The system of arrows in Figs. 1, 2, 3 are intended to represent the horizontal component of the moon's tide-producing force at various magnitude and direction to the horizontal forces at P', a point upon the same parallel of latitude as P, but 180° distant in longitude; or, what amounts to the same thing, they repeat them selves at any given point P every half lunar day, or 12h. 25m. 14s. on an average. But when the moon is not upon the equator, the forces are generally not the same at P and P', either in magnitude or in direction, and so do not ex actly repeat themselves every half lunar day. This alternation of the forces gives rise to a diurnal inequality in the tides. It will be no ticed that for places situated upon either side of the equator, the forces have, when the moon is upon the equator, a meridional component directed from the poles toward the equator, and that this component nowhere points from the equator toward the poles; consequently the ex istence of the moon causes the water (half-tide level) at the equator to be higher than it would otherwise have been. The moon's movement in declination, therefore, causes a fortnightly fluc tuation in half-tide level. Similarly the stm produces a semi-annual fluctuation.
4. Real Equilibrium If in the case of any body of water, its free period be several times smaller than the period of the tidal forces, say less than three or four hours, the surface will at every instant be normal to the plumb line as disturbed by these forces. In particular, consider the mean lunar semi-diurnal tide of a deep lalce situated in north latitude. Upon referring to Fig. 4 it may be inferred that high water will occur at a point south of the no-tide point when the moon is on meridian: at a point west of the no-tide point at 3 o'clock, lunar time; at a point north of it, at 6 o'clock; at a point east of it, at 9 o'clock. The no-tide point is the centre of gravity of the surface. This theory. nearly explains the tides found in Lake Superior and the semi-diurnal tide in the eastern portion of the Mediterranean Sea; it partially explains the semi-diurnal tides in the Gulf of Mexico and the Caribbean Sea, and the diumal tides in the Atlantic Ocean east of the United States.
5. Hypothetical Equilibrium If a spherical body like the earth were entirely cov ered by an ocean so deep that its free period of oscillation would be several times smaller than the period of the tidal forces, the tidal forces of the tnoon would cause the surface of the ocean to assume the form of an ellipsoid of revolution with the longer axis pointing to ward the moon's centre. For any zenith dis tance 8 of the moon the height of the tide above the undisturbed spherical surface becomes 1 M a (3 cos' 6 — 1) where a de 2 Re 1-1 aide notes the density of the water and de that of the earth. The numerical value .of M a4 1— — is .17 feet. The corresponding value for E the sun is 0.54 foot. If the a/de = 0, the range of the hypothetical lunar tide becomes 1.8 feet; if cr/ de=-- 1, the range becomes 4.4 feet. In case of the earth a / de--= A.
The hypothetical tide just described can be easily calculated for any time and place and is known as the uncorrected equilibrium tide. It bears no resemblance to the actual tide of our oceans.
6. Some Dynamical Questions Involved in the Because the requirements for equilibrium tides are seldom found in the oceans, their waters must be treated as aggre gates of heavy particles performing some kind of oscillatory motion. A progressive free wave in a canal has as its velocity of propagation V g h and for the maximutn velocity of the water particles A \IL where h denotes the undisturbed depth of the water and A the am plitude of the vertical movement. The longest or fundamental period of free oscillation of a rectangular area or sheet of water is twice length of sheet.
period— Vg k Sheets tapered or sharpened at the ends oscil late more rapidly than do rectangular ones of the same length, while sheets narrowed at the middle or broadened at the ends oscillate less rapidly. The free oscillations of a given body of water can often be approximately deter mined by comparing with a more simple body whose motion is lcnown. The given body need not have a strictly uniform depth nor be com pletely surrounded by land.
The general equations of motion for matter upon a rotating sphere show that a moving par ticle of unit mass is deflected or accelerated relatively to the earth's surface, toward the right in north latitude, toward the left in south latitude, as if by a force whose numerical value is velocity X0.0001458 sin (latitude), the veloc ity being expressed in feet per second and the force in poundals (Ferrel's law). This divided by 9 or 32.1722 gives the transverse slope which a river, or strait through which there is a cur rent, will assume on account of the deflecting force of the earth's rotation.
7. Hypothetical Dynamical The case of an equatorial canal encircling the earth is simple and instructive, although bearing no resemblance to any existing tidal body. If the depth of the water be greater than 67,000 feet, high water will occur when the moon is on meridian (above or below the horizon) ; if less than 67,000 feet, low water will occur when the moon is on meridian. For the depth 67,000 feet the range of tide will become very large. If this depth be greatly increased, the range will approach its equilibrium value, which is 1.8 feet for the lunar tide; if this depth be greatly di minished, the range will approach the value 0.000,026 h, h denoting the depth. In the latter case the amplitude of the horizontal displace ment will be 137 feet. For the depth of 10,000 feet the range of tide is 0.31 foot and the am plitude of the horizontal displacement 161 feet ; for the depth of 20,000 feet the corresponding quantities are 0.74 foot and 196 feet, respec tively. If friction proportional to the velocity be introduced, the effect will be to displace the crests of the lunar and solar wave with refer ence to the moon and sun, but by unequal amounts. To this has been attributed the age of the tide.