Popular Science Monthly/Volume 11/July 1877/The Tides I
By Professor ELIAS SCHNEIDER.
THERE has always been a difficulty in the minds of teachers, as well as in the minds of learners, to comprehend the theory of the tides as presented in our text-books. This theory fails to give a satisfactory account of the cause of the tides on the side of the earth most remote from the sun and the moon. According to this theory, at that part of the earth's surface which is turned away from the moon or from the sun, a less amount of attraction is felt by her waters than anywhere else on her surface; and the whole earth is therefore, in effect, drawn away from the waters on the far side of her, and thus, the water being left behind, a tide is produced on this side, as well as on the side at which the force of gravity acts directly. That so great an absurdity could have been accepted so long by our writers of textbooks, is truly marvelous. It is, indeed, so contrary to all known facts and laws of physics that, if no other influence were felt by the waters at the far side of the earth than attraction, there would be just the opposite effect produced to that alleged by this absurd hypothesis. This can be demonstrated by actual experiment, and as conclusively as any other fact coming within the reach of experimental philosophy. It has been proved experimentally that all bodies ou the surface of the earth at midnight are heavier than at any other hour of the twenty-four; and that when new moon occurs at midnight, this increase of weight or gravity felt by matter on this part of the surface of the earth is still greater. Now, if this theory were correct, attraction would produce just the opposite effect; that is, matter would weigh less at midnight than at other hours of the twenty-four. On the side of the earth facing the sun and moon, the weight of bodies is diminished, as it should be, according to the theory which I propose to establish in this article.
The truth of this fact is very easily accounted for. Suppose the earth were placed in such a position, in space, that she could not feel any of the sun's attraction, nor that of any other body. Then gravity would be equal on all parts of the earth's surface, on the supposition of its being a perfect sphere and at rest. But now bring her within the attractive influence of the sun. Then all particles of matter on the earth's surface most remote from the sun would feel the force of gravity of both the sun and the earth; and these two forces would act in the same direction and in the same straight line, directed through the centre of the earth to the centre of the sun. On the side facing the sun, these two forces would also act in the same straight line, but in opposite directions. Hence a decrease of weight on one side and an increase thereof on the opposite side of the earth. The same result follows between the earth and moon under a similar supposition. It is therefore not true that the least amount of attraction is felt by the waters of the earth at that part of her surface most remote from the sun or from the moon. It is indeed true that the sun and moon have less power of attraction on the particles of matter at this part of the earth's surface than they have on particles of matter facing them. But, as attraction diminishes as the square of the distance increases, this attractive force of these two bodies on any part of the earth's surface is not near so great as that of the earth herself on such part of her surface. Therefore, as these remote particles feel the attraction of sun and moon plus the attraction of the earth herself, they are drawn with greater force toward the centre of the earth than any other particles. Consequently, it cannot be true that the whole earth is drawn away from the waters, and that any tide is produced by the waters being left behind.
How, then, can we account for tides occurring on opposite sides of the earth at the same time? Let us see. In the first place, suppose the earth to occupy some place in space, and to be in a state of perfect rest. Then suppose the sun to come into position, and the earth to start on her journey of 68,000 miles an hour in her orbit around the sun; and suppose, too, that the earth rotates only once on her axis during one revolution around the sun. Then will the same side of her surface face the sun in every part of her orbit. Consequently, there will be a solar tide perpetually at a part of her surface, produced by centrifugal force, and at that part farthest from the sun. Night and solar tide will reign with unceasing steadiness at that one place; but there will be no motion of these piled-up waters. There they will stay, in a steady equilibrium, by the unceasing effect of centrifugal force, in the same manner as can be illustrated by swinging a hollow globe, partially filled with water, around the hand by means of a cord, or by swinging a bucket filled with the same liquid, and having for its bottom a piece of India-rubber, which bottom will bulge out when the bucket is swung around a centre, in the same manner as do the waters of the far side of the earth when she swings or sweeps around the central sun with a velocity of 68,000 miles an hour.
But there are always two solar tides occurring on opposite sides of the earth. The above explanation accounts only for the solar tide on the side of the earth farthest from the sun. How must we account for the fact that there is also one on her side facing the sun and occurring near noon? It is a well-known law of planetary motion that centrifugal and centripetal forces are precisely equal. By virtue of the first the earth seeks to fly from her centre of motion; by virtue of the second she has a tendency to fall into the central luminary; and everything on her surface is operated on in like manner. The particles of her water, moving very easily among one another, are therefore drawn readily away from her solid portions in opposite directions. On the one side the bulging out is caused by centrifugal, on the other by centripetal force. But, as these two forces are nearly equal in all parts of the earth's orbit, the tide-waves on opposite sides of her surface must also be nearly equal. The centrifugal force is produced by the revolution of the earth around the sun; the centripetal force is caused by the force of gravity lodged in the great central orb.
It must not be understood, however, that the earth in her orbital motion feels the effect of these two forces at her surface only. Every particle of the matter composing the earth feels both a centrifugal and a centripetal force while this planet moves around the central orb, and these two forces are precisely equal only at the centre of the earth. But the matter of her surface most remote from the sun feels a centrifugal force that is in excess of the centripetal force felt at this same point; and the matter of her surface facing the sun, being nearest to it, feels a centripetal force that is in excess of the centrifugal force felt at this same point. But these two excesses are equal; hence there are equal solar tides at these points, while at the earth's centre there is an exact balancing of the two forces.
Suppose the matter of the earth were all condensed into the volume of a cubic inch, and that this small volume were placed at the earth's present centre; then suppose it received an impulse carrying it forward with a velocity equal to that which moves the centre of the earth, and that it were influenced by the sun, according to the law of gravity. This small volume, though equal in mass to the entire mass of the earth, would then move in the same curve in which the centre of the earth moves, and with the same velocity. But suppose this solid inch of matter were to be placed 4,000 miles farther from the sun, namely, at that point in space where the earth's surface is most remote from the sun. This solid inch, or whole mass of the earth, would then move in a longer curve than it would when at her present centre, as under the first supposition. But completing, nevertheless, in this longer curve, one revolution in the same time in which one revolution is completed in the shorter curve, at the earth's centre, the centrifugal force would be much increased; and, the centripetal force being also diminished in the same ratio, this cubic inch of matter would either abandon the sun's companionship entirely or make a new orbit of motion. In like manner are the waters of the earth operated on by centrifugal force at this point of the earth's surface. They have a tendency to fly off in a line tangent to the earth's orbit.
Now, suppose again that this condensed matter of the earth were placed at that point in space where the earth's surface is nearest the sun, namely, 4,000 miles nearer the sun than the centre of the earth is: then the whole mass of the earth's matter would move in a shorter curve than when placed at the centre, but, completing one revolution in no shorter period, the centrifugal force would be diminished; and, being also nearer to the sun, the centripetal force would be much increased by the central power of attraction. Therefore, this body of matter would, under this supposition, also leave its orbit, but it would be drawn toward the sun, and probably plunge into it. In the one case, the centrifugal being greater than the centripetal force, the body would fly from its centre of controlling power; in the other case, the centripetal being equally superior to the centrifugal force, the body would also be drawn out of its orbit, but dragged to the centre of controlling power. These suppositions are made to show, by way of illustration, the excessive force of each kind over its opposite, at opposite sides of the earth. And these equally excessive forces, acting in such opposite directions, cause the opposite solar tides. The particles of water, moving easily among one another, are readily driven in opposite directions by these opposite forces. If the earth were entirely solid, then there could be no such bulging out of any of its matter, and therefore no tides.
A few words here in regard to the law of gravitation are in place. Every body of matter attracts every other body of matter, and with a force equal to the amount of matter each body contains; and this force diminishes as the square of the distance increases. Two bodies of equal mass approach each other equally; but, if one body contains four times as much matter as another, the smaller approaches the larger with a velocity four times as great as the larger does the smaller. Suppose two such bodies, being separated at a distance of 100,000 miles, attract each other with a certain known force: if this distance be increased to 200,000 miles, the force of attraction between these two bodies will be only one-fourth as great. In like manner, the earth, at the point farthest from the sun, feels a smaller degree of attraction than the matter at the centre. And, as the centrifugal force is also greater at this point than at the centre, there is here an excess of centrifugal over centripetal force, and sufficient, as can be ascertained by exact mathematical calculation, to produce a solar tide. And at that part of the earth's surface which is nearest the sun, or facing it, there is, according to the same law of gravity, an excess of centripetal over centrifugal force. Hence we have also a solar tide at this part of the surface of the earth.
I give one more illustration. Suppose the earth, at E (Fig. 1), is moving in a straight line toward E and with a velocity of 68,000 miles an hour; and suppose when she reaches E' she comes under the
attractive influence of the sun. She will then be deflected from her rectilineal course and move in a curvilineal orbit around the sun. That part of her surface turned away from the sun will be 8,000 miles farther from the attractive influence of the central orb than that part of her surface facing the sun. Hence this remote part will have a greater tendency to continue moving on in a straight line than any other part; and this tendency will show itself in the motion of its waters, by producing a tide. The waters will have a tendency to move in a line tangent to the orbit of the earth. The part of the earth's surface nearest the sun, being acted upon more powerfully by the gravitating influence of this central force than the remote part, will show a less tendency to move on in a line tangent to the earth's orbit. Hence there will be another tide produced by gravity directly.
I have thus far spoken only of the solar tides. It will be necessary also to say something of lunar tides, or what influence the moon has on the phenomena of the tides.
It is a well-known fact that there is a point between the earth and her moon called their centre of gravity. The distance between the centres of these two bodies is about 240,000 miles. A rough calculation brings the centre of gravity of these bodies about 2,687 miles from the centre of the earth, and 237,313 miles from the centre of the moon. This point describes the curve of an ellipse around the sun; and the earth and moon revolve around this point, while they both sweep through space in their majestic journey around the sun. It is therefore evident that the earth, in her ceaseless motions, is influenced by three different centrifugal forces. The one is produced by rotation on her axis; the other by her revolution around the sun; and the third by her revolution around the centre of gravity between herself and the moon.
Let us suppose that the earth and moon have no other motion in space than that of revolving around their common centre of gravity, and that the same side of the earth is always facing the moon. The earth will then feel a centrifugal force on her side farthest from the moon, and equal to the centripetal force felt on her side facing the moon. These two equal forces, acting in opposite directions, will cause tide-waves on opposite sides of the earth; and they will be produced in the same manner as the opposite ones, spoken of already, are produced by centrifugal and centripetal forces felt by the earth in her orbital motion around the sun.
Let us now place the earth and moon in their proper position with respect to the sun; and let us suppose the moon to be in conjunction with the sun, as at A Fig, 2. It is then new moon, and the moon's centre is 237,313 miles within and the earth's centre 2,687 miles outside the elliptic orb described by their centre of gravity. At this point of her path the earth feels, therefore, the greatest amount of centrifugal force on the side of her surface farthest from the sun. This large amount of centrifugal force is produced by axial rotation, by revolution around the sun, and by revolution around the centre of gravity already named. The direction of these three forces is in the same line. The motion of this part of her surface, which is in this line of direction, is therefore the most rapid; consequently, the centrifugal force felt here is also the greatest. Therefore, we have one of the highest tides when the moon is in conjunction with the sun; and, since centripetal is always equal to centrifugal force, the side of the earth facing the sun and moon at this point of her orbit must have an equally high tide at this time. The centripetal force here is produced by the gravity of both sun and moon acting jointly.
Let us now suppose the moon to be in quadrature, as at B. Then the two centrifugal forces, the one produced by revolution around the sun, the other by revolution around the centre of gravity of these two
bodies, do not act in the same line of direction, but at right angles with each other. The phenomena of solar and lunar tides are then about 90° apart; the solar being the smaller and the lunar the larger. Here the centres of both earth and moon are in the path described by their centre of gravity.
In the last place, let us suppose the moon and sun to be in opposition, as at C. Then, according to my theory, the earth feels, on her side farthest from the sun, an influence which diminishes the centrifugal force produced by her orbital revolution. For at this point the earth's centre is within and the moon's centre is without the elliptic path described by their centre of gravity. Here the revolution of the earth around this centre of gravity is contrary to her general motion around the sun. But what is thus lost in centrifugal force on her side turned away from the sun is more than made up by the gravity exerted directly on her by the moon. And, on the side of the earth facing the sun, she feels a centrifugal force produced by revolution around the centre of gravity of herself and the moon, and also a centripetal force produced by the gravitating influence of the sun. Hence there must be high tide also when sun and moon are in opposition.
It is a known fact that the solar are less than the lunar tides. How must we account for this fact? The sun is a body so large that the mass of the moon is not much more than a grain of sand in comparison with it. But it must also be remembered that gravity diminishes as the square of the distance increases; and as the moon is very near the earth, and the sun a great way off, the lunar influence is much more strongly felt in the phenomena of tides than the solar influence.
The amount of centrifugal force felt by a body moving in space around a centre depends, not only on the velocity with which it moves, but also upon the size of the curve in which it moves. If the circumference of the curve is very large, it differs not much from a straight line. If a body moves in space in the direction of a straight line, it feels no centrifugal force at all. If it is deflected from the direction of this straight line, only a very little, the circumference of the curve will be very long, and the centrifugal force will be small. But, if it is very much deflected, the curve becomes very small, and the body, turning around very "short corners," has a strong tendency to fly "off the track." In other words, in a short curve the centrifugal force is very great.
Now, let us make an examination of the orbital curve of the earth made in its motion around the sun. The length of the circumference of this curve is, in round numbers, about 570,000,000 miles. A straight line, 10,000 miles in length, tangent to this curve at one end, is only about .526 of a mile distant from the circumference at its other end. Therefore, the earth, moving in this orbital curve, feels rather a small amount of centrifugal force. But, in her motion around the centre of gravity between herself and the moon, she turns very "short corners," and hence under this influence she experiences a greater amount of centrifugal force than in her motion around the sun. For this reason, also, the lunar are greater than the solar tides.
If the earth had only one rotation in one revolution around the sun, there would be, as already stated, one solar tide by virtue of centrifugal force occurring at midnight, and another by virtue of centripetal force occurring at noon. That is, perpetual night and high tide would occur at one side, and perpetual day and high tide at precisely the opposite side of the earth. But now let us suppose the earth rotates on her axis once every twenty-four hours, and from west to east, as she actually does rotate: then there will be motion of the waters; but this motion will be only apparent motion, and from east to west. The real motion will be that of the solid portion of the earth that moves from west to east, and underneath these waves, though these waves do also acquire, by means of friction, a part of this motion; yet the centrifugal and centripetal forces are so much superior as to master the effect of this friction. This frictional force carries also these tide-waves so far eastward that they occur always several hours east of the meridian; that is, several hours after noon, and several hours after midnight.
It is a known fact that the waters of the tides rush up the rivers and small bays on the east coasts of all countries with great violence, but not up those on the west coasts. The reason of this is very evident. The west coasts turn away from the tide-waves; while the east coasts, moving with a velocity of nearly 1,000 miles an hour, in rotation, within all parts of the tropics, dash violently eastward against these waves. For this reason the waters, by resistance or inertia, appear to be driven violently westward up the streams and bays, while it is the mouths of these channels ploughing with violence into the tide-waves themselves.
It has been stated in this article that gravity is greater at that part of the earth's surface turned away from sun and moon than anywhere else. It may be asked, "How then can centrifugal force drive out the water above the usual level when its weight is increased?" This force acts in a line tangent to the earth's orbit, which tangent line, being perpendicular to the radius vector at perihelion and aphelion, and at all other points in the earth's orbit very nearly so, may be said to be at right angles with a line extending from this point of tangency through the centre of the earth to the centre of the sun. Therefore, the attractive power of the sun acts on matter, at the part of the earth most remote from it, in the direction of the radius vector; and centrifugal force acts on this same matter in a direction at right angles with the radius vector. Now, as was first demonstrated by Galileo, the motion of a body, produced by one force, is not destroyed by another force acting on this same body at right angles with it. The result of these two combined forces is only a change in the direction of motion. But, as has already been shown, centrifugal is always in excess of centripetal force at the place of the earth now under consideration. Hence this tide-wave at this side.
I conclude by saying that the great motions of the waters of the mighty deep are most assuredly the grandest ocular demonstrations of the rotation of the earth upon her axis, and of her revolution around the sun, that can be witnessed by the eyes of man.