• tides continuously raise and lower the sea surface, sometimes as much as several meters -- this requires large amounts of energy to move the water
• that energy is supplied as a result of the gravitational pull of the moon and sun and the rotations of the earth, moon, and sun; these rotations produce a centrifugal force which balances the gravitational forces, so that the planets do not fly away from or crash into each other
• the gravitational forces that contribute to tides were first described by Newton; Newton's law of gravitation states that every particle of mass in the universe attracts every other particle of mass with a force that is proportional to the product of the two masses and inversely proportional to the square of the distance between the two masses:

F = G * 

where G is the gravitational constant, M₁ and M₂ are the 2 masses, and d is the distance between them

• for spherical objects, the entire mass can be considered to be at the object's center of gravity and the distance d is measured between the centers of the two masses; thus even though the mass of the sun is greater than that of the moon, the gravitational force it exerts is less than that of the moon because it is farther away (it is 46% that of the moon)
• centrifugal forces are supplied by gravitational attraction between two orbiting bodies -- this force keeps the planets from flying off in a straight line
• as two bodies orbit together in space, they orbit around their common center of mass
• earth's mass is 81 times that of the moon -- therefore the common center of mass is 4,700 km from the earth's center or 1,640 km below the earth's surface (circumference is 12,680 km; radius is 6,340 km -- 6,340 - 4,700 = 1,640 km)
• all particles of mass within the earth are subject to the same centrifugal force, regardless of their position

• centrifugal forces and gravitational forces are not in perfect balance everywhere on the earth's surface
• this occurs because although all particles of mass within the earth are subject to the same centrifugal force, the gravitational force on each particle of mass within the earth varies with its distance from the moon or sun and the orbits of the planets are elliptical, not spherical
• on the side of the earth nearest the moon, the moon's gravity is strongest -- this is because the distance from the moon's center to the earth's center is 384,800 km and the earth's diameter is 12,680 km, so a point on the earth's surface nearest the moon is 378,460 km away vs. 391,140 km away for the opposite side of the earth
• remember, gravitational force is inversely proportional to the square of the distance: thus, the ratio of the moon's gravitational force between a point nearest the moon and farthest away is 378,460² divided by 384,800² or about 1 : 1.034 -- this means that the gravitational force is about 3% greater at the surface nearer the moon
• so there is a net excess gravitational force on the side of the earth nearest the moon and a net deficit on the side farthest away
• with solid objects, these imbalances are compensated for by changes in pressure gradients; however, water flows in response to the imbalances and these flows create the tides

I. Tide-Generating Forces

• at the point nearest the moon, there is a slight tidal pull directly toward the moon; this pull is compensated for by the earth's gravity, so paradoxically the tidal force has little effect
• similarly, at the point farthest from the moon, there is a slight tidal push away from the moon due to the greater centrifugal force and the lesser gravitational force; again this is compensated for by the earth's gravity, so there is little tidal force

• at other points on the earth's surface, the tidal pull is exerted in a direction different from that of the earth's gravity, so while the gravitational force is less it is not entirely compensated for and the result is that there is a large horizontal component generated along the earth's surface (called the tractive force) that results in a bulge of water being produced with a net movement towards the point closest and farthest from the moon

• because the earth spins, the point on the earth facing the moon is continuously changing so the locations of the tidal bulges also change
• a given point on earth that is initially at a crest or in the bulge (high tide) passes to a nonbulging area or trough (low tide), to another high tide, to another low tide, and back to the original position as the earth completes one revolution

• tides are classified into 3 general types and are based on lunar days, not solar days
• lunar days result because the moon revolves around the earth-moon center of mass once every 27.3 days in the same direction as the earth rotates; the period of earth's rotation with respect to the moon is 24 hrs and 50 min -- it takes the earth 50 minutes to "catch up" with the moon's movement; thus tides at many locations are almost an hour later each successive day and this is why the moon rises and sets almost an hour later each night

1. diurnal tides (daily tides) have one high tide and one low tide in each tidal day
2. semidiurnal tides (semidaily tides) have two hi and two low tides each tidal day thus a period of 12 hrs and 24.5 minutes; the tides are equal in height during each period
3. mixed tides have two high tides and two low tides each tidal day, but the heights of the two high tides/low tides are not equal -- there is a higher high tide, a lower high tide, a higher low tide and a lower low tide
• results because the moon's orbit is not in the plane of the earth's equator but is inclined at an angle of 28° on either side of the equator--the angle between the equatorial plane and the moon varies over a period of 27.2 from this maximum of 28° to 0° and back to 28°

• thus the two tidal bulges will be offset with respect to the equator and the effect at a given latitude will be unequal, particularly at mid-latitudes
• because of this, the only pure semidiurnal tides occur at the equator, while most diurnal and semidiurnal tides have some mixed characteristics

• the sun also plays a role in tidal forces -- like the moon, it produces tractive forces and two bulges, but its tide producing force is 46% that of the moon
• because the bulges produced by the sun sweep westward as the earth spins eastward, the solar tide has a semidiurnal period of 12 hrs
• like the moon, the sun is also declined with respect to the earth's equator and varies over a yearly cycle to 23° on either side of the equatorial plane
• consider the case when both the sun and moon have declinations of 0°; in this case, the tide-generating forces of the sun and moon are acting in the same direction
• the tides produced by both are added together; these tides have the greatest range between high and low water -- these are called spring tides and occur when the sun and moon are in conjunction (new moon) or in opposition (full moon); the moon is in syzygy here
• as the moon shifts its position relative to the earth each day, 7.38 days after the full or new moon, the moon, earth, and sun are aligned at right angles (moon's first and third quarters)
• the crests produced by the moon's tide will coincide with the troughs produced by the sun's tide; these will cancel each other and the range between high and low water will be low; these are the neap tides and occur when the moon is in quadrature
• the cycle of solar and lunar tides is 29.5 days (7.38 days * 4)

II. Dynamic Theory of Tides

• previous discussion of tides was idealized for an ocean-covered earth, but tides do not behave in the idealized manner described for the following reasons:
1. the wavelength of tidal waves is long relative to the depth in the oceans, so they behave as shallow water waves and their speed is governed by c = ; this means that the only way the idealized equilibrium tides (those that form the ellipsoidal bulges) can keep up with the moon's passage around the earth is if they occur in ocean depths > 20 km (they appear to travel at a speed of 1600 km/hr), but because oceans everywhere are < 20 km in depth, the tide is retarded with respect to the equilibrium tide
2. the presence of land masses prevents tidal bulges from circumnavigating the globe and the shape of the basins constrains the direction of tidal flows
3. the rotation of the earth on its axis is too rapid for the inertia of the water masses to be overcome in sufficient time to establish an equilibrium tide (which produces the ellipsoidal tidal bulges directed toward and away from the moon); thus there is a time lag in the ocean's response to the tractive forces (those that move parallel to the earth's surface and against which there are no centrifugal forces to compensate)
4. lateral water movements induced by tide generating forces are subject to the Coriolis force which deflects tidal flows with the sun (to the right in the Northern Hemisphere; to the left in the Southern Hemisphere)
• thus the dynamic theory of tides was formed to deal with how depths, configurations of the ocean basins, the Coriolis force, inertia, and frictional forces might influence the behavior of fluid subjected to rhythmic forces
• if you take the simplest situation -- the sun and moon at 0° declination and in syzygy (so that lunar and solar tides coincide) and consider that the equator moves more quickly than higher latitudes, then you can see that tides above 0° latitude will lag behind the equilibrium tide at the equator
• at low latitudes, the tides will lag 90° of longitude behind the equilibrium tide (the lag is limited to 6 hr 12 minutes on one side and 18 hr 36 minutes on the other side of the moon's passage overhead) -- these are called indirect tides
• at latitudes above 26° latitude, the tidal lag is less than 6 hrs 12 minutes; this lag continues to decrease with increasing latitude until it reaches 0 hrs 0 minutes at 65° latitude and tides will occur every 12 hours and 25 minutes -- these are direct tides
• both the ocean basin's configuration and the Coriolis force combine to form amphidromic systems where the crest of the tidal wave at high water circulates around an amphidromic point once during each tidal period; the tidal range is always 0 at the amphidromic point and increases outward from it
• co-tidal lines link all the points where the tide is at the same stage of its cycle and radiate outwards from the amphidromic point
• at approximately right angles to the co-tidal lines are co-range lines which join places having an equal tidal range; they form concentric circles around the amphidromic points
• the flooding tide is deflected to the right by the Coriolis force in the Northern Hemisphere, so water piles up on eastern sides
• the ebbing tide is deflected to the left and water piles up on the western side
• because of this, the amphidromic system is counterclockwise in the Northern Hemisphere, rather than clockwise; the reverse happens in the Southern Hemisphere