For most places the constituent of largest amplitude is the principal lunar semi-diurnal constituent M2, as would be expected from the equilibrium-form. When this is the case we may regard the elevation as approximately the average tidal elevation at the place in question, and the other constituents as producing inequalities. Where S2 is the pre dominating constituent the average interval between successive high waters will be 12 hr. Where M2 is the largest constituent and S2 comes next, as in British waters, we have the phenomenon of spring tides and neap tides. Spring tides occur when the phases of M2 and are the same so that S2 reinforces M2, and neap tides occur when these phases differ by i8o° so that S2 counterbalances M2 as far as it can. The phase of S2 gains on that of M2 at the rate so that the interval between successive spring tides is half a synodic month. While S2 reinforces M2 the period of the resultant tide is less than the average, and we have the phenomenon known as priming, whereas while S2 partly counterbalances M2 the period of the resultant tide is longer than the average and we have the phenomenon known as lagging. In the equilibrium-form it is easily seen that spring tides occur at new and full moon and neap tides at the quarters, but in the actual tides these phenomena usually occur a day or two later. Where M2 is the largest constituent and N2 comes next, as in certain Canadian waters, we have a phenomenon similar to springs and neaps except that it depends on the distance of the moon from the earth. Since the phase of M2 gains on that of N2 at the rate a — w, the period from one reinforcement to the next is an anomalistic month. In the equilibrium-form the reinforcement occurs at the time of lunar perigee, but this is not usually the case in the actual tides. Where M2 is the largest constituent and K2 comes next we have a similar phenomenon de pending on the declination of the moon. The presence of the diurnal constituents gives rise to the diurnal inequality, and when the combined diurnal constituents exceed the combined semi-diurnal constituents we may have only one high water in the day. Since the period of the M2 constituent is equal to the interval between two successive culminations (upper or lower) of the mean moon over any meridian, it follows that M2 high water at any place will always occur at the same interval after a culmination over a definite meridian.
There are other and older methods of describing the chief character istics of the tides at any place, besides that of specifying the harmonic constituents, though in general they are less satisfactory. The non harmonic constants for a particular place consist of such quantities as the average of all high water intervals, the average of all low water intervals, the average high water interval at springs, the average range, the average range at springs, and the average range at neaps. When one or two harmonic constituents predominate their constants may often be roughly determined from non-harmonic constants.
St. Lawrence there is a progression away from the Atlantic. Considering next the Indian Ocean we exclude the Bay of Bengal and the Arabian Sea. There is then a progression from Madagascar round by Somaliland, Ceylon, Sumatra and Java to Australia, the amplitudes being small. Along the east coast of the Pacific there is a progression both northwards and southwards away from Lower California, the southerly progression being very rapid round the Central American bay. Following the north erly progression round the northern and western boundaries of the ocean we come across a number of stretches, along each of which the constituent is nearly simultaneous. One is along the Kurile Islands and another from the south of Japan to the north-west of New Guinea; another is from the south-east of New Guinea to the north of New Zealand. Along the east coast of Japan there is a relatively slow southerly progression, and along the east coast of New Zealand there is a relatively slow north erly progression. Over the Southern Ocean the amplitudes are small. In the great oceans away from the Continental shores our direct ob servational knowledge is limited to the islands.
Distribution of Elevation Away from Shore.—(II). A convenient method of describing the distribution of the elevation in a harmonic constituent over an area of the sea is to give on a map the cotidal and lines. A cotidal line is one through all points of the sea with the same value of r, and a co-range line is one through all points with the same value of II. The records from accurate observations of elevations away from the shore, however, are so scanty that they do not form a sufficient basis for a description of the distribution of a harmonic con stituent. To give such a description we must have recourse to theory, and when the currents have been well observed this theory is supplied by the fundamental dynamical equations. These equations connect the elevation-gradients with the currents and the external forces, including those of friction. From a knowledge of the currents and a law for the frictional forces the elevation-gradients can be calculated. When the elevation is also known at a particular point, the directions of the co tidal and co-range lines and also the distance apart of neighbouring members of these lines can be calculated. Such conditions are fulfilled for many coasts. If the elevation-gradients can be calculated along a line which passes through one or more points at which the elevation is known, it is clear that methods can be devised by which the elevation can be calculated all along the line. In this way the distribution over the North Sea and northern part of the Irish Sea as shown in fig. 2 have been de termined (J. Proudman and A. T. Doodson, Merseyside [1923], Phil. Trans., A[1924])• Tides in small seas open to the ocean, are usually such that the actual elevation-gradients are large compared with those of the equilibrium form, for the equilibrium range of M2 only changes by 1.6 ft. from the equator to the poles. It follows that over such seas the direct effect of the generating forces is negligible, and this means that the tides of these seas are maintained by those of the oceans. It is on the water of the great oceans that the gravitational forces of the moon and sun generate the greater part of the tides. To simplify the argument we shall first neglect frictional forces; although often important near land, these forces are not as a rule predominant. The dynamical equations then express that the direction of the acceleration is the same as that of the downward surface-gradient, and this is perpendicular to the surface contour line. Now at the times of maximum and minimum current the acceleration is perpendicular to the current. We therefore see that at these times the surface contour lines have the direction of the current.