The action of the wind on the waters of the sea gives rise to problems which interest alike the physicist, the mathematician, the oceanographer, and the practical navigator. And these problems are far from having been satisfactorily solved. On their theoretical side they are of a peculiarly complex nature and on the observational side there is the difficulty that observations at sea must be made from the deck of a vessel the stability of which, especially when waves are running high, is of an altogether different nature from that of the physicist’s laboratory on land.
Because of the difficulties attending the observations on the waves of the sea, individual measurements are clearly not correct to the “second decimal place.” But from a number of observations statistical averages of a very helpful nature may be derived. And it is with this phase of the matter that we are here concerned.
It is clear that some time must elapse before a wind of given velocity can bring about the maximum wave development of which it is capable. That is, the duration of the wind enters as a factor in the relation between wave and wind. This factor, however, we may here disregard as being of academic rather than of practical interest. For the sea is rarely free from waves and swells. This means, therefore, that when a wind begins blowing it doesn’t start from scratch, so to speak, in regard to its wave-raising capacity. Furthermore, heavy winds do not, as a rule, arise suddenly, but follow winds of lesser velocity which have blown for some time. Hence the question of the duration of the wind, in connection with waves, while interesting, is but of secondary importance. From such studies as have been made of this question, it appears that the time required for a given wind to bring about its maximum wave development is to be measured in hours, with a day or two perhaps as the upper limit.
Close to the coast the relation between wind and wave is complicated by such factors as depth of water and fetch of wind. The waves of the sea due to the wind are, however, surface waves so that it is only very close to the coast that the depth of water may be insufficient for the full development of waves.
The fetch of the wind, or the extent of open water over which it blows, is a factor that has to be reckoned with for a considerable distance from the coast, as much as 1,000 miles perhaps. The formula used to express the relation between the height of waves and the fetch of the wind is purely empirical and was derived by Thomas Stevenson in the middle of the past century. He found that the height of waves appeared to be most nearly in the ratio of the square root of their distances from the windward shore. For strong winds he found the height of the waves in feet to be 1.5 times the square root of the fetch in nautical miles. If / is the fetch in nautical miles and h the height of the waves in feet, the formula gives, h = 1.5/f.
It must be emphasized, however, that Stevenson’s formula is only an approximate one. Stevenson himself found that for short distances a correction term was necessary, for the height of the waves observed under such conditions is greater than derived from the above simple formula. On the basis of this formula the heights of waves at given distances from the windward shore are as follows: 10 miles, 5 feet; 50 miles, 11 feet; 100 miles, 15 feet; 200 miles, 21 feet; 500 miles, 34 feet; 1,000 miles, 47 feet. Since in round numbers 50 feet may be taken as the highest waves found in the open sea, it follows that at distances greater than 1,000 miles the distance from the windward shore is not a factor affecting the height of waves.
In the open sea it is the strength of the wind that is the principal factor determining the height of the waves. To the navigator it is axiomatic that the stronger the wind the higher the waves. Various empirical formulas have been derived to express the relationship between wave height and wind velocity. The most widely used formula in this country is that due to Vaughan Cornish who developed it about twenty-five years ago. This states that the height of the waves in feet is equal to the velocity of the wind in statute miles per hour divided by 2.05. That is, if h is the wave height in feet and w the velocity of the wind in miles per hour, h=0.49w.
More recent investigations make it appear that the relation between wave and wind is not one of simple arithmetical proportion as expressed in the above formula. For moderate winds the factor appears to be less than for heavy winds. But even on the assumption of simple arithmetical proportionality it appears that the factor derived by Cornish will have to be changed. In a recent study of the question Dr. Erich Zimmermann derived the formula as follows, h = 0.65 w. And, since this is based on later data, it may be taken as more nearly representing the true relationship. This relationship therefore may be stated as follows: the height of the waves in feet is two-thirds the velocity of the wind expressed in statute miles per hour.
Accepting the latter formula, it follows that a wind of 80 miles per hour—a hurricane—will give rise to waves approximately 50 feet in height. As a matter of fact, in a hurricane the height of the waves may be much less, for the very violence of the wind tends to reduce the height of the waves by decapitating them—blowing off the tops of the crests into the troughs. It thus frequently happens that with severe storms the highest waves occur after the wind has begun to subside.
It appears quite certain that the height of regular waves met in the open sea is not in excess of 50 feet. This figure it must be noted, however, refers to the vertical distance between crest and trough of regular waves, and not to the height attained by intersecting waves or by breaking waves. There are well-authenticated cases of such waves attaining a height of 80 feet or more. In December, 1922, the SS. Majestic encountered waves in the North Atlantic which competent observers estimated to be 80 to 90 feet in height. And along the coast there are records of breaking waves attaining even greater heights. The lantern of Tillamook Rock Lighthouse, off the coast of Oregon, stands 130 feet above sea level; yet in severe storms rocks carried by the waves have been hurled through the lantern glass.
To the navigator the feature next in importance to wave height is wave length. And as regards this feature it must be noted that there is no fixed relation between the height and length of a wave, such that if one is known the other can be calculated. This arises from the fact that the form of the wave varies. As a general rule, however, the higher the wave the longer it is. From the data at hand the following rough rule may be formulated. For waves up to 5 feet in height the length is about 30 times the height; waves 6-10 feet in height, have a length 23 times the height; waves 10-20 feet in height have a length about 20 times the height; waves 20-30 feet in height have a length about 17 times the height; waves higher than 30 feet have a length about 14 times the height. It must be emphasized, however, that the above relations are only rough averages and that wide variations from these figures may be noted in individual cases. Zimmermann, using the observed data on the length and height of waves derived as an average the following formula, l=7.15/h4 in which l is the length of the wave and h the height, both in feet.
From the observations made on the length of waves in the open sea, it appears that wind waves several hundred feet in length are not at all uncommon. Storm waves 500 feet or more in length have been frequently measured, and there are trustworthy measurements of waves up to 2,500 feet in length. It appears probable, however, that such extraordinarily long waves result from the interference of several waves in such a manner that one or more intervening crests become obliterated.
The relation between wave height and wind velocity was found to be expressed by the simple formula, h = 0.65 w. But since no simple relation between wave height and wave length exists, there can be no simple relation between the length of a wave and the velocity of the wind. Here, too, we must be satisfied with roughly approximate values. Determining the relation between the lengths of the observed waves and the velocity of the wind Zimmermann derived the following average formula, 1 = 4^ in which l is the length of the wave in feet and w is the velocity of the wind in miles per hour.
As regards the velocity of wind waves in the open sea, it can be shown from theoretical considerations that this velocity is independent of the depth of the water but depends only on the length of the wave. The appropriate formula is v = /gl/2pi, in which v is the velocity of the wave, g the acceleration of gravity and x the ratio of the circumference of a circle to its diameter. Taking the average value of g as 32.172 feet per second, this formula becomes v = 2.26/l, v being in feet per second and l in feet. Approximately, therefore, the velocity of a wave in deep water is 2.25 times the square root of its length. A wave 400 feet in length therefore travels with a velocity of about 45 feet per second or a little more than 30 miles per hour; while a wave 1,600 feet in length travels with a velocity of a little more than 60 miles per hour.
Since there is no simple relation between the velocity of the wind and the length of a wave, there is no simple relation between wind velocity and wave velocity. From observations made on waves in the open sea, it is found that, as a general rule, well-developed waves travel with a velocity somewhat less than that of the wind to which they are due. However, waves have been observed which were traveling with a velocity greater than that of the wind. This appears to be due to the fact that so long as the velocity of the wind is increasing, the wave velocity will be less than that of the wind; so soon, however, as the waves have attained their maximum height with a wind of constant velocity, the energy of the wind is expended in increasing the length of the waves, and thus their velocity, until finally their velocity becomes greater than that of the wind.
On the basis of the observations that have been made on the relation between wave velocity and wind velocity, Zimmermann derived the following average formula, v = 3.08/w2, in which v is the wave velocity and w the wind velocity, both in statute miles per hour. From this formula it follows that the velocity of a wave is greater than that of the wind for wind velocities less than 29 miles per hour, and less than that of the wind for wind velocities above 29 miles per hour. Here again, it is necessary to emphasize the fact that since this formula is an average formula based on the observations to date, any conclusions based on it are necessarily of an approximate character.
It is to be noted, too, that all the formulas developed above relate to waves brought about by the wind. Occasionally a vessel will encounter one or more solitary high waves in otherwise smooth water. Such waves arise not from forces acting on the surface of the sea, as is the case with the wind waves discussed in the preceding paragraphs, but from forces acting beneath the sea. Submarine volcanic explosions, seaquakes, or other activities of the earth which result in sudden submarine displacements give rise to movements in the water which manifest themselves at the surface as waves. Such waves travel with a velocity depending on the depth of the water.
In conclusion it may be observed that the subject of waves is clearly one which can be forwarded by those who go down to the sea in ships. Observations on the length, height, and velocity of waves and their relations to the wind can be made without elaborate apparatus, the appropriate methods being described in any of the standard works on the sea.