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Waves and Tides
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Tides

The longest waves known in the ocean are those associated with the tidal movement, which manifests itself on the coast by the rhythmic rise and fall of the water and, particularly in sounds and narrow straits, by the regularly changing tidal currents. The wavelike character of the phenomenon is readily recognized by means of an automatic tide gauge, which records the actual sea level as a smooth curve with alternating maxima and minima. It is also recognized in connection with tidal currents in a strait, because these currents show a regularly alternating motion characteristic of waves. The rise and fall of the water and the accompanying currents should therefore be dealt with together, because they are only different manifestations of the same phenomenon. However, for practical reasons, it is of advantage to deal with them separately and, following common usage in English, to refer to the rise and fall of the water as the “tide,” and to the accompanying currents as “tidal currents.”

Since a knowledge of tides is of particular value to navigation, there exists an enormous literature dealing with the phenomenon, partly in the form of theoretical studies, partly in the form of extensive series of observations and discussions of them, and partly in the form of popular treatises. The tidal currents have also been dealt with extensively, although they are less readily observed than the tides and even more difficult to examine theoretically. The literature on tidal currents, however, is also very large. Only a brief outline of the tidal theories and of the character of tides and tidal currents will be presented here. (Reference made to Darwin, 1911; Krümmel, 1911; Marmer, 1932.)

Tide-Producing Forces. Tides are caused by the attraction of the moon and sun. In a given locality the tide is of a complicated character, but any tide can be shown to consist of a number of partial tides, each of which is related to the motion of the earth relative to the moon and the


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sun. In order to show the existence of these tides, it is necessary to discuss the character of the forces that produce tides.

Considering the moon and the earth only, let the mass of the earth be equal to unity, and the mass of the moon equal to m. The gravitational attraction of the moon at the center of the earth is then proportional to m/r2, where r is the distance from the center of the moon to the center of the earth (fig. 137). This distance remains constant, on an average, and the attraction of the moon on the earth must, therefore, on an average, be balanced by a centrifugal force that is directed away from the moon and is also proportional to m/r2.

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Schematic representation of the gravitational attraction between the earth and the moon and the centrifugal force which balances the attraction.


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The centrifugal force acting on any particle on the earth is the same, but the attraction of the moon varies. Consider a point at the surface of the earth lying on the line which joins the centers of the two bodies. At this point the attraction of the moon is proportional to m/(r − ρ)2, where ρ is the radius of the earth. The difference between the attraction of the moon at this point and the centrifugal force is proportional to m/(r − ρ)2m/r2, or to 2mρ/r3, because ρ/r is a small quantity. The numerical values are m = 1/81.45, r = 60.34ρ. Owing to this difference the moon's attraction tends to raise the surface of the earth at the point under consideration, but at the same point the attraction of the earth acts and is proportional to 1/ρ2. Thus the ratio between the disturbing force of the moon and the attraction of the earth is equal to 2mρ3/r3 = 1.176 × 10−7. The attraction of the earth, on the other hand, equals the acceleration of gravity, and thus the tide-producing force of the moon is only about 1.176 × 10−7 times the acceleration of gravity; that is, at the point under consideration the acceleration of gravity is reduced by the amount 0.000115 cm/sec2.

Consider next a point at the surface of the earth but opposite to the moon. At this point the attraction of the moon is less than the centrifugal force and is proportional to m/(r + ρ)2. The disturbing force is again found from the difference


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where the minus sign indicates that the force is directed away from the moon. Thus, the disturbing force is of the same magnitude as the one that was first considered. It is also directed away from the center of the earth and reduces the acceleration of gravity at that point by 0.000115 cm/sec2.

Now take a point at the surface of the earth at right angles to the line joining the centers of the moon and of the earth. At this point the attraction of the moon is of the same magnitude as the attraction at the center of the earth, m/r2, but is directed, in fig. 138, along the line PM. The centrifugal force is parallel, however, to the line EM, and the resultant of these two forces is directed toward the center of the earth and is proportional to mp/r3. In this case the disturbing force leads to an increase in the acceleration of gravity of 0.00058 cm/sec2.

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(A) Schematic representation of the tide-producing forces in a plane through the line joining the centers of the earth and the moon. (B) Distribution of horizontal tide-producing forces over the earth (according to G. H. Darwin). The moon is at zenith above the point marked Z.


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If a point were selected at random on the surface of the earth, one would find that generally the disturbing force forms an angle with the surface of the earth different from 90° and can therefore be considered as having a vertical component and a horizontal component along the earth's surface (fig. 138). The complete theory leads to the following equations for the vertical and horizontal components of the disturbing forces:

Here θ is the angle between the line joining the centers of the moon and the earth and the line from the center of the earth to the point on the earth's surface that is being considered. In fig. 138 are shown, according to G. H. Darwin (1911), the distribution of the horizontal components
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of the disturbing forces over the surface of the earth and their relative magnitudes.

Exactly similar reasoning can be applied if one considers the sun and the earth, but for numerical computations the mass of the sun and the distance between the sun and the earth must be entered. Using the same units as before, one finds that the mass of the sun is equal to 333,400 and that the distance is 23,484p. With these values, one finds that the maximum tide-producing force of the sun is only about 0.46 times that of the moon. The greater distance between the sun and the earth more than balances the effect of the greater mass of the sun.

By considerations of this nature, it is easily shown that no other heavenly body can produce tides on the earth. The closest planets are too small to have any effect, and the large planets are too far away.

The motion of the sun and the moon relative to the earth is so complicated that the system of tide-producing forces changes greatly in the course of time, but the patterns are repeated in regular sequence.

From the graphs in fig. 138 it is seen, for instance, that the field of the moon's tide-producing forces is symmetrical with respect to the poles of the earth if the moon stands above the Equator—that is, when the declination of the moon is zero. In this case, when the earth rotates around its axis, the components of the tide-producing forces will in all latitudes show two equally high maxima and two equally low minima during 24 lunar hours, a period that represents the time interval between two culminations of the moon, or 24.84 solar hours. To an observer on the earth the field of tide-producing forces appears to rotate around the earth in 12 lunar hours. The declination of the moon varies, however, during one month from about 28°S to about 28°N, meaning that at the greatest southern declination the moon passes through zenith in about lat. 28°S, and at the greatest northern declination the moon passes through zenith in about 28°N. In these positions of the moon the field of tide-producing forces is no longer symmetrical with respect to the poles of the earth. At the Equator the two diurnal maxima of the tide-producing forces still remain equal, but in all other latitudes one of the maxima will be less pronounced than the other during a complete revolution of the earth. To an observer on the earth the field appears to be composed of two fields, one that rotates twice in 24 lunar hours, and one that rotates once in the same time. Instead of continuing these reasonings, which would become very involved owing to the many variations in the relative positions of moon, sun, and earth, one can arrive at a complete picture by introducing a series of fictitious heavenly bodies that will each bring about a symmetrical field of tide-producing forces and by considering the total field as composed of all of these partial fields.

Assume first that there is an ideal moon that always remains in the equatorial plane and at any locality passes the upper meridian at intervals


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of 24.84 hours, or one lunar day. As already stated, the field of tide-producing forces due to such a moon would be symmetrical in respect to a plane through the poles of the earth. Relative to an observer on the earth, this field would have identically the same appearance after one half lunar day, or after 12.42 hours. The observer could represent the horizontal and vertical components of the tide-producing forces by means of equations of the type f = F sin (nt − κ), where F would be the amplitude of the tide-producing force, n would represent the so-called angular velocity of the tide, and κ would be a constant depending upon the value of f at the time t = 0. One hour is used as the unit of time, and, since the period in this case was 12.42 hours, the angular velocity will be

Similarly, one could introduce other fictitious bodies, but the resultant angular velocities of the principal components of the tide-producing forces can all be derived from a small number of characteristic velocities:

  1. The angular velocity of earth relative to the stars, g = 15.0411°.

  2. The angular velocity of the rotation of the moon around the earth, s = 0.5490°.

  3. The angular velocity of the movement of the long axis of the elliptic orbit of the moon, which completes one rotation in 8.85 years, p = 0.0046°.

  4. The angular velocity of the motion of the earth around the sun, e = 0.04107°.

The angular velocities of the different components which together form the actual field of tide-producing forces can be obtained by combinations of these. Thus the angular velocity of the field due to the ideal moon that was first considered is 2(gs) = 28.984°.

The complete analysis makes it possible to compute coefficients that are related to the intensities of these different partial fields. The significance of these coefficients will be explained. Analysis of tides shows that to these partial fields there are corresponding partial tides that have all received characteristic names and symbols. Table 70 contains the names of the more important semidiurnal, diurnal, and long-period partial tides, their symbols, their periods and angular velocities, and their so-called coefficients.

Thus, the components of the tide-producing forces in a given latitude can be represented by equations of the form


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We write M2(ϕ), S2(ϕ), and so on, in order to indicate that the components of the tide-producing forces are functions of latitude. It should be borne in mind that we are so far dealing with forces which, referred to a coordinate system on the earth, have three components, one vertical and two horizontal, and that equation (XIV, 23) tells only that each of these components can be represented as a sum of harmonic terms, the period lengths of which are determined by considering the relative motions of moon, sun, and earth. The next question is how these forces bring about the tides.

MOST IMPORTANT COMPONENTS OF THE TIDE-PRODUCING FORCES (According to Schureman, 1924)
Name of corresponding partial tide Symbol Period in hours Angular velocity Coefficient
Symbol In degrees per hour
Semidiurnal:
Principal lunar M2 12.42 2(gs) 28.9841 .4543
Principal solar S2 12.00 2(ge) 30.0000 .2120
Larger lunar elliptic N2 12.66 2g − 3s + p 28.4397 .0880
Luni-solar K2 11.97 2g 30.0821 .0576
Diurnal:
Luni-solar K1 23.93 g 15.0411 .2655
Principal lunar O1 25.82 g − 2s 13.9430 .1886
Principal solar P1 24.07 g − 2e 14.9589 .0880
Long-Period:
Lunar fortnightly Mf 327.86 2s 1.0980 .0783
Lunar monthly Mm 661.30 sp 0.5444 .0414
Solar semi-annual Ssa 2191.43 2e 0.0821 .0365

Theories of Tides. In the preceding section the tide-producing forces were discussed, but no reference to the actual tides was made except in the content of table 70, in which were listed the partial tides corresponding to the more important tide-producing forces. Two different theories have been advanced as to how these forces can bring about tides, of which the first, the equilibrium theory, is mainly of historical interest and has been replaced by the dynamic theory. The equilibrium theory, which was first developed by Newton, is so often referred to, however, that its principles should be mentioned.

Assume first that the rotation of the earth relative to the moon is such that the same side of the earth always faces the moon. In this case the field of tide-producing forces, which was derived on p. 547 and shown in fig. 138, would remain stationary relative to the earth. The tide-producing forces would lead to a permanent reduction of the acceleration of gravity at the points of the earth nearest to and farthest away


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from the moon, and to a permanent increase of the acceleration of gravity in the plane at right angles to the line joining the centers of the two bodies. If the earth were completely covered by water, the free surface of the water would be in equilibrium only when raised slightly at the points nearest to and farthest away from the moon and when lowered slightly halfway between these two points.

Since the reduction of the acceleration of gravity at the points nearest to and farthest away from the moon is 2mρ/r3, the value of the acceleration of gravity is in the same units: 1/ρ2 − 2mρ/r3. This reduced value is found at a small distance h above the undisturbed surface, where the acceleration of gravity is 1/(ρ + h)2 = 1/ρ2 − 2h/ρ3. Therefore

If at the point nearest to and farthest away from the moon the undisturbed surface were raised to this height, there would be equilibrium, and the greatest elevations of the new equilibrium surface above the undisturbed surface would be 35.6 cm. Similarly, the greatest depressions would be half that amount, or 17.8 cm.

The equilibrium theory assumes that these flood protuberances are actually formed and that the highest portions of the protuberances lie at the points nearest to and farthest away from the moon. However, the earth does not always have the same side turned toward the moon, but rotates relative to the moon once in 24.84 hours. Each flood protuberance therefore appears to travel once around the earth in 24.84 hours, and, because there are two protuberances, the time interval between the passages of flood protuberances will be 12.42 hours. Similar flood protuberances will be caused by the sun and by the irregularity of the motion of the moon and the sun, and the actual tides should appear as the combined result of such a series of protuberances, the heights of which would be proportional to the coefficients in table 70. The obvious criticism that can be directed at this theory is that the movement of a flood protuberance over the surface of the earth cannot take place unless water masses actually change position, but consideration of the movement of the water has been completely disregarded. Other objections to the theory need not be discussed here.

The dynamic theory is based on the fact that only the horizontal tide-producing forces are of importance to the movement of the water. The vertical tide-producing forces are unimportant because they can be considered as consisting of very small periodical variations of the acceleration of gravity. It has previously been shown that the distance between isobaric surfaces in the sea depends upon the acceleration of gravity, and it is evident that variations of the latter must lead to variations of the distances between isobaric surfaces. Where the depth to the bottom is


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about 5000 m, the variations in gravity due to the tide-producing forces would, however, lead only to a maximum variation of about 0.085 cm in the distance between the sea surface and the 5000-decibar surface. Since this variation is far too small to be observed as a tide, the horizontal tide-producing forces must be considered. The problem then consists in determining what types of motion in the oceans will arise under the influence of periodically varying horizontal forces that are distributed over the ocean in a given manner. From this point of view the tides must be considered as waves that are induced by rhythmical forces and therefore have the same periods as the forces.

The general equations of the dynamic theory, which were developed by Lagrange, lead to problems mathematically so difficult that they have not yet been solved so far as the tides of the oceans are concerned. Instead, the application of the dynamic concept has followed two different lines. In the first place, the theory of tides in basins of defined geometrical shape has been developed, particularly by Proudman and Doodson; in the second place, tides in natural basins have been studied mainly by Sterneck and Defant by methods of numerical integration of the hydrodynamic equations. Both methods have helped toward an understanding of the observed phenomena, but here only the latter approach will be dealt with, because it leads to direct comparisons between theoretical results and observed conditions, and because it does not require any lengthy mathematical presentation.

Tides in relatively small bodies of water that are in communication with the open sea have much in common with seiches, and can therefore be discussed in much the same manner. The tides, however, differ from the seiches in the respect that they are forced oscillations of periods which must coincide with the periods of the impulses by which they are maintained, whereas the periods of the free oscillations depend only upon the geometrical shape of the bay.

The similarities and differences between tides and seiches are brought out by considering the oscillations in a long, rectangular bay of constant depth. Any wave in such a bay must fulfil the equations of motion and continuity. It was shown on p. 538 that, when the earth's rotation and friction are neglected, these equations are satisfied by

These equations must always hold, and, in addition, certain boundary conditions must be satisfied. These conditions depend upon whether one considers a free oscillation or an oscillation of the same period as an oscillation of the body of water with which the bay communicates, a cooscillation. In the case of a free oscillation in a bay of length l the boundary conditions were x = 0, ξ = 0, and x = l, ξ = 0, and from these


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the period of the longest free oscillation was found, Tf = 4l/c. In the case of a cooscillation in a bay the period is that of the tide in the open sea, Tc = 2π/σ, and the boundary condition takes the form x = 0, ξ = 0, x = l, η = Z cos (σt + ∊), or, at the closed end of the bay, the horizontal displacements are zero, and at the open end the vertical displacements coincide with those in the adjacent sea. The latter boundary condition leads to a determination of the arbitrary constant, a: Putting and introducing a new variable, y = x/l, equations (XIV, 23) are obtained in the form These equations represent the cooscillating tide. They show that if v, the ratio between the periods of the free and the tidal oscillations, has the values v = 1, 3, 5 …, the amplitudes of the cooscillating tide become infinite. In this case resonance occurs. It is also evident that the number of nodes of the forced wave depends upon the value of v. Fig. 139A (Defant, 1925) shows the character of the cooscillating tide in such a basin as has been dealt with for different values of v. It should be observed that Defant introduces a value of v which is one half of the value defined by (XIV, 25).

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(A) Character of cooscillating tides in bays corresponding to different values of v. (B) Character of independent tides in bays corresponding to different values of v. (According to Defant.)


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For the independent tide which is produced directly by the tidal forces, it is necessary to add on the right-hand side of the equation of motion (XIV, 9) a periodic force: x = f cos (σt + ∊). With the boundary conditions x = y = 0, ξ = 0, and x = l, or y = 1, η = 0, one obtains

Again, resonance occurs if v = 1, 3, 5, …; otherwise the number of nodes depends on the value of v. Fig. 139B (Defant, 1925) shows the character of the independent tide corresponding to different values of v.

So far, a rectangular bay of constant depth has been considered. For a bay of irregular shape and varying depth, one can start from the equations of motion and continuity in the form (XIV, 16) and can determine the character of the cooscillating and the independent tides by means of a numerical integration similar to the one developed by Defant (p. 539) for determining the free oscillation of the water in a basin (Defant, 1925). In all cases the character of the forced oscillation depends upon the relation of the period of the forced oscillation to that of the free. It can be shown that in general the increase of the range of the tide due to narrowing and shallowing is relatively small, and that very high tides at the ends of bays are, as a rule, a result of resonance.

In the preceding discussion, tide-producing forces have been considered which act in the direction of the long axis of the basin, and it has been assumed that at any given time the force is the same at all cross sections. If the basin is large, it may have to be taken into account that transverse forces exist, causing transverse oscillations, or that at a given time the force varies in the longitudinal direction.

The introduction of friction leads to further complications. In a free oscillation the period is increased by the effect of the friction, but the period of the forced oscillation must always remain equal to that of the force and cannot be altered by friction. The result is that no complete resonance can develop, because the greater the amplitude of the wave becomes, the greater is the effect of friction and the greater the increase of the period of the free oscillation. The ratio v can in these circumstances never remain at the value, 1, 3, 5, …, but can stay near one of these values.

Another effect of friction alters the character of the forced waves. In the absence of friction the cooscillating tide can be considered as a wave in a basin of constant depth which proceeds with constant amplitude and is totally reflected at the closed end of the bay. In the presence of friction the amplitude of the “incoming” wave decreases in the direction of progress, and the reflected wave has therefore a smaller amplitude


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than the incoming. The result is that the oscillation loses its character of a simple standing wave, but can instead be considered as composed of two standing waves of a phase difference of π/2 which are superimposed on each other. In the extreme case in which the incoming wave is completely destroyed by friction, so that the amplitude at the closed end is zero, the cooscillating tide has the character of a progressive wave that is subjected to extreme damping (p. 575).

Even greater complications arise when the effect of the rotation of the earth is considered. Such consideration is necessary in most cases, since a comparison of the magnitude of the acting forces shows that the deflecting force cannot be neglected when dealing with tidal phenomena, because their period length is of the order of magnitude of a pendulum day (p. 520).

In order to get some idea of the modifications that arise owing to the rotation of the earth, it is necessary to return to the complete equations of motion and to consider not only the vertical and horizontal displacements, but also the horizontal velocities to which no attention has been paid so far. If the depth is constant, if friction is neglected, and if γ = 2ω sin ϕ, the equations of motion and continuity take the form

Two solutions of these equations can easily be written. Consider first an infinitely long canal of constant width b. If the x axis is placed in the direction of the canal, the boundary conditions y = 0, vy = 0, and y = b, vy = 0 must be fulfilled, because at the walls of the canal the motion can be in the x direction only. With these boundary conditions, one obtains in the Northern Hemisphere

where, as previously, c = formula [Equation].

This solution, which was first given by Lord Kelvin (Lamb, 1932), defines a wave, the Kelvin wave, which proceeds in the x direction with the velocity c and is characterized by great amplitudes on the right-hand side and by small amplitudes on the left-hand side. At high water, when the current flows in the direction of progress, the wave crest slopes down from right to left, and the component of gravity acting down that slope is exactly balanced by the deflecting force of the earth's rotation acting in the opposite direction. At low water the directions of the slope and the current are reversed. The forces again balance each other, and the


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same is the case at any time between high and low water and between low and high water.

Another solution, given by Sverdrup (1927), is applicable to a wave that proceeds in an unlimited sea, provided that γ = 2ω sin ϕ can be considered constant:

where s = γ/σ and where now

These equations define a wave that has horizontal crests like an ordinary long wave, but a velocity of progress that has been increased in the ratio formula [Equation]. At the same time the motion of the water particles is no longer alternating back and forth, but is rotating, because, in addition to the velocities in the direction of progress (the longitudinal velocities), transverse velocities also exist. The ratio between the maximum longitudinal and transverse velocity is 1/s, and the latter reaches its maximum one fourth of a period after the longitudinal velocity was at its maximum. The character of these currents will be dealt with later, and attention will be focused here on the wave itself.

The velocity of progress of the wave becomes infinite when s = 1 and imaginary when s > 1. Now, s = γ/σ = (Tt/12) sin ϕ, where Tt is the period of the tide in hours. Thus, c becomes infinite when Tt = 12/sin ϕ, or, if sin ϕ = 1, when Tt = 12; that is, on a disk which rotates once in 24 hours, waves of this type cannot exist if the period of the wave is greater than half the period of rotation of the disk. Applied to the earth, the meaning is that waves of this type cannot exist if their period is longer than one half pendulum day. This result is not so significant as it appears to be, because the solution is valid only for an unlimited body of water, and on the rotating earth the limitation of the ocean will make impossible a complete development of this wave type. If a wave proceeds in the longitudinal direction of an ocean, the transverse velocities must vanish along the coasts, but may develop at some distance from the coast. The wave must be of an intermediate type between the Kelvin wave and the wave described here, and can be considered as composed of longitudinal and transverse oscillations, the nature of which cannot yet be expressed analytically.

In a narrow sea, transverse oscillations of the sea level will be present even if no conspicuous transverse currents are developed. The periodic


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variations of the longitudinal currents automatically give rise to a periodic transverse oscillation corresponding to that characteristic of the Kelvin wave. In a standing wave, horizontal currents reach their greatest values at the nodes, and the amplitudes of the transverse oscillation will therefore be maximal at the nodes. The tide will be zero only at the central point of the nodal line; on the right-hand side of the bay, looking toward the closed end, high water will occur one quarter period before high water at the closed end, and on the left-hand side it will occur one quarter period after. At the antinode the transverse oscillations disappear. These conditions are represented schematically in fig. 140. The reasoning is based on the assumption that the deflecting force associated with the longitudinal currents is balanced by a transverse slope, as was the case for a Kelvin wave.

In the absence of transverse oscillations the tide will be zero along a nodal line, but in the presence of transverse oscillations the tide will vanish at a single point only. Points that have high or low water at the same hour can be joined by lines which are called cotidal lines. These cotidal lines all meet at a point where the tide vanishes, which is called an amphidromic point. An amphidromic point can be caused either by the effect of the earth's rotation or by the interference of two tidal waves.

If a standing oscillation exists in a bay, there are no cotidal lines, because high water occurs at the same time everywhere. However, when transverse oscillations are present which differ from the longitudinal by one quarter period, an amphidromic point is developed that lies on the original nodal line (fig. 140). To the right of the center, high water occurs at the time t = 0, to the left at the time t = ½T, and the other cotidal lines are spaced between these. Thus, the rotation of the cotidal lines around an amphidromic point that is caused by the earth's rotation is, in the Northern Hemisphere, counterclockwise (as shown in fig. 140), but it is clockwise in the Southern Hemisphere. This characteristic may be used for deciding whether or not an observed amphidromic point may be due to the rotation of the earth.

The methods of analysis that have been set forth here have been applied mainly by Sterneck and Defant to the tides in “adjacent seas,” such as the North Sea, the Adriatic Sea, the Red Sea, and others. In the case of the North Sea, from which numerous current measurements are available, other methods developed by Proudman and Defant have been used for deriving a complete picture. The considerations have even found application to the Atlantic Ocean, the principal tides of which have been studied by Sterneck and Defant. Their conclusions will be dealt with when the tidal currents are discussed. In the other oceans, application of the dynamic theories presents enormous difficulties and has not been attempted. The above reasonings are purely qualitative, but G. I. Taylor has succeeded in giving an exact solution for a bay of rectangular


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shape and constant depth (see Defant, 1925, 1929). In several aspects this solution supports the above reasoning and shows particularly the arrangement of the cotidal lines.

A generalization of the observed conditions has been made by R. A. Harris (1894–1907), who divided the oceans into “oscillating areas,” the period of free oscillation of which would be about 12 or 24 lunar hours if they were enclosed by solid boundaries. The areas were selected so that the observed difference in the time of high water at both ends of the area would be approximately 6 or 12 hours. Within each area one or more nodal lines would be present. This division of the oceans into “oscillating areas” brings a surprising consistency in the otherwise confused picture of the tides, but, as pointed out by G. H. Darwin, it can hardly be accepted as a physical explanation because, among other factors, the rotation of the earth has been disregarded.

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Schematic representation of transverse oscillations in a bay in the Northern Hemisphere leading to the development of an amphidromic point.


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The Character of the Tides. The tide-producing forces can be computed with great accuracy, but the response of the oceans to these forces is too complicated to be determined. Experience has shown, however, that the tides can be considered as composed mainly of a series of harmonic oscillations, or partial tides, having the periods of the tide-producing forces. To these terms must be added, in many localities, annual and semiannual terms which are not related to astronomical forces but which are ascribed to the effect of prevailing winds or changes of sea level due to heating and cooling (p. 459). These long-period tides


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are called meteorological tides, in contrast to the astronomical. In localities where the wave of the tide is deformed by friction, particularly in shallow bays, it may be necessary to introduce higher harmonic terms, the periods of which are fractions of the periods of the principal partial tides. Thus, in any given locality the tide can be presented by means of a formula similar to equation (XIV, 22), adding, if necessary, the meteorological tides and the higher harmonic terms. The periods of the astronomical tides are the same as those of the forces. However, the coefficients of the different terms are mostly not proportional to those representing the forces, but depend upon the relative importance at the locality in question of the waves produced by the various forces. Similarly, the phase angles differ from the phase angles of the forces.

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Tide curves at spring tide (left) and neap tide (right).


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The most important tide-producing forces are those of nearly semi-diurnal and nearly diurnal period, for which reason the character of the tide in any locality depends mainly upon the relative heights and phase angles of the partial tides corresponding to these forces. Let us first consider the partial tides due to the semidiurnal lunar forces of a period 12.42 hours and to the semidiurnal solar force of a period of 12 hours. The former will give rise to two high waters and two low waters in 24.84 hours (one lunar day), and the latter to two high waters and two low waters in 24 hours (one ordinary day). Thus, the high water due to the moon will be retarded 0.84 hours every day, or about 50 minutes relative to that caused by the sun. After coinciding on a given day, the partial high waters will move apart, to coincide again when the lunar tide has been retarded 12 hours, or after 12/0.84 = 14.3 days. The semidiurnal tide will be great when the two partial tides coincide and small when they counteract each other (fig. 141). The large tides are called spring tides, and the small tides are called neap tides, and in a locality in which the semidiurnal components are dominating, spring and neap tides come at intervals of 14.3 days. Where the tides are of the semidiurnal type, the ratio between the ranges of the lunar and solar partial tides may be fairly close to the theoretical ratio 1:0.47 (table 70 p. 550), in which case the ratio between the ranges of spring and neap tides is fairly close to 1.47:0.53 = 2.77.


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If the tides followed the tide-producing forces, the lunar and solar semidiurnal tides would coincide at full and new moon and would be opposite in phase at the quarters of the moon. In most localities the tides lag somewhat behind the tide-producing forces, and spring tides occur a day or two after full moon and a day or two after new moon. The time difference between the meridian passage of full or new moon and the occurrence of the highest high water is called the age of the tide, and is given in days.

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Examples of different types of tides resulting from simultaneous diurnal and semidiurnal tidal components.


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The close relation of the tide to the moon is also demonstrated by the fact that, where a semidiurnal tide prevails, high water always comes at nearly the same number of hours after the moon passes through the upper or lower meridian—that is, after the moon is due south or due north. The mean time difference between the meridian passage of the moon and the occurrence of the next high water is called the mean high water lunitidal interval, and is measured in hours.

Diurnal partial tides greatly complicate the picture, because the tide will depend upon the relative magnitudes of the semidiurnal and diurnal components and upon the time at which these components reach their maxima. Figure 142 illustrates a few possible combinations. Figure 142A shows an ordinary “mixed type” of tide in which one of the two high waters of the day is much higher than the other, and one of the


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two low waters is much lower. Also, the time interval between the higher low water (l) and the lower high water (h) is much shorter than the time interval between the lower low water (L) and the higher high water (H). This tide is characterized by a diurnal inequality. The tides of the Pacific coast of the United States are of the mixed type, showing a considerable diurnal inequality, but the tides of the Atlantic coast are more nearly of the semidiurnal type.

Figure 142B illustrates a case in which the inequality is found in the high waters only, the two low waters being equally low. Figure 142C illustrates a case in which only one high and one low water occur during the day, because the other high and low water melt together into a period of several hours with nearly constant water level. This particular phenomenon is called the “vanishing” tide.

The examples in fig. 142 were constructed by combining diurnal and semidiurnal tides of different heights and different phase angles. It is evident that different combinations lead to other tide curves in which the diurnal inequality of the tide appears more or less pronounced. The diurnal inequality of the tide varies during a month, because the distribution of the moon's tide-producing force over the earth varies with the declination of the moon. The tides that display the greatest diurnal inequalities are called the tropic tides, since they occur when the moon's declination is at its maximum or at its minimum—that is, when the moon is nearly above the Tropic of Cancer or of Capricorn.

Taking partial tides of other periods into account, and considering the fact that the tide-producing forces vary with the distance from the earth of the moon and the sun, one finds that a nearly unlimited number of possible types of tides exists and that, in any given locality, the type of the tide may change considerably during one month. As a rule, however, the tide has the same characteristics at neighboring stations on an open coast. Detailed information as to the character of the tides in different localities is found in several of the books listed at the end of this chapter.

Several terms used for describing the tide have been defined, but a few more must be added. Mean sea level is the plane about which the tide oscillates (Marmer, 1927). It is determined from tidal observations by averaging the tabulated hourly heights of the tide over a period of several years. Mean sea level does not coincide, as a rule, with an equipotential surface, because, where permanent currents are present, the sea surface always slopes at right angles to the current (p. 391). Mean sea level may also rise or drop along a coast, as is evident from results of precision leveling along the Atlantic coast of the United States (p. 677). Daily, weekly, monthly, and yearly sea level can be derived from observations during a day, a week, a month, or a year. For daily sea level, it is necessary to state how the average has been computed (see Marmer, 1927).


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Mean high water is the average height of all high waters over several years, and, similarly, mean low water is the average height of all low waters over several years. The half-tide level lies exactly half way between mean high water and mean low water, and differs as a rule from mean sea level. In localities where the tide shows a considerable diurnal inequality, mean higher high water and mean lower low water are computed from the highest and lowest tides of each day.

In bays and in seas which communicate with the ocean through a relatively narrow opening, the tide may differ from that in the ocean, because the shape of the bay or the adjacent sea may favor the development of certain components of the tide. Modifications due to the rotation of the earth may also arise.

The tide in the English Channel represents an example of the latter modification. The tide there has in part the character of a progressive wave which enters the continental shelf from the Atlantic Ocean and which, as it advances, takes the appearance of a Kelvin wave, with small ranges on the left-hand side, the south coast of England, and great ranges on the right-hand side, the northwest coast of France. On the coast of France the Bay of St. Malo is particularly famous for its large tides, because in the inner part of the bay the range of the spring tide is up to 12 m (39 feet). This enormous range is in part attributed to the narrowing of the bay and the shoaling of the bottom.

The largest known tides occur in the Bay of Fundy, where, in Noel Bay, spring-tide ranges up to 15.4 m (50.5 feet) have been measured. However, this tide can be accounted for in a different manner. In the Bay of Fundy, high water occurs nearly simultaneously all around the bay and, furthermore, it has been found that the strongest tidal currents flow into the bay when the water is rising most rapidly, and out of the bay when the water is falling fastest. These features—the increase in the range of the tides toward the end of the bay, the simultaneous occurrence of high and low water all over the bay, and the maximum currents at mean water—all indicate that in the Bay of Fundy one has to deal with a standing wave.

The range of the tide is, however, not zero at the opening of the bay, for which reason the tide must be of the cooscillating type (fig. 139), and the great increase of the range toward the head of the bay must be due to resonance. An exact computation of the period of free oscillations of the waters of the Bay of Fundy has not been undertaken, but according to rough estimates this period lies between 13 and 11.6 hours (Defant, 1925). Such a period means that in the Bay of Fundy the ratio v = Tf/Tc (p. 553) is probably sufficiently close to unity to bring about resonance, but the increase of range toward the end of the bay may also be augmented by the narrowing and the shoaling of the upper part.


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Owing to the rotation of the earth, the tides are larger on the southeastern than on the northwestern shores.

The tides in the Bay of Fundy are remarkable because of their great range, but the increase of the range from the opening to the end of the bay is not greater than in some other localities. The tides of the Adriatic Sea have been examined very thoroughly by Defant and Sterneck, who have found that there the cooscillating tide dominates. The longitudinal semidiurnal tide shows a node at a distance from the opening of about three quarters of the length of the sea, and inside the node the range of the semidiurnal tide increases rapidly. On the other hand, the range of the diurnal tides increases regularly from the opening toward the end, and is, at the end, about four times as great as the range at the opening. The range of the Bay of Fundy tide is also increased about fourfold from opening to end, and this increase represents therefore no exceptional case. By studying fig. 139 it is easily seen that the range of the cooscillating tide must increase from the opening to the end of a bay if the ratio v lies between 0 and 2, and that the increase must be the greater the closer v is to unity. In the Adriatic Sea the transverse oscillations due to the earth's rotation have also been studied, and excellent agreement has been obtained between observations and theory. Interested readers are referred to Defant (1925).

Attempts to represent the tides of a large area on charts encounter considerable difficulty, because the character of tides is known from coasts and islands only, and the data can be combined in many different ways. The first comprehensive representation was prepared by Whewell, who in 1833 published a map of the cotidal lines of all oceans (Marmer, 1926). As explained earlier (p. 557), cotidal lines are lines which join points having high water at the same time, referred to Greenwich or some other standard meridian. Later, charts of cotidal lines were prepared for several smaller areas, and for the Atlantic Ocean the cotidal lines of the semidiurnal and the diurnal tides have been plotted separately. Charts dealing with the partial tides are preferable, because the character of the cooscillating and independent partial tide of any body of water depends on the period of that tide. Similarly, charts showing the range of the tide have more rational meaning when they present partial tides, and charts of this nature have been prepared for some adjacent seas. When dealing with tidal ranges, one should use data from well-exposed stations, since the tide in bays and estuaries may be distorted.

Owing to the uncertainty involved in arriving at a general representation of the tides, it is often necessary to interpret the available data in a certain manner in order to arrive at a consistent picture. The most outstanding example of such interpretation is Harris's division of the oceans into “oscillating areas,” which was mentioned on p. 558. However,


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a satisfactory presentation of the tides of the oceans has not yet been given, with the possible exception of the tides of the Atlantic Ocean, to which we shall return.

Analysis and Prediction of Tides. Any observed tide curve can be represented, as has already been stated, by means of a series of harmonic terms, the periods of which correspond to the periods of the astronomical and meteorological tides (p. 550). This is true regardless of how complicated the tide is. The coefficients of the different terms can at any given localities be determined with great accuracy by means of harmonic analysis if sufficient data are available. Methods employed in harmonic analysis are described by Schureman (1924). Here it will be mentioned only that the harmonic analysis of tidal data is a complicated process, because many periods have to be considered, some of which differ but little in length. Methods have been developed, however, which permit fairly rapid calculations.

The tide curve can be reproduced with a high degree of accuracy if a sufficient number of harmonic terms have been evaluated. These terms can be used for computation of future tides, since the tide is one of the few geophysical phenomena that repeat themselves with nearly astronomical regularity. The process of computation consists in calculating the tide corresponding to each single term, using the empirically determined amplitudes and phase angles and finally adding the terms, thus constructing a predicted tide curve. The calculation and addition is now made by means of specially constructed tide-predicting machines for preparing tide tables. These machines are operated by the U. S. Coast and Geodetic Survey, Washington, D. C., the British Admiralty, London, and the Deutsche Seewarte, Hamburg. The tide tables, which are issued for each year, give advance information as to the time and height of high water and low water of all commercially important ports of the world. In addition, information is given as to the time difference between high water at principal ports and high water in neighboring localities.

A popular description of the tide-predicting machine of the U. S. Coast and Geodetic Survey is given by Marmer (1926), and a technical description is given by Schureman (1924). Experience has shown that the deviations of the actual tides from the predicted values are mostly small and are due to nonperiodic disturbances caused by winds and shifts in currents.


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