Equations of Motion.--The dynamical theory of tides introduced by Laplace in the 18th century has given much insight into the problem of tides in the ocean. The following assumptions and approximations are commonly introduced:

(1) The water is assumed to be homogeneous; (2) vertical displacements and velocities of the water particles are assumed small in comparison to the horizontal displacements and velocities; (3) the water pressure at any point in the water is given adequately by the hydrostatic law; i.e., it is equal to the head of water above the given point; (4) all dissipative forces are neglected; and (5) the ocean basins are assumed rigid (as if there were no bodily tide), and the gravitational potential of the tidally displaced masses is neglected.

If assumptions (1) and (3) are valid it can readily be shown that the tidal currents at any locality are uniform with depth.

Take a to denote the radius of the earth and g the acceleration of gravity at its surface. Position on the earth's surface will be
designated by the longitude  and the co-latitude (angular distance from the north pole to the point under consideration)  . Let h denote the depth of the ocean at any point, so that h is a function of  and  . Also, let  denote the elevation of the free surface at any point of the ocean above its mean level, and u and v the southward and eastward components of the tidal current, so that these three quantities are functions of , as well as of the time t. Then the equation of continuity, which expresses the condition that the volume of any mass of water must remain constant, may be written in the form (1)

The equations of motion are formed by equating the southward and eastward components of acceleration of the water particles to the sum of all the forces per unit mass acting on the water in the southward and eastward directions, respectively. The acceleration consists of three parts: the local acceleration; the field acceleration; and the Coriolis acceleration resulting from the earth's rotation. As an example, the southward component of acceleration is equal to

It may be noted that these equations are linear in u, v,  and with the result that the tide-generating forces may be separated into several constituents, the sum of whose effects is exactly equivalent to the effect produced by the sum total of all the tide-generating constituents. Indeed, as has already been pointed out, this is the basis of the harmonic development of the tide. Let us review under what conditions the ocean tide is truly linear and under what conditions this linearity breaks down. Unless the elevation of the sea surface above its mean position is always small as compared to the mean depth h, equation (1) becomes nonlinear. Similarly, if the field acceleration becomes so large it is no longer negligible as compared to the local acceleration equations (2) become nonlinear. This is most likely to occur in regions of large tidal currents with a large spatial rate of change (current shear). Nonlinear dissipative forces may be another source of nonlinearity. All these effects are quite small in the vast oceanic regions but tend to become larger in regions of shallow water and in the vicinity of the coast line, and consequently are referred to as shallow-water tides.

As to the other approximations, that of neglecting all dissipative mechanisms (4) is not a very good one, but more will be said about this point in the section on Tidal Friction. The bodily tides of the earth invalidate to a certain extent assumption (5) by virtue of a small variable movement of the sea floor and of the earth's own gravitational field. If , designates the variable elevation of the sea floor above its mean position, equation (1) will stand unaltered if we take  to denote the elevation of the sea surface relative to the sea floor, but in the formulas for pressure gradients  must be replaced by +. The disturbance n the gravitational forces brought about by the displacements of water and solid earth must be accounted for by adding a term , to . Inhomogeneities in the sea water have been neglected according to assumption (1), but may have a more important effect in tidal motions than has been anticipated. At any given time of the year a large part of the ocean is covered by a nearly homogeneous layer of light, warm water 50 to 200 meters thick. Such stratification of the sea water may influence the vertical structure of the tidal current with resulting deviations from their generally assumed uniformity with depth. Bottom friction will also introduce vertical shear in the tidal current.

Suppose now that we consider the ocean's response to one harmonic constituent of the equilibrium tide. If we take a long-period constituent,  will have the form of a standing wave over the entire earth. A constituent of any other species will have the form of a progressive wave rotating relative to the earth about its axis in the westward direction. Such progressive waves can, however, be expressed as the sum of two standing waves separated in longitude by 90°, and separated in time also by 90°. In any case, therefore. it will suffice to express  in the form

Nonlinear Distortion (Shallow-Water Tides)-At places where the tidal elevation becomes appreciable as compared to the water depth, or other nonlinear effects occur, the principle of superposition breaks down. One might think that under these circumstances the concept of harmonic constituents would be completely useless, but it has been found possible to retain this concept provided that additional constituents, the so-called shallow-water tides, be included. The regions where this phenomenon occurs are generally confined to rivers, estuaries and shallow gulfs. The shallow-water constituents are not produced by the direct action of the tide-generating forces, but by the distortion, attributable to shoal water, of already existing tidal constituents which have been formed in the ocean. In fact, if the region of distortion is small enough (as it usually is), the tide-generating forces have little or no direct effect on the shallow-water tide.

 

It is no longer correct to consider separately the reactions to individual equilibrium constituents, but let us take the simple example of only one equilibrium constituent in order to gain insight into the problem. The response of a nonlinear system to the harmonic forcing constituent will not be harmonic, but at least it will be periodic, having the same period as the forcing or primary constituent. If  is the period of the primary constituent, it is well known from the theory of Fourier series that a periodic function of this period can be expressed as the sum of purely harmonic terms having speeds a etc. Extending this idea to the actual situation one should expect to find shallow-water constituents having periods of one-half, one-third, etc., of those of the major tidal constituents. The shallow-water constituents obtained in this way are called overtides because of their analogy to overtones in acoustics. The same symbol is used to designate these overtides as is used to designate the equilibrium constituent which gives rise to them, and again the subscript designates the species (approximate number of periods in a day) to which the overtide belongs. For example the constituent M₂ gives rise to the series of overtides M₄, M₆, etc., and K₁ gives rise to the overtides K₂, K₈, etc. In some cases an overtide will have exactly the same period as another equilibrium constituent and it will be impossible to separate the two by harmonic analysis. For example, K₂represents the lunisolar semidiurnal constituent, but an overtide of exactly the same period as K₂ is produced by nonlinear distortion of the lunisolar diurnal constituent K₁. In this and in similar cases the symbol K₂ (referring to the actual tide) represents the combined effect of both the linear reaction to the equilibrium constituent K₂and the overtide.

The theory of progressive shallow-water waves of finite height has been used to estimate the relationship between the amplitudes of the overtides and primary tide. The amplitude of the nth overtide is found to be proportional to the nth power of the amplitude of its primary at any given place, as the amplitude of the primary varies in time. For example, if the range of the semidiurnal tide is twice as great during the spring tides as during the neap tides, the range of the quarter-diurnal overtide resulting from distortion of the semidiurnal tide will be approximately four times as great during the spring tides as during the neaps. The range of the 6-diurnal overtides will be eight times as great during the spring tides, etc. As to the 19-year nodal variation in amplitude, the node factors f of the overtides bear a relation to those of the primary as illustrated in the following example in the case of the overtides of the constituent M₂:

The above relations seem to be confirmed fairly well in most cases by tidal observations in spite of the fact that assumptions in the theory from which they have been derived are far from fulfilled. For example, in many regions where overtides are important, such as some rivers and estuaries, the tidal wave has more nearly the form of a standing wave than that of a progressive wave, but the theory for standing shallow-water waves of finite height has not been adequately developed. Also, the nonlinear effects caused by one constituent cannot, strictly speaking, be superimposed on those caused by another.

The nonlinear interaction between two or more primary constituents can be accounted for by the inclusion of compound constituents. In the case of two interacting primary constituents having speeds al and 0?2, a series of compound constituents will result, having speeds etc., including all possible combinations of the form , where a and b are integers. The relationship of the amplitudes of the compound constituents to those of their primaries has been approximated by assuming that the system responds like a power-law filter; i.e., like a system whose output is equal to a polynomial function of the input. In this way the relative importance of the compound constituents of any given species as well as the dependence of their node factors on those of the primary constituents can be evaluated. No general conclusions can be drawn concerning the relative importance of two compound constituents belonging to different species.

TABLE II--Shallow-Water Constituents

Symbol	Speed	Speed in °/hr.	Node factor
2SM2		31.016
MNS2		27.424
MK3		44.025
MK3		42.927
M4		57.968
MS4		58.984

Table II includes the more important shallow-water constituents, both overtides and compound constituents. In the table, only two of each of the semidiurnal, terdiurnal and 4-diurnal shallow-water constituents are given for illustration, but there are ports for which many others would be needed to give an accurate representation of the tide. In fact for some ports the number of these constituents that would be needed is so high that the harmonic method becomes impractical. It may be noted that the two terdiurnal constituents listed in Table II are given the same symbol. Actually there is some confusion as to the designation of the compound constituents. The designations given in Table II are based on the nomenclature used by the United States coast and geodetic survey.