TIDES. 1. Introductory. Persons living along the seashore are familiar with a semi daily rising and falling of the waters, which, although generally amounting to only a few feet, often suffices to cover and bare by turns wide stretches of the sea-shore. Without tide tables, or any knowledge of the moon's posi tion, the approximate time of the tide can be foretold from its observed time of occurrence on a previous day, by allowing 50 minutes for its daily retardation. Toward and after the time of Caesar, the Romans were well acquainted with ocean tides as the writings of Caesar, Sen eca, Pliny the Elder, Claudianus, and others clearly show. By referring the tides to the attraction of the moon and Newton took the first important step in their explanation. Since his time important investigations of the laws of the tides have been made by Laplace, Airy, Kelvin, G. H. Darwin, H. Poincare and many others.
2. Definitions, etc. The tide is the periodic rising and falling of oceanic and other large bodies of water, due mainly to the attraction of the moon and sun as the earth rotates upon its axis. Remarkable stages of the water level at a given place, whether due to earthquakes, gales or other causes which probably have no definite law of recurrence, although popularly known as (tidal waves ,D cannot be regarded as belonging to tidal phenomena. The rising and falling is accompanied by, and depends upon, lateral or horizontal movements of the waters called tidal currents or tidal streams or the flow and ebb. Their periodic character distinguishes them from ordinary ocean currents. The tide rises until it reaches a maximum height called high water and then falls until it reaches a minimum height called low water. The differ ence in height between a high and a low water is called a range of tide. At most ports two high waters and two low waters occur each lunar day. The average length of a lunar day is 24 hours 50 minutes 28 seconds. The inter val of time between the transit of the moon across the meridian and the occurrence of high or low water is called a lunitidal interval. The average value of the high-water lunitidal inter val at any seaport is sometimes called the cor rected establishment to distinguish it from the high-water lunitidal interval on the days of new and full moon, which is known as the es tablishment, or the vulgar establishment, of the port.
At the times of new and full moon the tidal forces of the moon and sun act in the same direction, whereas at first and last quarters they oppose each other. When they unite their forces we have spring tides, characterized by large ranges of the tide; when they are opposed, neap tides, having small ranges. The spring and neap tides usually occur soon after the cor responding phases of the moon. The interval is called the retard or age of the tide, or age of the phase inequality, and is usually less than 60 hours. The lunitidal intervals have their mean values at the times of spring and neap tides; the tides occur a fraction of an hour earlier between spring and neap tides, and later between neap and springs. Other things being equal, the range of tide is greater than usual by about one-sixth part when the moon is near perigee and about as much less than usual when near apogee. An increase or de crease of about one-tenth part of the range oc curs when the moon is near the equator or near its point of extreme declination, respectively.
Diurnal inequalities among the four tides of a day are due to the presence of a diurnal wave or partial tide, whose period is approxi mately 24 hours. The cause of this wave lies in the fact that if the moon is north or south of the equator, its tidal forces are somewhat different both in magnitude and in direction when two times half a lunar day apart are compared.
3. The Tidal Forces. All particles of the earth (the seas included) will continue to oc cupy positions fixed relatively to one another if no other forces are impressed upon them than the following: The earth's attraction; its centrifugal force of axial rotation; and a force acting upon all of its particles alike, for exam ple, the centrifugal force due to the revolution of the earth about the centre of gravity of earth and moon. If an extraneous force does not act upon all particles alike, then motions will be set up in the yielding parts. The at traction of the moon upon a given particle (near the surface, say) is along a line drawn at any given instant) from the particle to the moon's centre; its intensity, which is inversely proportional to the square of the distance, and its local direction (that is, direction with re as the earth rotates upon its axis. The attrac tion of the moon upon a particle at the earth's centre (or upon the earth as a whole) is along a line drawn from the earth's centre to that of the moon; it is independent of the earth's axial rotation. Because the action of the moon upon the surface particle differs from its action upon the particle at the earth's centre there re sults a tendency to produce motion relatively to the earth's centre. A consideration of this tendency will enable us to answer the question why there should be two high waters each lunar day, instead of only one high water. In a sin gle sentence, the reason is that the moon at tracts the waters on the hemisphere facing the moon more powerfully than it does the earth; but attracts the earth in general more power fully it does the waters on the far ther side of the earth. The difference be tween the action of the moon at any point of the ocean and its action on the centre of the earth is the tide-producing force at the specified point. It is not difficult to show that, to higher powers of the small quantity air, the vertical and horizontal components of the moon's tide-producing force are very nearly M 3 M as ( g 3 cos' 1) and g sin 2 8 respec Eri 2 E tively, where M denotes the mass of the moon, E that of the earth, g the force of gravity, a the mean radius of the earth, r the places on the earth's surface. The arrows lo cated upon the same small 'circle are supposed to be of equal length, and all arrows are sup posed to lie in a system of great circles which meet in a point directly under the moon and, of course, in the antipodal point. At these two points and along a great circle midway between them the length of the arrows is zero; in ocher words, the force vanishes. The system of ar rows is fixed with respect to the moon, and so sweeps over the surface of the earth as the moon performs her apparent daily revolution. The system shifts somewhat when the moon is north or south of the equator. At any point P on the earth's surface, the moon being upon the equator, the horizontal forces are equal in distance between centres of earth and moon, and 0 the zenith distance of the moon rected for parallax. The numerical value of M as when r has its mean value, is E 0.000000056; and so the vertical force has a range of 0.000000168g, as has also the horizon tal force. The solar tidal force is 46 per cent that of the lunar. The tides are mainly due to the horizontal component of the forces. These are the forces which deflect a plumb line, although by an amount so small that it can hardly be measured. The deviation, in case of the moon, amounts to only 0."017 either way from the mean vertical. For a sufficiently deep body of water extending 163 nautical miles along the equator the range of tide at either end will be one inch.