Failure to warn the student that the force diagrams are different depending on whether the plane of the diagram is parallel to, or perpendicular to, the plane of the moon's orbit. If continents are shown on the earth, that's a clue. If part of the orbit of the moon is shown, that tells you that the diagram is in its orbital plane.
But do students always notice these details? Neglect of tensile properties of solid and liquid materials. Neglecting to mention that liquids under stress physically move toward a lower-stress configuration. Failure to specify the baseline earth shape against which a tide height is measured. They are trying to get by "on the cheap". So why are there tidal bulges on opposite sides of earth? For a while we will set aside the complications of the actual earth, with continents, and look a the simpler case of an initially nearly spherical earth entirely covered with an ocean of water.
If this Earth rotates on its axis there's equatorial bulge of both earth and water, but we will treat this as a "baseline" shape upon which tidal bulges due to the earth and sun are superimposed. The ocean's shape is produced by the Earth's gravitation and its axial rotation. The distortions of this baseline shape are called tidal effects and are entirely due to the gravitational forces of the moon and sun acting upon the earth. The stress-producing effects of a non-uniform gravitational field acting on an elastic body are called tidal forces.
Tidal forces are vector quantities, and may be drawn as arrows in a diagram, but the interpretation of such a diagram is different from that of a diagram of the gravitational forces themselves. Therefore textbooks should always specify which is being depicted.
The distortion of water and earth that we call a "tidal bulge" is the result of deformation of earth and water materials at different places on earth in response to the combined gravitational effects of moon and sun. It is not simply the size of the force of attraction of these bodies at a certain point on earth that determines this. It is the variation of force over the volumes of materials water and earth of which the earth is composed. Some books call this variation the differential force or tide-generating force TGF or simply tidal force.
Let's concentrate on the larger effect of the moon on the earth. To find how it distorts shapes of material bodies on earth we must do the calculus operation of finding the gradient of the moon's gravitational potential a differentiation with respect to length upon each part of the earth. If this procedure is carried out for all places around the earth, a diagram of tidal forces can be constructed, which would look something like this: Tidal forces due to a satellite moon. The liquid-filled cavity in the rock below them is stretched and squeezed as the tides deform the solid earth, and the pressure rises and falls on their gauges twice each day.
One can now easily visualize how these shape-distorting stresses produce tidal bulges at opposite sides of the earth. The deformation of the earth's crust reaches equilibrium when the internal elastic forces in the solid crust become exactly equal to the tidal forces.
The deformation of the water reaches equilibrium when it moves to minimize its potential energy. Tidal forces have radial components and tractive tangent to the earth's surface components. The radial components stretch or compress solid materials radially. The tractive components stress solid materials laterally, and, in the case of liquid materials, can physicaly move them significanty. At about There the tidal forces are directed tangentially.
At this point there's no component of tidal force to increase or decrease radial compression stress, and the radius of the earth there is nearly the same as the radius of the unstressed earth. Most of the earth, crust and mantle, behaves as an elastic solid. The earth's radius is 6, km. The crust is less than 10 km thick, the mantle km thick. Only the core and the oceans behave as a liquid.
Fluids can flow when forces are applied to them. They strongly resist compression or expansion. Water is very nearly incompressible and is clearly not rigid.
So the tidal bulges in water arise because some water has moved toward the bulges from elsewhere, that is, from other regions of the ocean. This should not be surprising, for we know that water moves from higher to lower pressure regions in all situations, moving toward a condition of equilibrium at lowest possible potential energy.
For a liquid body, tractive forces dominate, but the end result is still two tidal "bulges" when equilibrium is achieved. The tangential components of tidal force push liquid material toward the highest part of the tidal bulgs. This necessarily depresses the ocean surface elsewhere outside of those bulges.
Tidal forces do not change the density or volume of water, they just move it. How does this apply to the real earth? The real earth has a solid crust with thin layers of ocean bounded by continents. The solid earth tides are dominated by the compressive-expansive radial components of the tidal forces.
But the large oceans are dominated by the tractive tangential components of the tidal forces. The mantle of the earth behaves, in this context, like a solid elastic body. In either case, at equilibrium, the gravitational forces on each portion of matter are balanced by internal tensile forces.
The tidal bulges are primarly due to small motion of large volumes of water over the earth's surface. The tidal bulges in the ocean should not be thought of as due to "lifting" of water, or due to compression and decompression of water. They are the result of water moving toward the regions of the tidal bulge. But do not think of "moving" as something like converging ocean currents rushing into the bulge. A tidal bulge is maintained by small displacements of huge amounts of water, over a huge area.
Also, the tidal bulges in the ocean are very small, seemingly insignificantly small, compared to the radius of the earth. But over the huge area of one of the oceans, the tidal bulges contain a huge amount of water. We have discussed these using the conceptual model of a stationary earth-moon system without continents, with a uniform depth ocean covering its entire surface. We do this to emphasize that these tidal bulges are not due to rotation, but simply to the variation of the moon's gravitational field over the volume of the earth.
When we add continents to this model, the ocean bulges reflect from shorelines, setting up currents, resonant motion and standing waves. Standing waves of a liquid in a shallow basin have regions of high amplitude variation antinodes and regions of zero amplitude variation nodes. So it's not surprising that in oceans we see some places where the tidal variations are nearly zero. All of this ebb and flow of the water surface affects ocean currents as well. Yet it is all driven by the tidal forces due to the moon's changing position with respect to earth.
Coastal topography sea-floor slope and mouths of rivers and bays can intensify coastal water height fluctuations with respect to the solid land. In fact, these effects are usually of greater size than the tidal bulges would be in a stationary earth-moon system—sometimes ten times higher than the tidal bulge.
But most important is the fact that this whole complicated system, including the coastal tides, is driven by the tidal bulges discussed above, caused by the moon and sun. It is a tribute to the insight of Isaac Newton, who first cut through the superficial appearances and complications of this messy physical system to see the underlying regularities that drive it. Even when we look at this more realistic model, including the Earth's rotation, it is the rotation of continents and their coastal geometry with respect to the tidal bulges that gives rise to the complicated water level variations over the seas.
It is not some mysterious effect of "centrifugal force" or "inertial effects" as some textbooks would mislead you to think. We have ignored the stress due to the gradient of the earth's own potential field, because it is nearly the same strength anywhere on the surface of the earth.
We have also ignored the equatorial bulge of the earth, for we are treating that as the baseline against which the tidal effects are compared. If all you want is the reason there are two tidal bulges, you needn't read further. I've sketched out an even shorter treatment as a model for textbooks that have no need to go into messy details.
A picture of tidal forces. Tidal forces. Remember, when you see this diagram of tidal forces, that it shows not the gravitational forces themselves, but the differential force, often called the tide-generating force. Similar pictures are found in other textbooks, but one must be careful not to mix the several different interpretations of the picture.
These include: The picture shows simply the tide-generating forces on the earth due to the combined gravitational forces due to earth and moon. An inertial coordinate system is assumed, so there's no inclusion of centrifugal forces in the discussion.
Nor should there be. The picture may also be thought of as showing the vector sum of the gravitational forces due to earth and moon and the centrifugal force, when the analysis is done in a rotating coordinate system. To avoid the messy details of talking about rotating coordinates and centrifugal forces, some books shortcut all this by loosely defining tidal forces as the difference between the actual lunar gravitational force at a point on earth and the lunar gravitational force at the center of the earth.
Sometimes they call the latter the "average force" due to the moon. This produces a picture very like that above. This interpretation may be justified, if properly explained. To see how this approach works when done well. See Bolemon []. In any of these interpretations, similar force summation is happening throughout the volume of the earth. Tidal forces stress and push the materials of the earth earth and water , distorting the earth's shape slightly—into an ellipsoid.
These diagrams are necessarily exaggerated, for if drawn to scale, the earth, even with tidal bulges, would be smoother than a well-made bowling ball. Quincey has a good discussion of this, with diagrams. We can see from this photograph of earth from space, that all of the distortions due to rotation, mountains and ocean trenches, and tides, are really very tiny relative to the size of the earth.
Keep this photo in mind as you look at the drawings, which are necessarily greatly exaggerated. Exercise: How closely does the earth compare with a bowling ball. For the necessary data about bowling balls, see Bowling ball specifications. Accordidng to this source, the diameter of a bowling ball 13 lb.
That's a 0. The difference between earth's polar and equatorial diameters is 23 km, or 0. By bowling ball standards, the earth doesn't quite meet the required roundness tolerance due to its equatorial bulge and polar flattening. But this departure from sphericity is still too small to be noticed in photographs. Mountains and ocean trenches are much smaller, and tides far smaller still. Some photographs of the earth from space are computer synthesized composites of many photographs taken from orbiting earth satellites near the earth.
The photos that are the best direct evidence of earth's roundness are unmanipulated single photos taken from a great distance, as from the moon, taken with a well-corrected camera lens. The equilibrium theory of the tides. Our simple analysis above also showed the importance of the relaxation of earth materials to achieve an equilibrium between gravitational forces and cohesive forces of materials.
In more detailed analysis, we find that the figure shape of our idealized earth model at equilibrium consists of two bulges nearly oriented in alignment with the moon. Underneath this equilibrium profile, the earth turns on its axis once a day, so the bulges move with respect to geography.
It is the surface profile of the bulges that moves once a day, not the entire mass of those bulges. An alternative treatment of this is called the equilibrium theory of the tides. It is carried out in a coordinate system rotating about the barycenter of the earth-moon system. In this coordinate representation, the solid earth and the moon are considered stationary in equilibrium with respect to each other.
In this model we can treat the earth-moon system as if it were an inertial system, but only at the expense of introducing the concept of centrifugal force , technically called a "fictitious" force to distinguish it from "real" forces that are due to physical interactions between material bodies.
This is handy when the measurements of a problem are with respect to a rotating frame of reference and the desired results are measurements with respect to that same frame of reference.
The rotating earth is such a convenient frame of reference. Typically one chooses a polar coordinate system fixed on the earth. It turns out that when this is done, the centrifugal force on a mass anywhere on or within the earth is, at every instant, of constant size and direction. So it cannot raise tides, nor can it deform the shape of material objects.
Only real forces can do that. The centrifugal seduction. So what about those centrifugal forces that so many elementary textbooks and websites make such a fuss about? You'll notice we never mentioned them in our simple explanation above. Should we have? No, they are only appropriate when doing the analysis in a rotating reference frame. We digress for a look at how some textbooks create confusion about centrifugal forces. Let's be very clear about this.
The only real physical forces that act on the body of the earth are: The gravitational forces between each part of the earth and every other part, and the gravitational forces on parts of the earth due to the moon, sun, and the nearly negligible forces due to more distant bodies in the solar system.
Internal tensile forces within the materials of the earth. If a textbook mentions centrifugal forces without defining non-inertial systems and without telling the reader that this term has meaning only when using a non-inertial reference frame, you can reasonably suspect that the book may also be deceiving you in other ways. As a result, we often hear students who have been so misled ask, "Why doesn't the motion of the earth around its barycenter give rise to centrifugal forces that might cause tides?
In an inertial reference frame, the monthly motion of the earth is such that each piece of earth moves in a circle. At any instant all of these circles have the same radii and all radii are parallel.
This force field of parallel and equal forces has no spatial gradient, and cannot raise a tide. This figure, from French, shows the geometry. The dotted arcs A, B, and C have the same size and same radius. At any instant all of their radii are parallel. In a non-inertial rotating reference frame, in which the earth and moon are both stationary, the same conclusion is reached even if fictitious forces could raise tides. A more detailed account follows.
We ignore the effects of the earth's rotation about its own axis. The equatorial bulge it produces is the baseline against which tidal variations are referenced. We are now focusing on the effects due only to the earth-moon system. We are also, still, assuming an idealized earth covered entirely with an ocean of constant depth. Therefore coastlines, ocean depth variations, and resonance phenomena are not issues.
The motion of the earth about the earth-moon center of mass the barycenter causes every point on or within the earth to move in an arc of the same radius.
This is a geometric result that some books totally ignore, or fail to illustrate properly. Every point on or within the earth experiences a centripetal force of the same size and direction at any given time. A force of constant size and direction throughout a volume cannot give rise to tidal forces as we explained above. The size of the net centrifugal force is the same as the force the moon exerts at the earth-moon center of mass the barycenter , where these two forces are in equilibrium.
So the bottom line is that centrifugal forces on the earth due to the presence of the moon are not tide-raising forces at all. They cannot be invoked as an "explanation" for any tide, on either side of the earth or anywhere else. So why do we find them used in "explanations" of tides in elementary-level books?
Could it be because these text's authors are often misled by their own pretty diagrams? Once they launch into the rotating coordinate mode and start talking about centrifugal forces, they seem to forget that the earth's own gravitational field is still present and acting on every portion of matter on earth. They also forget that the non-uniformity of moon's gravitational field over the volume of the earth is alone sufficient to account for both tidal bulges, bulges that would be essentially the same if the earth-moon system were not moving, and the earth and moon were not moving relative to each other.
The daily rotation of the earth underneath these "tidal bulges" causes the bulges to move around the earth's surface each day. And all of these deformations sit "on top" of the equatorial bulge that extends all the way around the earth, and is due to the earth's axial rotation.
Details and complications. The oceans don't cover the entire earth, but "slosh around" daily within the confines of their shores. Timing of the ocean tidal bulges even at mid-ocean can depart considerably from the idealized tides we have described. Reflections from shores can set up interference patterns farther out in the ocean. Coastal tides have considrable local variations due to difference of shoreline slope, and ocean currents. But the driving force for all of these complications is still those two "daily" lunar tides 12 hours 25 minutes apart , which we have explained above, combined with the two much smaller daily tides 12 hours apart due to the gravitational field of the sun.
More misleading textbook illustrations. An oceanography textbook has this diagram that at least shows centrifugal forces of equal size. One is tempted to think "This book has it right! Then on the very next page we see this diagram in which the author identifies one tide as being from gravitation, the other from inertia. Comparing the two pictures, one sees that they contradict. The one that shows forces clearly suggests that the moon's gravitational force is responsible for both tides.
Unfortunately, like so many other books, this book fails to tell the student the origin of these centrifugal forces, and fails to emphasize that they are not "real" forces, but only a useful device to do problems in rotating coordinate systems.
Here the chickens come home to roost. Misunderstanding of centrifugal effects originates in some elementary-level physics textbooks. Nowhere does this book even suggest that rotating coordinate systems are being assumed. Other lunar misconceptions. We mentioned frictional drag of water on the ocean floor. These forces dissipate rotational energy of the earth.
Similar drag effects act within the solid crust of the earth as well, since it is stressed by tidal forces. This causes the tidal bulges to arrive a little "early" compared to the time of the moon's crossing the observer's meridian. This diagram illustrates the effect but exaggerates its size. The earth's rotation and the moon's revolution are both counter-clockwise as seen from above the N pole.
At most places on earth, there are two high tides each day. With each passing day, the high tides occur about an hour later. The moon rises about an hour later each day, too actually, 54 minutes later.
Since the moon pulls up the tides, these two delays are connected. As the earth rotates through one day, the moon moves in its orbit. A point on the earth must move a little farther than one rotation to line up with the moon again. Paste it onto heavy paper or tagboard, and cut around it. You make our reporting better.
Facebook Twitter LinkedIn Email. Tags Environment Ask Sam. Sam Evans-Brown. Murrow awards, one national Murrow, and the Overseas Press Club of America's award for best environmental reporting in any medium.
He studied Politics and Spanish at Bates College, and before reporting was variously employed as a Spanish teacher, farmer, bicycle mechanic, ski coach, research assistant, a wilderness trip leader and a technical supporter. See stories by Sam Evans-Brown. Related Content. Ask Sam: Do Mosquitoes Sleep? This week, we're buzzing, because
0コメント