Introduction

The preceding sections have dealt with the types of motion in the ocean that bring about transport of water masses in a definite direction during a considerable length of time. They have also dealt with the random motion, the turbulence, which is superimposed upon the general flow. Besides these types, one has also to consider the oscillating motion characteristic of waves. In general, this motion manifests itself to the observer more by the rise and fall of the sea surface than by the motion of the individual water particles.

Waves have attracted attention since before the beginning of recorded history, and in recent years they have been the subject of extensive theoretical studies. Surveys of our knowledge as to the character of ocean waves have been presented by Cornish (1912, 1934), Krümmel (1911), Patton and Marmer (1932) and by Defant (1929). Lamb (1932) has discussed the hydrodynamic theories of waves, and Thorade (1931) has given a comprehensive review of the theoretical studies of ocean waves and has compiled a long list of literature covering the period from 1687 to 1930.

Our understanding of the waves of the ocean, how they are formed and how they travel, is as yet by no means complete. The reason is, in the first place, that actual observations at sea are so difficult that the characteristics of the waves cannot easily be determined. In the second place, the theories that serve to bring the observed sequence of events in nature into intimate connection with experience gained by other methods of study are still incomplete, particularly because most theories are based on classical hydrodynamics, which deal with wave motion in an idealized fluid. Here will be presented only a brief review of the best-established facts concerning waves and of some of the more outstanding theoretical accomplishments. Readers who wish to gain further insight are referred to some of the above-mentioned books.

In order to classify waves, it is necessary to introduce certain definitions. Wave height, H, is defined as the vertical distance from trough to crest, whereas wave amplitude, a, is one half of that distance (fig. 128).

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Wave period, T, is defined as the time elapsed in a fixed locality between the occurrence of one wave crest and the occurrence of the next. Wave length, L, is the horizontal distance from crest to crest or from trough to trough. The term “wave velocity,” c, is applicable to a sequence of uniform waves passing a given locality. The wave velocity is equal to the distance traveled by the wave in 1 sec. If a wave progresses in the x direction and if the crest was at the locality x = 0 at the time t = 0, it will have advanced one wave length when a time equal to the wave period has elapsed, or when t = T. Thus, the wave advances the distance L in the time T, and therefore c = L/T.

For a wave the amplitude of which is small compared to the wave length, the height of the free surface, η, at a given locality can be represented by means of a simple harmonic function,

For a progressive wave of small amplitude, the variation in time and space of the free surface can be written A standing wave (fig. 128) can be considered as composed of two progressive waves traveling in opposite directions, and the term “velocity of progress” retains, therefore, a definite meaning.

In wave motion, two types of velocity have to be considered: the velocity of progress of the wave itself and the velocity of the individual water particles. The water particles move back and forth, in circles, or

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in ellipses, regardless of whether the wave itself is of the standing or the progressive type.

The rise and fall of the free surface can be ascribed to convergence and divergence of the horizontal motion of the water particles. Within a progressive wave (fig. 128, also fig. 98, p. 426) the horizontal flow at the wave crest is in the direction of progress, and at the trough it is opposite to the direction of progress. Convergence therefore takes place between the crest and the trough, and there the surface rises. Within a standing wave (fig. 128) the horizontal velocity is zero at every point at the time when the wave reaches its greatest height. During the following half period the horizontal velocity is directed from the crest to the trough, causing divergence below the crest and convergence below the trough, for which reason the crest will sink and the trough will rise. This process continues until the positions of the crest and of the trough become interchanged, and during the following half period the horizontal motion is reversed. The vertical velocity is always zero halfway between the crest and the trough, where the wave has nodes. The horizontal velocity is always zero at the crests and troughs, where the wave has antinodes or loops. Evidently a vertical wall can be inserted at the antinode without altering the character of the wave, because no horizontal motion exists at the antinode.

From a different point of view, waves can be classified as forced or free waves. A forced wave is a wave that is maintained by a periodic force, and the period of a forced wave must always coincide with the period of the force, regardless of the dimensions of the basin or of frictional influence. A free wave, on the other hand, represents one of the possible oscillations of a body of water if this body is set in motion by a sudden impulse. The period of a free wave depends on the dimensions of the basin and on the effect of friction. Later on, these types will be dealt with more fully.

When ocean waves are concerned, gravity and Corioli's force are the two important forces to be considered. For waves of a few centimeters in length the surface tension of the water has to be taken into account, but such waves are of no consequence in the sea. We shall therefore deal with gravity waves only and shall at first neglect Corioli's force.

A rational division of gravity waves into two classes can be made when considering the relation between wave velocity, wave length, and the depth to the bottom. The wave velocity can with sufficient accuracy be represented by means of the equation of classical hydrodynamics:

where h is the depth to the bottom. If the depth to the bottom exceeds about half a wave length, tanh 2π(h/L) can be put equal to unity, and

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formula (XIV, 3) is reduced to In this case the velocity of the wave is independent of the depth but dependent upon the wave length.

If, on the other hand, the depth is small compared to the wave length, tanh 2πh/L can be replaced by 2π;h/L, so that c = . Thus, if the depth is small compared to the wave length, the velocity of the wave depends only on the depth to the bottom and is independent of the wave length. The latter waves are called long waves, whereas the former, the velocity of which is independent of depth, are called short waves, or surface waves. For water of any given depth the transition takes place within a narrow range of wave lengths, for which reason the classification is a very satisfactory one.

The physical reason for the difference between the surface waves and the long waves has been explained in simple words by H. Jeffreys (Cornish, 1934). Jeffreys points out that within surface waves the individual water particles near the surface move in circular orbits, but that the radii of these orbits, and therefore the velocities, decrease rapidly with depth. Theoretically the diameter of orbits at a depth of one half the wave length is only one twenty-third of the corresponding diameter at the surface. Regardless of the actual depth the character of the wave therefore remains unaltered if the depth to the bottom is greater than that short distance. Direct observations for substantiating this conclusion have not been made, but experience on submarines shows that in deep water a moderate wave motion decreases rapidly with depth and becomes negligible at a short distance below the surface, say at a depth of 30 m.

In shallow water the fact that no vertical motion can exist at the bottom modifies the character of the waves. At the bottom the motion can be only back and forth, and, if the depth is small compared to the wave length, the motion will remain nearly horizontal at all depths. Actually, the orbits of the single water particles will be flat ellipses that become more and more narrow when approaching the bottom, and at the bottom they degenerate into straight lines.

In a sea of variable depth the transition from short to long waves begins when the depth to the bottom becomes less than half the wave length, or where h < ½L. Since L = T²g/2π (p. 525) it is possible to establish a relation between the critical depth at which the transformation begins to take place and the wave period:

This formula is preferable to one containing the wave length, because the

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wave period is more easily measured. The longest periods of ocean surface waves are seldom more than 10 to 12 sec, and h is therefore of the order of 100 m, meaning that the transformation of surface waves of periods 10 to 12 sec begins when depths less than 100 m are encountered.

Waves of tidal period, on the other hand, always have the character of long waves. If the rotation of the earth is disregarded, their velocity of progress is equal to = L/T if h/L is so small that tanh 2πh/L equals 2πh/L. The equation , = L/T can be written

For the semidiurnal tide, T is about 44,700 sec. Even with h = 10,000 m = 10⁶ cm, one obtains h/L = 1/1380, and this value is sufficiently small to make tanh 2πh/L equal 2πh/L. Waves of tidal period have therefore the character of long waves.

For waves of short periods the rotation of the earth can be disregarded, as can be shown by comparing the accelerations of the moving particles with Corioli's force. If Corioli's force is very small compared to the accelerations, it can be disregarded, because it is then negligible compared to the other forces, the resultant of which represents the accelerations.

Corioli's force is proportional to 2ω sin ϕ ν v, where ν is the horizontal velocity, and the acceleration, dv/dt, is proportional to (2π/T)v. The ratio between Corioli's force and the acceleration due to the wave motion is therefore proportional to (T/2π)2ω sin ϕ or to (T/T_e)2 sin ϕ, where T_e is the period of rotation of the earth. If the period is measured in hours, one can write (T/T_e)2 sin ϕ = (T/12) sin ϕ. Now, 12/sin ϕ is equal to one half pendulum day (p. 437), and it can therefore be stated that the earth's rotation will be of importance to wave motion if the period of the wave approaches the length of one half pendulum day. For ocean surface waves the wave period T is always a very small fraction of half a pendulum day, for which reason the deflecting force is negligible beside the other forces, but long waves may be of tidal period, in which case the period length is of the same order of magnitude as a pendulum day, meaning that the deflecting force is of the same importance as other acting forces. Corioli's force will therefore be introduced when dealing with these waves.

Another noteworthy characteristic of surface waves is that below the depth to which motion of particles is perceptible the pressure remains constant when the waves pass. A pressure gauge placed on the bottom, if the depth were great enough, would not show any effect of surface waves, regardless of their height. The reason is that below the crest of the wave the acceleration is directed downward and will therefore counteract the effect of the acceleration of gravity, but below the trough the acceleration is directed upward and will be added to the acceleration

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of gravity. Consequently, under the crest a column of water exerting a given pressure will be of greater height than under the trough, and, at the level at which the motion is imperceptible, the difference in height equals the wave height. At and below this depth a pressure gauge will not record any surface waves. The above statements can be exactly verified by means of the general equations of motion as applicable to surface waves.

For waves of long periods the vertical accelerations, on the other hand, can be neglected, because the vertical displacements require a very long time. Consider a surface wave of period 10 sec and height 1 m, and a long wave of semidiurnal tidal period 44,700 sec and height 1 m. The ratio of the average vertical accelerations during the time when a water particle near the surface moves from its lowest to its highest position is inversely proportional to the square of the ratio of the wave periods, or in the wave of tidal period the vertical accelerations are about 5 × 10⁻⁸ times the vertical accelerations within the surface wave—that is, they are negligible. Consequently, when a long wave passes, the pressure at any given level is proportional to the height of the water, and a pressure gauge at the bottom gives a true record of the passing wave.

Some of the most outstanding characteristics of ocean surface waves and long waves can be summarized as follows:

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In the following discussion the characteristics of the wave types of the oceans will be dealt with more fully.