whitecaps close together, carrying over a distance equal to the wave |
|
and carry along; breaking, rolling waves are forming. |
|
height. Slight traces of wind streaks. |
31-37 |
Continual rolling waves; wind carries along all wave crests for a distance |
14-16 |
Clearly defined wind streaks whose lengths are equal to about 10 wave |
|
equal to one-half wave length; scud or foam streaks. |
|
lengths. Light flurry patches. |
38-43 |
Well-defined waves form on the heavy |
17-19 |
Long, well-defined streaks; waves and streaks coming from same direction. |
|
seas; scud or foam streaks; waves and seas breaking and rolling. |
Another choppy situation occurs when a long series of waves strikes a steep cliff or vertical structure and rebounds. The natural horizontal motion of each crest is stopped; the resulting extra pressure is expended in splashing, spurting some of the water up in the air, and when it comes down it starts a reverse or counteracting oscillation away from the wall. For a moderate distance this combination makes a very irregular chop.
You may have watched some of the water thrown high into the air, several times the height of the waves as they roll in and strike the rocks or cliff. It is not the velocity of the water in the waves which causes that. The water in waves does not move as fast as a stream of water from a hose nozzle which would squirt up to such height. Instead, it comes from the great momentum of moving tons of water, striking the obstruction with such an impact or hammer blow that the resulting pressure throws part of the water to considerable height. This horizontal momentum and impact of waves should be kept clearly in mind, also, for safe design of waterfront structures.
Tide rips.—In the basic waves we have discussed previously, the water goes around and around the same circular orbit in the same place. But when a series of waves passes along the surface of any current, the whole mass of revolving water is, of course, carried along with that current. In the time between two crests an opposing current will carry the water a certain distance against the wind, and during this time interval the water is making one complete circle. As a result, when the particular unit of water we are watching next reaches the top of its circle and becomes the top of the next crest, it will be some distance from the place where it started its last revolution. The length of the waves will be shortened from their normal length by exactly the distance the opposing current carried that revolving water during the time between crests. So the effect of an opposing current on a series of waves is to jam all the crests closer together, with the same height and velocity of rotating water, except to the extent they expend their energy in turbulence in this boiling chop of short, steep tide rips, which may have come from waves that were only moderate outside of this local run of current.
You may apply this current effect to our previous discussion of following seas when entering a river mouth, where shoaling water makes the waves shorter and much sharper at the crests, with great differences in horizontal velocities in half the wave length. When there is any natural run-off or ebbing tide from the river, this opposing current will still further reduce the distance between crests, bring these differences in motion of the water still closer together, and make the crests that much sharper without affecting the height of the waves except to make them curl and break more.
Several locations have disagreeable reputations for rough water—the English Channel where tides ranging from 20 to 35 feet add currents to further roughen any foul weather sea, also our great Gulf Current that turns off Cape Hatteras. The shorter, steep waves from bucking a fairly uniform or straight current are unpleasant enough, but currents in restricted inland waters contribute several other factors to the crazy turmoil of tide rips often experienced under adverse conditions. With numerous variables and special cases, it is beyond our power to account satisfactorily for all the peculiar phenomena, but we can analyze a couple of the better known motions that throw regular waves into a jumbled disorder. The speed of the crests shown in Table I seems sufficient in most cases to pass through any such narrows against a rather strong tidal flow, if these further confusions did not break them up and dissipate their energy.
Usually in such passages there are some differences of current velocity in rather short distances crosswise of the channel. Each change of velocity, compared to water alongside, retards some lines or parts of the wave series differently than others, with resulting irregular spacing and interference of crests. Also some swirling motion is present in most inlets, caused by the irregular banks, points, and bends. These carry some of the revolving chunks of water laterally, sidewise, into the path of other rotating masses that are entirely out of rhythm due to encountering somewhat different current. So the typical inland tide rips are made up of pitching water where each peak seems completely out of step with all the other heaving lumps near it. Naturally, the clash of all the different circular orbits fairly near each other spends the force or momentum of the waves, so that little, if any, of the original regular wave series ever gets past a turbulent current. Approaching waves from a broad sound often bank-up at the mouth of such an inlet where the regular waves meet the current, without ever reaching the stronger current in the neck. These rips are mean for smaller craft, though too short to bother a ship of much size.
Taking the reverse situation where wind and current both act in the same direction, a moderate current tends to lengthen the distance between crests and makes the waves less severe. Yet even here, a strong current evidently has sufficient force and variety of actions to flatten out and destroy a large part, if not all, of the easy undulations of the regular wave form.
The speed of a boat gives it some of the effects of a current. Running into the waves they appear shorter and sharper for you meet more of them each minute. Turning then to run with the waves, all but the experienced are convinced the wind and waves have suddenly moderated.
Other items.—Brief comments can cover some other uses of this wave information. First, knowing that the waves become shorter in shallow water, we can now understand how and why they turn in toward the shore line. While the outer ends of waves, some distance from shore in deep water, continue at their original speed of crest, the ends in shoaling water nearer the land travel slower and slower so that the whole series swings in toward the beach. This same effect explains how waves passing around both ends of an island will have their lines of march changed by retarding the landward ends of the series, so they will cross beyond the island. Captain Eddie Rickenbacher recently noted this fact in his advice for aviators downed at sea. In the open ocean, this condition with waves coming from two different angles may inform a person on a life raft of the existence of a near-by island he cannot see.
We wish to offer a better explanation of ocean currents than the old theory of “friction” of the wind on the water. Figure 14 shows pressure of the wind pushing down and forward just behind the crest, also an upward and forward pull in front of each crest. The greater part of these forces is exerted in developing the rotational velocity of the water dose to the surface. This produces and increases the waves. But the smaller horizontal component of these forces will tend to move the water along with the wind. Gradually, over considerable periods of time, this can drive masses of open water at moderate speeds as ocean currents. However, we believe it can be shown that the great equatorial currents are not caused merely by the trade winds; that these ocean currents and trade winds are both caused chiefly by the same forces which produce the ocean tides; and that the temperature convection currents near the equator, deflected by the effects of the earth’s rotation and curvature, are only a contributing influence to these great air and water currents flowing from east to west on both sides of the equator.
Some may be interested in the depth effect of waves on shallow water. There is a point near each end of an ellipse called a focus. As in deep water, the disturbance of the water or orbit of rotation becomes smaller at lower levels. In shallow water the height of the elliptical orbit is reduced faster than its length, but the focal length or distance remains constant for any location and wave series, so that water at the bottom moves back and forth in a straight line equal to the distance between foci in the orbit of the surface wave above. Therefore, waves on shallow water often stir up the mud.
One must not consider all waves unfriendly. There are times when they are Heaven-sent lifesavers for small craft, especially for yachtsmen in unfamiliar reaches. In waters thickly studded with rocks, many of them just below the surface, navigating with unusual accuracy in clear calm weather from an accurate large scale chart still leaves uneasy moments. Add limited visibility and it becomes very difficult. And where the Coast and Geodetic Survey people spilled the pepper can on the chart to indicate areas full of reefs and rocks, with some notation that locations are uncertain, that is a very good place to be from during a dead calm. A long comfortable ground swell on these locations will spot the unseen rocks at dangerous levels for vessels of very moderate draft by making creamy crests and breakers. The safe depth shown by this means depends, of course, on the over-all height of the swells. When the rocks are only a few feet below the surface they will cut the rotation of the waves and show as small shoals. Then stay out of white water and you are much safer than with a flat ocean.
Conclusion.—With any new subject a second reading generally gives a much more useful grasp. So, if your interests may ever take you into a contest with the elements at sea, let us suggest that you review this whole subject in the near future. For those who have had no opportunity for that long, thorough experience of earlier days, we sincerely hope it will contribute to safety and enhance enjoyment of the sea.
1. Philosophical Transactions of the Royal Society of London for 1863.
2. Transactions of the American Society of Mechanical Engineers, 1892.