§ 9. Tidal Friction
All solid bodies yield more or less to stress; if they are perfectly elastic they regain their shapes after the stresses are removed, if imperfectly elastic or viscous they yield to the stresses. We may thus feel certain that the earth yields to tide-generating force, either with perfect or imperfect elasticity. Chapter VIII. will contain some discussion of this FIG. 4. - Tide-Predicting Instrument.
subject, and it must suffice to say here that the measurement of the minute elastic tides of the solid earth has at length been achieved. The results recently obtained by Dr 0. Hecker at Potsdam constitute a conspicuous advance on all the previous attempts.
The tides of an imperfectly elastic or viscous globe are obviously subject to frictional resistance, and the like is true of the tides of an actual ocean. In either case it is clear that the system must be losing energy, and this leads to results of so much general interest that we propose to give a short sketch of the subject, deferring to chapter VIII. a more rigorous investigation. It is unfortunately impossible to give even an outline of the principles involved without the use of some technical terms.
In fig. 5 the paper is supposed to be the plane of the orbit of a satellite M revolving in the direction of the arrow about the planet C, which rotates in the direction of the arrow about an axis perpendicular to the paper. The rotation of the E Tidal planet is supposed to be more rapid than that of the satellite, so that the day is shorter than the month. Let Friction us suppose that the planet is either entirely fluid, or has an ocean of such depth that it is high-water under or nearly under the satellite. When there is no friction, with the satellite at m, the planet is elongated into the ellipsoidal shape shown, cutting the mean sphere, which is dotted. The tidal protuberances are drawn with much exaggeration and the satellite is shown as very close to the planet in order to illustrate the principle more clearly. Now, when there is friction in the fluid motion, the tide is retarded, and high-tide occurs after the satellite has passed the meridian. Then, if we keep the same figure to represent the tidal deformation, the satellite must be at ' M, instead of at m. If we number the four quadrants as shown, the satellite must FIG. 5. be in quadrant t. The protuberance P is nearer to the satellite than P', and the deficiency Q is farther away than the deficiency Q'. Hence the resultant action of the planet on the satellite must be in some such direction as MN. The action of the satellite on the planet is equal and opposite, and the force in NM, not being through the planet's centre, must produce a retarding couple on the planet's rotation, the magnitude of which depends on the length of the arm CN. This tidal frictional couple varies as the height of the tide, and as the satellite's distance. The magnitude of the tidal pro tuberances varies inversely as the cube of the distance of the satellite, and the difference between the attrac tions of the satellite on the nearer and farther pro tuberances also varies inversely as the cube of the distance. Accordingly the tidal frictional couple varies as the inverse sixth power of the satellite's distance. Let us now consider its effect on the satellite. If the force acting on M be resolved along and perpendicular to the direction CM, the perpendicular component tends to accelerate the satellite's velocity. It alone would carry the satellite farther from C than it would be dragged back by the central force towards C. The satellite would describe a spiral, the coils of which would be very nearly circular and very nearly coincident. If now we resolve the central component force along CM tangentially and perpendicular to the spiral, the tangential component tends to retard the velocity of the satellite, whereas the disturbing force, already considered, tends to accelerate it. With the gravitational law of force between the two bodies the retardation must prevail over the acceleration.' The action of Velocity tidal friction may appear somewhat paradoxical, but it is the exact converse of the acceleration of the linear and angular velocity and the diminution of distance of a satellite moving through a resisting medium. The latter result is generally more familiar than the action of tidal friction, and it may help the reader to realize the result in the present case. Tidal friction then diminishes planetary rotation, increases the satellite's distance and diminishes the orbital angular velocity. The comparative rate of diminution of the two angular velocities is generally very different. If the satellite be close to the planet the rate of increase of the satellite's periodic time or month is large compared with the rate of increase of the period of planetary rotation or day; but if the satellite is far off the converse is true. Hence, if the satellite starts very near the planet, with the month a little longer than the day,'as the satellite 1 This way of presenting the action of tidal friction is due to Sir George G. Stokes.
recedes, the month soon increases so that it contains many days. The number of days in the month attains a maximum and then diminishes. Finally the two angular velocities subside to a second identity, the day and month being identical and both very long.
We have supposed that the ocean is of such depth that the tides are direct; if, however, they are inverted, with low-water under or nearly under the satellite, friction, instead of retarding, accelerates the tide; and it would be easy by drawing another figure to see that the whole of the above conclusions would hold equally true with inverted tides.
Attempts have been made to estimate the actual amount of the retardation of the earth's rotation, but without much success. It must be clear from the sketch just given that the effect of tidal friction is that the angular motion of the moon round the earth is retarded, but not to so great an extent as the earth's rotation. Thus a terrestrial observer, who regards the earth as a perfect timekeeper, would look on the real retardation of the moon's angular motion as being an acceleration. Now there is a true acceleration of the moon's angular motion which depends on a slow change in the eccentricity of the earth's orbit round the sun. After many thousands of years this acceleration will be reversed and it will become a retardation, but it will continue for a long time from now into the future; thus it is indistinguishable to us at present from a permanent acceleration. The amount of this true acceleration may be derived from the theories of the motions of the moon and of the earth when correctly developed. Laplace conceived that its observed amount was fully explained in this way, but John Couch Adams showed that Laplace had made a mistake and had only accounted for half of it. It thus appeared that there was an unexplained portion which might be only apparent and might be attributed to the effects of tidal friction.
The time and place of an eclipse of the sun depend on the motions of the moon and earth. Accordingly the records of ancient eclipses, which occurred centuries before the Christian era, afford exceedingly delicate tests of the motions of the moon and earth. At the time when Thomson and Tait's Natural Philosophy' was first published it was thought that all the numerical data were known with sufficient precision to render it possible to give a numerical estimate of the retardation of the earth's rotation. But the various revisions of the lunar theory which have been made since that date throw the whole matter into doubt. It seems probable that there is some portion of the acceleration of the moon's motion which is unexplained by gravitation, and may therefore be attributed to tidal friction, but its amount is uncertain. We can only say that the amount is very small. It is, however, not impossible that this smallness may be due to counteracting influences which tend to augment the speed of the earth's rotation; such an augmentation would result from shrinkage of the earth's mass through cooling. However this matter may stand, it does not follow that, because the changes produced by tidal friction in a man's lifetime or in many generations of man are almost insensible, the same must be true when we deal with millions of years. It follows that it is desirable to trace the effects of tidal friction back to their beginnings.
We have seen above that this cause will explain the repulsion of a satellite from a position close to the planet to a more remote distance. Now when we apply these considerations to the moon and earth we find that the moon must once have been nearly in contact with the earth. This very remarkable initial configuration of the two bodies seems to point to the origin of the moon by detachment from the earth.
Further details concerning this speculation in cosmogony are given below in chapter VIII.3 § t o. Bibliography. - Many works on popular astronomy contain a few paragraphs on the tides, but the treatment is generally so meagre as to afford no adequate idea of the whole subject. A complete list of works both general and technical bearing on the theory of the tides, from the time of Newton down to 1881, is contained in vol. ii. of the Bibliographie de l'astronomie by J. C. Houzeau and A. Lancaster (1882). This list does not contain papers on the tides of particular ports, and we are not aware of the existence of any catalogue of works on practical observation, reduction of observations, prediction and tidal instruments. The only general work on the tides, without mathematics, is George Darwin's Tides and Kindred Phenomena in the Solar System.' This book treats of all the subjects considered in the present article (with references to original sources), and also others such as seiches (q.v.) and the bore (q.v.).
The most extensive monograph on the tides is A Manual of Tides by Mr Rollin A. Harris, published by the United States Coast Survey in a series of parts, of which pt. i. appeared in 1897, and pt. iv.
' See that work (ed. 1883), § 830; P. H. Cowell, M. N. R. Ast. Soc. (1905), lxv. 861.
For a discussion of the subject without mathematics, see G. H. Darwin's Tides. 4 London (1898) and with important changes (1901, 191 1); (Boston, 1898); translations: German, by A. Pockels (Leipzig, 1902, 1911); Italian, by G. Magrini (Turin, 1905), with appendices by translator; Magyar, byRado von Kovesligethy (Budapest,1904),with appendices by translator.
B in 1904. This work contains an enormous mass of useful work, and gives not only complete technical developments both on the theoretical and practical sides but also has chapters of general interest. The present writer feels it his duty, however, to dissent from Mr Harris's courageous attempt to construct the cotidal lines of the various oceans.
This work contains the most complete account of the history of tidal theories of which we know. Laplace's admirable history of the subject down to his own time has been summarized in § 7. Dr Giovanni Magrini has an appendix to his translation of Darwin's book, entitled La Conoscenza della marea nell'antichita, founded on the researches of Dr Roberto Almagia. Dr Almagia. himself gives the results of his researches more fully in a memoir, presented to the Accademia dei Lincei of Rome (5th series, vol. v. fascic. x., 1905, 1 37 PP).
Another monograph on tides, treating especially the mathematical developments, is Maurice Levy's La The'orie des marees (Paris, 1898). Colonel Baird's Manual of Tidal Observation (1886) contains instructions for the installation of tide-gauges, and auxiliary tables for harmonic analysis. Airy's article on " Tides and Waves " in the Ency. Metrop., although superseded in many respects, still remains important. Harris's Manual contains a great collection of results of tidal observations made at ports all over the world.
The article " Die Bewegung der Hydrosphare " in the Encyklopeidie der mathematischen Wissenschaften (vi. I, 1908) gives a technical account of the subject, with copious references. The same article is given in English in vol. iv. (1911) of G. H. Darwin's collected Scientific Papers; and vols. i. and ii. contain reprints of the several papers by the same author referred to in the present article.
Since the date of the 9th edition of the Ency. Brit. some technical discussion of the tides has appeared in textbooks, such as H. Lamb's Hydrodynamics.' That work also reproduces in more modern form Airy's investigation of the effects of friction on the tides of rivers. We are thus able to abridge the present article, but we shall present the extension by Hough of Laplace's theory of the tides of an oceancovered planet, which is still only to be found in the original memoirs.
II
[[Tide-Generating Forces § I T]]. Investigations of Tide-Generating Potential and Forces. - We have already given a general explanation of the nature of tide-generating forces; we now proceed to a rigorous investigation. If a planet is attended by a single Forces. satellite, the motion of any body relatively to the planet's surface is found by the process described as reducing the planet's centre to rest. The planet's centre will be at rest if every body in the system has impressed on it a velocity equal and opposite to that of the planet's centre; and this is accomplished by impressing on every body an acceleration equal and opposite to that of the planet's centre.
Let M, m be the masses of the planet and the satellite; r the radius vector of the satellite, measured from the planet's centre; p the radius vector, measured from same point, of the particle whose motion we wish to determine; and z the angle between r and p. The satellite moves in an elliptic orbit about the planet, and the acceleration relatively to the planet's centre of the satellite is (M-}-m)/r 2 towards the planet along the radius vector r. Now the centre of inertia of the planet and satellite remains fixed in space, and the centre of the planet describes an orbit round that centre of inertia similar to that described by the satellite round the planet but with linear dimensions reduced in the proportion of m to M-Fm. Hence the acceleration of the planet's centre is m/r 2 towards the centre of inertia of the two bodies. Thus, in order to reduce the planet's centre to rest, we apply to every particle of the system an acceleration m/r 2 parallel to r, and directed from satellite to planet.
Now take a set of rectangular axes fixed in the planet, and let M i r, Mgr, Mar be the co-ordinates of the satellite referred thereto; and let gyp, np, p be the co-ordinates of the particle P whose radius vector is p. Then the component accelerations for reducing the planet's centre to rest are - mM i /r 2, - mM 2 /r 2, - mM 3 /r 2; and since these are the differential coefficients with respect to pt, pn, of the function - mP - - Men-f-M3f), and since cos z=Mlle-Men-+-M31', it follows that the potential of the forces by which the planet's centre is to be reduced to rest is - m p cos Z.
r 1 The theory as presented in the Mecanique celeste is unnecessarily difficult, and was much criticized by Airy. Before the publication of the 9th and Toth editions of the Ency. Brit. it was necessary for the student to read a number of controversial papers published all over the world in order to get at the matter.
Now let us consider the other forces acting on the particle. The planet is spheroidal, and therefore does not attract equally in all directions; but in this investigation we may make abstraction of the ellipticity of the planet and of the ellipticity of the ocean due to the planetary rotation. This, which we set aside, is considered in the theories of gravity and of the figures of planets. Outside its body, then, the planet contributes forces of which the potential is M/p. Next the direct attraction of the satellite contributes forces of which the potential is the mass of the satellite divided by the distance between the point P and the satellite; this is ‘ 11r 2 -{- p 2 - 2rp cos z} To determine the forces from this potential we regard p and z as the variables for differentiation, and we may add to this potential any constant we please. As we are seeking to find the forces which urge P relatively to M, we add such a constant as will make the whole potential at the planet's centre zero, and thus we take as the potential of the forces due to the attraction of the satellite m tr 2 -{- p 2 - 2rp cos zi r It is obvious that in the case to be considered r is very large compared with p, and we may therefore expand this in powers of p/r. This expansion gives us m 3-P + p2 2 P + p 2 P 3 +. .. (, where P I = cos z, P2 = cos t z - 2 i P3 = cos 3 z - 2 cos z, &c. The reader familiar with spherical harmonic analysis of course recognizes the zonal harmonic functions; but the result for a few terms, which is all that is necessary, is easily obtainable by simple algebra.
Now, collecting together the various contributions to the potential, and noticing that '- 'r °' P1= mp cos z, and is therefore equal and opposite to the potential by which the planet's centre was reduced to rest, we have as the potential of the forces acting on a particle whose co-ordinates are p, pn, pf P - f - -(2 cos 2 z2)-}-' - 4 3 (I cos 3 z-2 cos z) -{- ... (I) The first term of (I) is the potential of gravity, and the terms of the series, of which two only are written, constitute the tide-generating potential. In all practical applications this series converges so rapidly that the first term is amply sufficient, and thus we shall generally denote V=2 p2 (cos t z - a) (2) as the tide-generating potential.2 At the surface of the earth p is equal to a the earth's radius.
§ 12. Form of Equilibrium. - Consider the shape assumed by an ocean of density a, on a planet of mass M, density S and radius a, when acted on by disturbing forces whose potential is a solid spherical harmonic of degree i, the planet not being in rotation.
If S i denotes a surface spherical harmonic of order 1, -such a potential is given at the point whose radius vector is p by V Si. (3) In the case considered in § II, 1=2 and S i becomes the second zonal harmonic cos 2 z - 3.
The theory of harmonic analysis tells us that the form of the ocean, when in equilibrium, must be given by the equation p= a -}-eiSi. (4) Our problem is to evaluate e i . We know that the external potential of a layer of matter, of depth e i S i and density o, has the value I / /I p i+l eiSi. Hence the whole potential externally to the planet and up to its surface is i al i + 1 2r 3 a> Si - i - 2 i - }- I (p/ ei Si. (5) The first and most important term is the potential of the planet, the second that of the disturbing force, and the third that of the departure from sphericity.
Since the ocean must stand in a level surface, the expression (5) equated to a constant must be another form of (4). Hence, if we put p = a -}-e i S i in the first term of (5) and p=a in the second and third terms, (5) must be constant; this can only be the case if the The reader may refer to Thomson and Tait's Natural Philosophy (1883), pt. ii. §§ 798-821, for further considerations on this and analogous subjects, together with some interesting examples.
coefficient of i vanishes. Hence on effecting these substitutions and equating that coefficient to zero, we find 3ma 2 pro-a - o. are‘ +- 1 - r 3 + 21+ lei But by the definitions of S and a we have M=1-7r&a 3 =ga t , where g is gravity, and therefore 3ma2 2gr3 e i = 3c I - (21+ I)6 In the particular case considered in § II we therefore have p - a[1-4- Im3%/Sb3 (cos2z3)?
as the equation to the equilibrium tide under the potential V = 2r, p2 (cos 2 z - 3) .
If a were very small compared with the attraction of the water on itself would be very small compared with that of the planet on the water; hence we see in the general case that 1/ (I - 3c 2T+1)S/ is the factor by which the mutual gravitation of the ocean augments the deformation due to the external forces. This factor will occur frequently hereafter, and therefore for brevity we write Q b, - I - (21+I)S' and we may put (6) in the form _ 3ma2 e t 2gr3 bi Comparison with (5) then shows that V=gbi (id is the potential of the disturbing forces under which p =a-f-e i S i (I I) is a figure of equilibrium.
We are thus provided with a convenient method of specifying any disturbing force by means of the figure of equilibrium which it is competent to maintain. In considering the dynamical theory of the tides on an ocean-covered planet, we shall specify the disturbing forces in the manner expressed by (io) and (II). This way of specifying a disturbing force is equally exact whether or not we choose to include the effects of the mutual attraction of the ocean. If the augmentation due to mutual attraction of the water is not included, b i becomes equal to unity; there is no longer any necessity to use spherical harmonic analysis, and we see that if the equation to the surface of an ocean be p= a+S, where S is a function of latitude and longitude, it is in equilibrium under forces due to a potential whose value at the surface of the sphere (where p=a) is gS. In treating the theory of tidal observation we shall specify the tide-generating forces in this way, and then by means of " the principle of forced vibrations," referred to in § 7 as used by Laplace for discussing the actual oscillations of the sea, we shall pass to the actual tides at the port of observation.
In this equilibrium theory it is assumed that the figure of the ocean is at each instant one of equilibrium under the action of gravity and of the tide-generating forces. Lord Kelvin has, however, reasserted' a point which was known to Bernoulli, i but has since been overlooked, namely, that this law Theory. of rise and fall of water cannot, when portions of the globe are continents, be satisfied by a constant volume of water in the ocean. The necessary correction to the theory depends on the distribution of land and sea, but a numerical solution shows that it is practically of very small amount.
§ 13. Development of Tide -generating Potential in Terms of Hour - Angle and Declination. - We now proceed to develop the tidegenerating potential, and shall of course implicitly (§ 12) determine the equation to the equilibrium figure.
We have already seen that, if z be the moon's zenith distance at the point P on the earth's surface, whose co-ordinates referred to A, B, C, axes fixed in the earth, and at, an, a;', cos z =M1 - { - rtM2 where M I, M2, M3 are the moon's direction cosines referred to the same axes. Then, with this value of cos z, } cos 2 z-3 = 2 ?JMiM2+ 2 27J2M,2 2 M 22+2n1"M2M3+2EJM1M3 2 I Thomson and Tait, Nat. Phil. § 807. G. H. Darwin and H. H. Turner, Proc. Roy. Soc. (1886).
The axis of C is taken as the polar axis, and AB is the equatorial plane, so that the functions of, i, are functions of the latitude and longitude of the point P, at which we wish to find the potential. The functions of M1, M2 i M3 depend on the moon's position, and we shall have occasion to develop them in two different ways - first in terms of her hour-angle and declination, and secondly (§ 25) in terms of her longitude and the elements of the orbit.
Now let A be on the equator in the meridian of P, and B 90° east of A on the equator. Then, if M be the moon, the inclination of the plane MC to the plane CA is the moon's easterly local hour-angle. Let ho =Greenwich westward hour-angle; l =the west longitude of the place of observation; A = the latitude of the place; S = moon's declination: then we have M I =cos cos (ho - l), M2 = - cos sin(ho - l), M3=sin S, E=cos A, r t =o, i =sin A.
Also the radius vector of the place of observation on the earth's surface is a. Whence we find V - 3M(12 a cos t X cos 2 S cos 2 (ho - l) + sin 2X sin cos 6 cos (ho - 1) - sin e %) (a - sin2X) (13) The tide-generating forces are found by the rates of variation of V for latitude and longitude, and also for radius a, if we care to find the radial disturbing force.
The westward component of the tide-generating force at the earth's surface, where p =a, is dV/a cos Xdl, and the northward component is d VladX; the change of apparent level is the ratio of these to gravity g. On effecting the differentiations we find that the westward component is made up of two periodic terms, one going through its variations flour- twice and the other once a day. The southward cornponent has also two similar terms; but it has a third very small term, which does not oscillate about a zero value. This last term corresponds to forces which produce a constant heaping up of the water at the equator; or, in other words, the moon's attraction has the effect of causing a small permanent ellipticity of the earth's mean figure. This augmentation of ellipticity is of course very small, but it is necessary to mention it.
If we consider the motion of a pendulum-bob under the influence of these forces during any one day, we see that in consequence of the semi-diurnal changes of level it twice describes an ellipse with major axis east and west, and the formula when developed shows that the ratio of axes is equal to the sine of the latitude, and the linear dimensions proportional to cos' 6. It describes once a day an ellipse whose north and south axis is proportional to sin cos 2X and whose east and west axis is proportional to sin 23 sin A. Obviously the latter is circular in latitude 30°. When the moon is on the equator, the maximum deflexion occurs when the moon's local hour-angle is 45°, and is then equal to 3 3 m a cos A. 2M r This angle is equal to o0174" cos A. Attempts actually to measure the deflexion of the vertical have at length proved successful (see Seismometer).
III. - Dynamical Theory Of The Tides § 14. Recent Advances in the Dynamical Theory of the Tides. - The problem of the tidal oscillation of the sea is essentially dynamical. In two papers in the second volume of Liouville's Journal (1896) H. Poincare has considered the mathematical principles involved in the problem, where the ocean is interrupted by land as in actuality. He has not sought to obtain numerical results applicable to any given configuration of land and sea, but he has aimed rather at pointing out methods by which it may some day be possible to obtain such solutions. Even when the ocean is taken as covering the whole earth the problem presents formidable difficulties, and this is the only case in which it has been solved hitherto.2 Laplace gives the solution in bks. i. and iv. of the Mecanique celeste; but his work is unnecessarily complicated. In the 9th edition of the Ency. Brit. we gave Laplace's theory without these complications, but the theory is now accessible in H. Lamb's Hydrodynamics and other works of the kind. It is therefore not reproduced here.
In 1897 and 1898 S. S. Hough undertook an important revision of Laplace's theory and succeeded not only in introducing the effects of the mutual gravitation of the ocean, but Lord Kelvin's (Sir W. Thomson's) paper on the gravitational oscillations of rotating water, Mag. (October 1880), bears on this subject. It is the only attempt to obtain numerical results in respect to the effect of the earth's rotation on the oscillations of land-locked seas.
2 7 2 - 2 ? - M1 2 +M2 2 - 2 M3 2 (12) 3 3 (io) also in determining the nature and periods of the free oscillations of the sea.' A dynamical problem of this character cannot be regarded as fully solved unless we are able not only to discuss the " forced " oscillations of the system but also the " free." Hence we regard Mr Hough's work as the most important contribution to the dynamical theory of the tides since the time of Laplace. We shall accordingly present the theory briefly in the form due to Mr Hough.
The analysis is more complex than that of Laplace, where the mutual attraction of the ocean was neglected, but this was perhaps inevitable. Our first task is to form the equations of motion and continuity, which will be equally applicable to all forms of the theory.
§ 15. Equations of Motion. - Let r, 0, 4, be the radius vector, colatitude and east longitude of a point with reference to an origin, a polar axis and a zero-meridian rotating with a uniform angular velocity n from west to east. Then if R, H, be the radial, colatitudinal and longitudinal accelerations of the point, we have R dt2 - r (d8) 2 - r sine (dn) 2 41, - r dt (r' j) - r sin 0 cos 0 (d +n) H - r sin° dt [r2 sin20 (d +n) 1. If the point were at rest with reference to the rotating meridian we should have R = - n 2 r sin 0, E= - n 2 r sin 0 cos 0, H =o.
When these considerations are applied to the motion of an ocean relative to a rotating planet, it is clear that these accelerations, which still remain when the ocean is at rest, are annulled by the permanent oblateness of the ocean. As then they take no part in the oscillations of the ocean, and as we are not considering the figure of the planet, we may omit these terms from R and. This being so we must replace (+n)' as it occurs in R and E by (j4,) +2ndt Now suppose that the point whose accelerations are under consideration never moves far from its zero position, and that its displacements E, rt sin 0 in colatitude and longitude are very large compared with p its radial displacement. Suppose, further, that the velocities of the point are so small that their squares and products are negligible compared with n 2 r 2; then we have dr dp dt - dt a very small quantity; sin' 0d - dt (n sin 0), do di rc- - dt' Since the radial velocity always remains very small it is not necessary to concern ourselves further with the value of R, and we only require the two other components which have the approximate forms, V=' - 2n sin 0 cos 0 d - t' dt 2 H =sin 0a t 2 +2n cos B d t We have now to consider the forces by which an element of the ocean is urged in the direction of colatitude and longitude. These forces are those due to the external disturbing forces, to the pressure of the water, surrounding an element of the ocean, and to the attraction of the ocean itself.
If e denotes the equilibrium height of the tide, it is a function of colatitude and longitude, and may be expanded in a series of spherical surface harmonics e i. Thus we may write the equation to the equilibrium tide in the form.
r=a+e=a+Eei. Now it appears from (io) and (i i) that the value of the potential, at the surface of the sphere where p= a, under which this is a figure of equilibrium, is V = Egbiei.
We may use this as specifying the external disturbing force due to the known attractions of the moon and sun, so that e i may be regarded as known.
But in our dynamical problem the ocean is not a figure of equilibrium, and we may denote the elevation of the surface at any moment of time by b. Then the equation to the surface may be written in the form r=a+b=a+Ebi, where 1, denotes a spherical harmonic just as e i did before.
1 Phil. Trans., 189 A, pp. 201-258 and 191 A, pp. 139-185.
The surface value of the potential of the forces which would maintain the ocean in equilibrium in the shape it has at any moment is Egb i b i. Hence it follows that in the actual case the forces due to fluid pressure and to the attraction of the ocean must be such as to balance the potential just determined. Therefore these forces are those due to a potential - Egb i b i. If we add to this the potential of the external forces, we have a potential which will include all the forces, the expression for which is - gEb i (b i - e i). If further we perform the operations d/ad9 and d/a sin 0d4 on this potential, we obtain the colatitudinal and longitudinal forces which are equal to the accelerations, and H.
It follows, then, from (i¢) that the equations of motion are dt2 - 2n sin e cos e at = - a E bi (F9 (bi - ei) 05) sin Bat2 +2n cos e B dt = - a sun Eb i d (b - e)It remains to find the equation of continuity. This may be deduced geometrically from the consideration that the volume of an element of the fluid remains constant; but a shorter way is to derive it from the equation of continuity as it occurs in ordinary hydrodynamical investigations. If t be a velocity potential, the equation of continuity for incompressible fluid is Sr-1 e (r 2 d-sin 0 50 SO) +5Bd (r si n Bd95r5c6) +S4 'd? (r r sine d5r SB) =0. The element referred to in this equation is defined by r, 0, 0, r+Sr, 0+50, 0-{-14,. The colatitudinal and longitudinal velocities are the same for all the elementary prism defined by 8, 4,, 0+50, 0+4, and the sea bottom. Then = dt' r s i n ed4 =sin 0- and, since the radial velocity is db/dt at the surface of the ocean, where r = a -}-y, and is zero at the sea bottom, where r = a, we have d? - r=a db Hence, integrating with respect to from TIT y dt r = a+y to r = a, and again with respect to t from time t to the time when b,, n all vanish, and treating y and Il as small compared with a, we have ba sin 0 - } - d e (y sin e)+ 4 (yr t sin 0) =0. (16) This is the equation of continuity, and, together with (15), it forms the system which must be integrated in the general problem of the tides. The difficulties in the way of a solution are so great that none has hitherto been found, except on the supposition that y, the depth of the ocean, is only a function of latitude. In this case (16) becomes i d 0 d0`'."sin 8) +yd? = o. (17) § 16. Adaptation to Forced Oscillations. - Since we may suppose that the free oscillations are annulled by friction, the solution required is that corresponding to forced oscillations. Now we have seen from 03) that e (which is proportional to V) has terms of three kinds, the first depending on twice the moon's (or sun's) hour-angle, the second on the hour-angle, and the third independent thereof. The coefficients of the first and second vary slowly, and the whole of the third varies slowly. Hence e has a semi-diurnal, a diurnal and a long-period term. We shall see later that these terms may be expanded in a series of approximately semi-diurnal, diurnal and slowly varying terms, each of which is a strictly harmonic function of the time.
Thus according to the usual method of treating oscillating systems, we may make the following assumptions as to the form of the solution e = Eei = Ee i cos(2nft+ sfi + a) b = bE i =Eh i cos(2nft+ + a) = Eb i x i cos(2nft +so +a) rt = Eb i y i sin(2nft+s0 +a) where e i, h i, x i, y i are functions of colatitude only, and e i, h i are the associated functions of colatitude corresponding to the harmonic of order i and rank s. For the semi-diurnal tides s = 2 and f is approximately unity; for the diurnal tides s = i and f is approximately 2; and for the tides of long period s=o and f is a small fraction. Substituting these values in 0E7) we have [ š th o)+ y y + d9(y b ixi s i n 9 s bi i h i a = o. Then if we write u i for h i - e i, and put m=n'a/g, from (18) in 05) leads at once to f 2 Eb;x i - f sin 0 cos 9Eb i y i = 4 m d B Ebiui, r . (20) f 2 sin 0Eb i y i +f cos 0Eb i x i = - 4m Ebiui. ¢m sin o (18) (14) (19) substitution (Eb i x i) (-cos 2 0)=-1 4m [d i . " -r" f sin B biu i? (Ebiyi) sin e 0(f2-cos" 0) _ c - 4m [ f B de zb i ui+ sin e`eiui Then substituting from (21) in (19) we have d y(sin d +f cos 0?b i u i) sin o f' -cos 2 0 [-° d+ sy Sln 0 biuiJ J sin 0 (f 2 -cos 2 0) This is almost the same as Laplace's equation for tidal oscillations in an ocean whose depth is only a function of latitude. If indeed we treat b i as unity (thereby neglecting the mutual attraction of the water) and replace Zu i and le i by u and e, we obtain Laplace's equation.
When u i is found from this equation, its value substituted in (21) will give x i and yi.
§ 17. Zonal Oscillations. - We might treat the general harmonic oscillations first, and proceed to the zonal oscillations by putting s=o. These waves are, however, comparatively simple, and it is well to begin with them. The zonal tides are those which Laplace describes as of the first species, and are now more usually called the tides of long period. As we shall only consider the case of an ocean of uniform depth, y the depth of the sea is constant. Then since in this case s=o, our equation (22), to be satisfied by u i or hi - ei, becomes sin od-biui +-4y a sin 0 h i =o.
f 2 - cos 2 0 This may be written d Ebiui+ 4 y a sin B 28f re_ sin od0+A =0, (23) where A is a constant.
Let us assume hi= C i P i, ei = EiPi where P i denotes the ith zonal harmonic of cos 8. The coefficients C i are unknown, but the E i are known because the system oscillates under the action of known forces.
If the term involving the integral in this equation were expressed in terms of differentials of harmonics, we should be able to equate to zero the coefficient of each dP i /de in the equation, and thus find the conditions for determining the C's.
The task then is to express f 2 - co s 2 0 ° P i sin Bdo in differentials sin o J z of zonal harmonics.
It is well known that P i satisfies the differential equation d e (sin 4Ç) +1(1+I)P i sin o=o. (24) Therefore f P i sin odor -i(i+I)sin o de , and 2 sin 0 Pi sin Bdo = -. 2 1 (f cos 0) ei (f )dPi I 2 dPi I de - i(i+1) sin ode Another well-known property of zonal harmonics is that sin O dP -= 221+ }- i) (Pi+i-Pi-1) If we differentiate (25) and use (24) we have 2) 21 + 1 ( d'de 1 - d ?8 1) +i(i+I)P i si n 0=0 . (26) Multiplying (25) by sin 0, and using (26) twice over, dP i(i+1) _ 1 dP 2_ dP i 1 dPi_dPi_2 sin g e x 21+ 22+3 (de do) + 21 - (de do) S f2 0 dPi_2 Therefore ore P sin Bdo - 5 f 2 - I 2 dPi I dPi +2 i(i +I)+(21 -I)(21 +3) "do + (21 +I)(21 +3) de This expression, when multiplied by 4ma/-y and by C i and summed, is the second term of our equation.
The first term is ?bi(C4-Ei) Vii. In order that the equation may be satisfied, the coefficient of each dP i /do must vanish identically. Accordingly we multiply the whole by y/4ma and equate to zero the coefficient in question, and obtain 4 a z-E:)? - (21 - ) y (21-3) t i(i+I) m (C (22-1)(22+3) C$ Ci+2 (2 +3) (22 (27) This equation (27) is applicable for all values of i from 1 to infinity, provided that we take Eo, C_1, E_ 1 as being zero.
We shall only consider in detail the case of greatest interest, namely that of the most important of the tides generated by the attraction of the sun and moon. We know that in this case the equilibrium tide is expressed by a zonal harmonic of the second order; and therefore all the Ei, excepting E2, are zero. Thus the equation (27) will not involve E i in any case excepting when 1=2.
If we write for brevity _ f 2 - 1 2 biy Li -i(i+1)+ (2 i- I) (21+3) 4ma' the equation (27) is C i+2 Ci - 2 - o. 28 (21+3)(22+5) LtiCi+( 21-3)(21-) () Save that when 1=2, the right-hand side is b 2 -yE 2 /4ma, a known quantity ex hypothesi. The equations naturally separate themselves into two groups in one of which all the suffixes are even and the other odd. Since our task is to evaluate all the C's in terms of E2, it is obvious that all the C's with odd suffixes must be zero, and we are left to consider only the cases where 1=2, 4, 6, &c.
We have said that Co must be regarded as being zero; if however we take Co = - 3b2yE2/4ma, so that Co is essentially a known quantity, the equation (28) has complete applicability for all even values of i from 2 upwards. The equations are It would seem at first sight as if these equations would suffice to determine all the C's in terms of C2, and that C2 would remain indeterminate; but we shall show that this is not the case. For very large values of i the general equation of condition (28) tends to assume the form C i+2 + Ci-2 + i ma 2 y Cc = o.
By writing successively i+2, 1+4, id-6 for i in this equation, and taking the differences, we obtain an equation from which we see that, unless Ci/Ci+2 tends to become infinitely small, the equations are satisfied by Ci = Ci+2 in the limit for very large values of 1.
Hence, if C i does not tend to zero, the later portion of the series for h tends to assume the form Ci(Pi+Pi+2+Pi+4...). All the P's are equal to unity at the pole; hence the hypothesis that Ci does not tend to zero leads to the conclusion that the tide is of infinite height at the pole. The expansion of the height of tide is essentially convergent, and therefore the hypothesis is negatived. Thus we are entitled to assume that C i tends to zero for large values of i.
Now writing for brevity ai /(21+I) (22+3)2(22+5), we may put (28) into the form Ci-2/Ci - Li - Ci/Ci+2 (21-3) (21 -I)- (21+I)(21+3) By successive applications of this formula we may write the righthand side in the form of a continued fraction.
Let Then we have or Thus C 2 = 3.5K2Co; C4=3.5.7.9K2K4Co; C6 = 3. 5.7. 9 . I I.13K 2 K 4 K 6 Co, &c. If we assume that any of the higher C's, such as C14 or C16, is of negligible smallness, all the continued fractions K2, K4, Ks, &c., may be computed; and thus we find all the C's in terms of Co, which is equal to-3b 2 yE 2 /4ma. The height of the tide is therefore given by lt = Zh i cos(2nft+ a) = - 4m 2 Y E 2{3.5 K P +3.5. 7.9K2 K 4 P 4 + ... } cos (2nft + a). It is however more instructive to express h as a multiple of the (21) (22) dPi, (25) ai-2 ai +2 Li - L i+2 - Li+4 - Ci -2 /Ci ai -2 (21-3)(211) - Ki' Ci/C12 = (21 - (21 - I) (22+ I)Ki.
- I - 4maE(us+ei) =o.
sin 0 (21 - I)(22+I) de Co - 0 5.7 -L4C4+ II.13 - .
ai equilibrium tide e, which is as we know equal to E P cos (2nft+a). Whence we find b = 4 2a P2 { 3.5K P +3.5.7.9K K P +3.5 ...13K K K P ... } m The number f is a fraction such that its reciprocal is twice the number of sidereal days in the period of the tide. The greatest value of f is that appertaining to the lunar fortnightly tide (Mf in notation of harmonic analysis), and in this case f is in round numbers 1/28, or more exactly f 2 = .00133. The ratio of the density a of sea-water to S the mean density of the earth is 18093; which value gives us b2 = I - 5 = 89144.
The quantity m is the ratio of equatorial centrifugal force to gravity, and is equal to 1/289. Finally, /a is the depth of the ocean expressed as a fraction of the earth's radius.
With these numerical values Mr Hough has applied the solution to determine the lunar fortnightly tide for oceans of various depths. Of his results we give two: - First, when =7260 ft. = 1210 fathoms, which makes y/4ma =1/40, he finds b =p { 2669P -1678P4+. 0485P6 - o081P + 0009Pia - 0001P ... }. If the equilibrium theory were true we should have e b= P {P2}; thus we see how widely the dynamical solution differs from the equilibrium value.
Secondly, when y =58080 ft. =9680 fathoms, and y/4ma =1/5, he finds b=P {7208P - 0973P4+ o048P6 - 0001P8... }.
From this we see that the equilibrium solution presents some sort of approximation to the dynamical one; and it is clear that the equilibrium solution would be fairly accurate for oceans which are still quite shallow when expressed as fractions of the earth's radius, although far deeper than the actual sea.
The tides of long period were not investigated by Laplace in this manner, for he was of opinion that a very small amount of friction would suffice to make the ocean assume its form of equilibrium. In the arguments which he adduced in support of this view the friction contemplated was such that the integral effect was proportional to the velocity of the water relatively to the bottom. It is probable that proportionality to the square of the velocity would have been nearer the truth, but the distinction is unimportant.
The most rapid of the oscillations of this class is the lunar fortnightly tide, and the water of the ocean moves northward for a week and then southward for a week. In oscillating systems, where the resistances are proportional to the velocities, it is usual to specify the resistance by a " modulus of decay," namely the time in which a velocity is reduced by friction to e1 - or 1/2.78 of its initial value. Now in order that the result contemplated by Laplace may be true, the friction must be such that the modulus of decay is short compared with the semi-period of oscillation. It seems certain that the friction of the ocean bed would not reduce a slow ocean current to one-third of its primitive value in a day or two. Hence we cannot accept Laplace's discussion as satisfactory, and the investigation which has just been given becomes necessary. (See § 34).
§ 18. Tesseral Oscillations. - The oscillations which we now have to consider are those in which the form of surface is expressible by the tesseral harmonics. The results will be applicable to the diurnal and semi-diurnal tides Laplace's second and third species.
If we write a=s/f the equation (22) becomes d (sin cos o) Eb u (0- cos 8d +s 2 cosec o) Ebiui do s2 - a' cos' 0 52-a2 COS' 0 +- sin 0 2h = 0. (29) If we write D for the operation sin d, the middle term may be arranged in the form a cot 0 (D +a cos o) (Eb i u i) Ebiui s 2 -02 cos' 0 sin 0' Therefore on multiplying by sin 0 the equation becomes (D -a cos o) [(D± cos ') (biu)] - (' E b i u i ) + m sin BEh = o. (30) We now introduce two auxiliary functions, such that Eb h - e -Ebiui =(D -0- cos 0)T+ (s 2 - a' cos' 0)4>. (31) It is easy to prove that (D+a cos 0)(D -a cos 8) =D 2 -s'+a sin' o + (s'-a' cos' 0), (D-a cos 0)(D+a cos 0) = D 2 -s 2 -a sin' 0+ (s 2 - a' cos' o). Also (D +a cos 8) (s 2 -a 2 cos' 0) I = (s 2 - a 2 cos t 0) (D +a cos B) +2a 2 sin 2 8 cos 0 (33) Now perform D+a cos 0 on (31), and use the first of (32) and (33), and we have (D +a cos 0)(Eb i u i) =(D2-s2-1-0- sin e 0+s 2 - a 2 cos28) ?.
} (s 2 -a 2 cos t 8)(D-{-a cos 0)4)+2a 2 sin 2 0 cos 0 1. (34) The functions 4, and are as yet indeterminate, and we may impose another condition on them. Let that condition be (D 2 - s 2 +0- sin' 0) _ -202 sin e 0 cos 0 c b. (35) Then (34) may be written (D+a cos 0) (niui) 1 +(D+a cos 8)43. s 2 -a 2 cos' 0 Substituting from this in (30), and using the second of (32), the function disappears and the equation reduces to (D 2 - $ 2 - a sin' 8)(1)+ 4 y a sin' e 2;14=0. (36) c os 8 Since by (35) -a 2 cos' 0 - 2 sin' (D2-s' +a sin' 0) 4,, (31) may be written Eb u = [D - a cos 8+1 s n 6 sin' 8)] 4 Y+s 2 4). (37) The equations (35), (36) and (37) define and 4), and furnish the equation which must be satisfied.
If we denote cos 0 by the zonal harmonics are defined by _ i Py 2` I 2 ?(dµ) (µ2 1)i ' The following are three well-known properties of zonal harmonics: [(I _,.,2) dP2]-i(i-} I)Pi =o, (38) (i+I)P - (21+I)µP +iP _ =0, (39) dPi+i dPi_i dµ - µ = (21-i- d (40) If Pi s? are the two tesseral harmonics of order i and rank s, it is also known that Pi = (1 - / 22) z8 a BPi µ Let us now assume h i = C:P, 'e =E:P:, = EaP R, '43= E1331P:.
These must now be substituted in our three equations (35), (36), (37), and the result must be expressed by series of the Pi functions. It is clear then that we have to transform into Ps functions the following functions of PR, namely in' (e D 2 - s 2 cr sin' 8)PL, cos OP:, s [D_ cos o+2 cos ° 0 (D 2 -s 2 +cr sin 2 0)]P:. If we differentiate (38) s times, and express the result by of the operator D, we find ( D 2 - s 2) p.4 +i (i + I) P 4 sin' 8 =0. Again, differentiating (39) s times and using (40), we find (i-s+ I)P:+1- (21+I) cos 0 P i+(i + s) P ti 1=o. Lastly, differentiating (41) once and using (38), (40) and 2(i-s+I) s (1+1)(2+S) DPi = i+ Pi+1 21+I P i - 1 By means of (42), and (44) we have sin' 0(D'-s2ta sin 8)Pi=[-2(2+ I) Ea1F, P:= s s cos 8 Pt121+I P-+ 21+I Pi+it [D-a + 12 s 2 +a sin'8)]F - (i-s+1)[a - I)1P8+1 s in 60°(D2 2(21+I) (i+s)[a+(2+I)(i+2)1 s P i-1 Therefore the equations (35), (3 6), (37) give i+s i s+I [a{_i(i+I) +a}P:+2a2s 21 + IP8-1+ 21+I P ti+1 ] =0, M[0:1 -i(i+I) -a}P;+ y 732a C;Pi] =o, a [c:_ E)P+a (i -s 2(2[Z+I(i-1)1p: +1 +(2+s){a"i"(2+I) (2-}'2)1Pa - ist`s2P:] =o. 2(22+I) (41) (32) means (43) Since these equations must be true identically, the coefficients of P. in each of them must vanish. Therefore ai{a-2(2+I)}+2r 2 2 } ?i+ 1 2 +31+131 _ 122- S I -f i{a+2(2+I)I+ ? a - C a =o, (Ci -E;)-1-ai - 1(1-s)[a+(1-I)(i-2)] bi 2(21 - I) 'a ' If we eliminate the a's and g's from the third equation (45), by means of the first two, we find
8 L'C3 C8 - yb E° (46) 4ma +771+2 i t2 - 'i, ' a s2 (22 - S2) [a+ (2 - I)(2-2)] L - 1)1(422 - I)[a - (1 - I)i}[a+2(2+I)] [(i+I)2-s2][a+(i+2)(i+3)I 'Ybi + 14(11 - 0 2 - '11 a - (i + I) (i+2)][a+i(i+1)]_ 4ma - (2-s)(i- s- I) - (210(213)[ ° -- (i _ -(i+s+i)(2+s+2) ni+2 (21+3) (21 +5)[0- (i+ I) (i+2)1 In the case of the luni-solar semi-diurnal tide (called in the notation of harmonic analysis) we have i = 2, s =2, a =2. Hence it would appear that these formulae_ for Ly and E fail by becoming indeterminate, but i and s are rigorously integers, whereas a depends on the " speed " of the tide; accordingly in the case referred to we must regard terms involving (i-s) as vanishing in the limit when approaches to equality with i (11). For this particular case then we find and to=o. 5 The equation (46) for the successive C's is available for all values of i provided that C_ 1, E_ 1 , Co, E 0 are regarded as being zero.
As in the case of the zonal oscillations, the equations with odd suffixes separate themselves from those with even suffixes, so that the two series may be treated independently of one another. Indeed, as we shall see immediately, the series with odd suffixes are satisfied by putting all the C's with odd suffixes zero for the case of such oscillations as may be generated by the attractions of the moon or sun.
For the semi-diurnal tides i=2, s=2, and f is approximately equal to unity. Hence the equilibrium tide is such that all the Ei, excepting E2, are zero.
For the diurnal tides i=2, s= i, and f is approximately equal to I. Hence all the Ei, excepting El, are zero. Since in neither case is there any E with an odd suffix, we need only consider those with even suffixes.
In both cases the first equation among the C's is -L 2 8 C +na C 71 ' 2 E 2 = 2 or r)It follows that if we write tZC C = -4 ma E2 (s=2 or I), the equation of condition amongst the C's would be of general applicability for all even values of i from 2 upwards.
The symbols E g o, 77 3 2 do not occur in any of the equations, and therefore we may arbitrarily define them as denoting unity, although the general formulae for t and n would give them other values. Accordingly we shall take o C 0 = CO= -4maE2 (S = 2 Or i). With this definition the equation O(S=2 or I) is applicable for i =2, 4, 6, &c.
It may be proved as in the case of the tides of long period that we may regard C2/C: +2 as tending to zero. Then our equation may be written in the form =Li ?iC Gi+2' and by successive applications the right-hand side may be expressed in the form of a continued fraction. Let us write r Li- L:+2 - Hence our equation may be written i C Ca H n: TJ 1l C3 It follows that H a H:H3 H3H:H 3 Ca 2 ' a 2 6 a s s ' 174.77 8 4, 7 g s ' Then since we have defined n2 and 4maE2, all the C's are expressed in terms of known quantities. Hence the height of tide b is given by b=Zh i cos(2nft+S4h+a) -4 b2 8 3 8 H2H4 H2H,2Hs ma E2 cos(2nft+ s? a) [HP +__ 1 _ :+ But the equilibrium tide c is given by c=E2P2 cos (2nft +4+ a).
Hence we may write our result in the following form, which shows the relationship between the true dynamical tide and the equilibrium tide. - _ 7b2 3 3 1 4 11 13 H 8 H 8 H s b 4ma ?12 P 2-i- 77 4 From a formula equivalent to this Mr Hough finds for the lunar semi-diurnal tide (s=2), for a sea of 1210 fathoms (y l 4 ma 40) b = 3 Io 39 6P9 + '5799 8P 1 - ' 19273 P s + 03054 1... .
This formula shows us that at the equator the tide is " inverted," and has 2.4187 times as great a range as the equilibrium tide.
For this same ocean he finds that the solar semi-diurnal tide is " direct " at the equator, and has a range 7.9548 as great as the equilibrium tide.
Now the lunar equilibrium tide is 2.2 times as great as the solar equilibrium tide, and since 2.2 X2.4187 only 5.3, it follows that in such an ocean the solar tides would have a range half as great again as the lunar. Further, since the lunar tides are " inverted " and the solar " direct," spring tide would occur at quarter moon and neap tide at full and change.
We give one more example from amongst those computed by Mr Hough. In an ocean of 9680 fathoms (y/4ma =1/5), he finds p 2 I. 7646P1- 06057P1:+.001447P6...
At the equator the tides are " direct " and have a range of 1.9225 as great as the equilibrium tide. In this case the tides approximate in type to those of the equilibrium theory, although at the equator, at least, they have nearly twice the range.
We do not give any numerical results for the diurnal tides, for reasons which will appear from the following section.
§ 25. Development of Equilibrium Theory of Tides in Terms of the Elements of the Orbits
Within the limits at our disposal we cannot do more than indicate the processes to be followed in this development. We have already seen in (2) that the expression for the moon's tide-generating potential is V=2Y3 m p 2(cos2 z-3), and in (12) that cos' z - = 2 07 M i M 2 + 21' 2 n2 M1 2 M22 + 2? .M 2 M 3 -12t;-M1M3 + 3 5 2 +f 2 - 22 M12 +M22-2M32, 2 3 3 where Mi, M2 i M3 and, r t, Iare respectively the direction cosines referred to axes fixed in the earth of the moon and of a place on the earth's surface at which the potential V is to be evaluated. At such a place the radius vector p is equal to a the earth's radius.
Let the axes fixed in the earth be taken as follows: the axis C the north polar axis; the axis A through the earth's centre and a point, on the equator on the same meridian as the place of observation; the axis B at right angles to the other two and eastward of A. Then if A be the latitude of the place of observation =cos A, n ' =sin A.
With these values we have cos' z - 3 = z cos' A(M12 - M22) +sin 2A. M1M2 +2(i - sin ' A) (M12+M22-2M32).
In fig. 7 let ABC be the axes fixed in the earth; XYZ a second set of axes, XY being the plane of the moon's orbit; M the projection of the moon in her orbit; I = ZC, the obliquity of the lunar orbit to the equator; x =AX =BCY; I=MX, the moon's longitude in her orbit measured from X, the descending node of the equator on the lunar orbit, hereafter called the " intersection." Then M I =cos l cos x+sin 1 sin x cos I =cos 2 -1/ cos (x i - --sin' 21 cos (x +l), M2 = - cos l sin x+sin 1 cos x cos I= - cos' 21 sin (x - l) - sin' ZI sin (x+l), Ma = sin 1 sin I --=.2 sin 11 cos zI sin 1. When these expressions are substituted in M, 2 - M2 2, M1M3, M I ' +M22-2M32. it is clear that the first will have terms in the cosines of 2(x - l), 2x, 2(x+l); the second in sines of x-2l, x, x+21; and the third in cos 2l, together with a term depending only on I. Now let c be the moon's mean distance, e the eccentricity of her orbit, and let X = [c(I y e2)1:114,, y e')] IM21 z = [c(I r e2) ] 2M3, m and T = c3Then we have for the lunar tide-generating potential at Tide- the place of observation generating z 1 z (X2 Potential. V= ( 1 - e21a[ 2 cos A. (X - z Y) + 2A.XZ +1(1 - sin' A) (X2+Y22Z2) (47) The only parts of this expression which are variable in time are the functions of X, Y, Z.
To complete the development the formulae of elliptic motion are introduced in these functions, and terms which appear numerically negligible are omitted. Finally, the three X-Y-Z functions are obtained as a series of simple time-harmonics, the arguments of the sines and cosines being linear functions of the earth's rotation, the moon's mean motion, and the longitude of the moon's perigee. The next step is to pass, according to the principle of forced oscillations, from the potential to the height of tide generated by the forces corresponding to that potential. The X-Y-Z functions being simple time-harmonics, the principle of forcedoscillations allows us to conclude that the forces corresponding to V in (47) will generate oscillations in the ocean of the same periods and types as the terms in V, but of unknown amplitudes and phases. Now let 3E 2 - V, 3EZ, ( - 221 2) be three functions having respectively similar forms to those of x2_ y2 XZ (X2+Y' - 2Z2) (1 - e 2) 3 ' (1 - e2)3' and (I - e2) 3 but differing from them in that the argument of each Height of of the simple time-harmonics has some angle subtracted Tide at any from it, and that the term is multiplied by a numerical port. factor. Then, if g be gravity and h the height of tide at the place of observation, we must have 7 h = T - 2 2 COS' A (JC'- ') sin 2A3E .Z+ (3 - Sin'A) 3(X' +12'-2x')].(48) The factor Ta e /g may be more conveniently writtenM (iYa, where M is the earth's mass. It has been so chosen that, if the equilibrium theory of tides were fulfilled, with water covering the whole earth, the numerical factors in the E-'-Z functions would i H, be each unity and the alterations of phase would be Tide of zero. The terms in 0 €2+ 1 z2 _ 2 2) require special Tide of Lon g consideration. The function of the latitude being a - sin' A, it follows that, when in the northern hemisphere it is high-water north of a certain critical latitude, it is lowwater on the opposite side of that parallel; and the same is true of the southern hemisphere. It is best to adopt a uniform system for the whole earth, and to regard high-tide and high-water as 2 FIG. 7.
consentaneous in the equatorial belt, and of opposite meanings outside the critical latitudes. We here conceive the function always to be written --sin e X, so that outside the critical latitudes high-tide is low-water. We may in continuing the development write the X-' -X functions in the form appropriate to the equilibrium theory with water covering the whole earth, for the actual case it is only then necessary to multiply by the reducing factor, and to subtract the phase alteration As these are unknown constants for each place, they would only occur in the development as symbols of quantities to be deduced from observation. It will be understood, therefore, that in the following schedules the " argument " is that part of the argument which is derived from theory, the true complete argument being the " argument " -K, where is derived from observation.
Up to this point we have supposed the moon's longitude and the earth's position to be measured from the " intersection "; but in order to pass to the ordinary astronomical formulae we must measure the longitude and the earth's position from the vernal equinox. Hence we determine the longitude and right ascension of the " intersection " in terms of the longitude of the moon's node and the inclination of the lunar orbit, and introduce them into our formulae for the 3E-11-X functions. The expressions for the functions corresponding to solar tides may be written down by symmetry, and in this case the intersection is actually the vernal equinox.
The final result of the process sketched is to obtain a series of terms each of which is a function of the elements of the moon's or sun's orbit, and a function of the terrestrial latitude the place of observation, multiplied by the cosine of an angle which increases uniformly with the time.
We shall now write down the result in the form of a schedule; but we must first state the notation employed: e, e, _ eccentricities of lunar and solar orbits; I, w =obliquities of equator to lunar orbit and ecliptic; p, p, =longitudes of lunar and solar perigees, a, a, =hourly increments of p, p,; s, h =moon's and sun's mean longitudes; a, n=hourly increments of s, h; t= local mean solar time reduced to angle; y -r t =15° per hour; X _ latitude of place of observation; E, v=longitude in lunar orbit, and R.A. of the intersection; N=longitude of moon's node; i= inclination of lunar orbit. The " speed " of any tide is defined as the rate of increase of its argument, and is expressible, therefore, as a linear function of y, i, a, a; for we may neglect c'7, as being very small.
The following schedules, then, give h the height of tide. The arrangement is as follows. First, there is a universal coefficient 2 M (° -c ') 3 a, which multiplies every term of all the schedules. Secondly, there are general coefficients, one for each schedule, viz. cos 2 X for the semi-diurnal terms, sin 2X for the diurnal, and 2 -2 sin' X for the terms of long period. In each schedule the third column, headed " coefficient," gives the functions of I and e. In the fourth column is given the mean semi-range of the corresponding term in numbers, which is approximately the value of the coefficient in the first column when I = ee; but we pass over the explanation of the mode of computing the values. The fifth column contains arguments, linear functions of t, h, s, p, v, . In [A, i.] 2t+2(h-p) and in [A, ii.] t-1--(h-v) are common to all the arguments. The arguments are grouped in a manner convenient for subsequent computation. Lastly, the sixth is a column of speeds, being the hourly increases of the arguments in the preceding column, estimated in degrees per hour. It has been found practically convenient to denote each of these partial tides by an initial letter, arbitrarily chosen. In the first column we give a descriptive name for the tide, and in the second the arbitrarily chosen initial.
The schedule for the solar tides is drawn up in precisely the same manner, the only difference being that the coefficients are absolute constants. In order that the comparison of the importance of the solar tides with the lunar may be complete, the same universal coefficient 2 M a) ' a is retained, and the special coefficient for each term is made to involve the factor ?'. Here 3 m 3,, m being the sun's mass. With =81.5, T'=460352.17226' To write down any term, take the universal coefficient, the general coefficient for the class of tides, the special coefficient, and multiply by the cosine of the argument. The result, taken with the positive sign, is a term in the equilibrium tide, with water covering the whole earth. The transi tion to the actual case by the introduction of a factor E and a delay of phase (to be derived from observation) has been already explained. The sum of all the terms is the complete expression for the height of tide h.
It must be remarked that the schedule of tides is here largely abridged, and that the reader who desires fuller information must refer to the Brit. Assoc. Report for 1883, or vol. i. of G. H. Darwin's Scientific Papers, or to Harris's Manual of Tides. A. - Schedule of Lunar Tides. Universal Coefficient = 2 M (a) 3a.
§ 32. Diurnal Tides
These tides have not been usually treated whence (57) COS 2 (s - 5) (55) with much completeness in the synthetic method. In the tidetables of the British Admiralty we find that the tides at some ports are " affected by diurnal inequality"; such a statement may be interpreted as meaning that the tides are not to be predicted by the information given in the so-called tide-table. The diurnal tides are indeed complex, and do not lend themselves easily to a complete synthesis. In the harmonic notation the three important tides are K 1, 0, P, and the lunar portion of K 1 is nearly equal to 0 in height, whilst the solar portion is nearly equal to P. A complete synthesis may be carried out on the lines adopted in treating the semi-diurnal tides, but the advantage of the plan is lost in consequence of large oscillations of the amplitude through the value zero, so that the tide is often represented by a negative quantity multiplied by a circular function. It is best, then, only to attempt a partial synthesis, and to admit the existence of two diurnal tides. One of these will be a tide consisting of K 1 and P united, and the other will be O.
We shall not give the requisite formulae, but refer the reader to the Brit. Assoc. Report for 1885. A numerical example is given in the Admiralty Manual for 1886.
§ 33. On the Reduction of Observations of Highand Low-Water. 1 - A continuous register of the tide or observation at fixed intervals of time, such as each hour, is certainly the best; but for the adequate use of such a record some plan analogous to harmonic analysis is necessary. Observations of highand low-water only have, at least until recently, been more usual. In the reduction the immediate object is to connect the times and heights of highand low-water with the moon's transits by means of the establishment, age and fortnightly inequality in the interval and height. The reference of the tide to the establishment is not, however, scientifically desirable, and it is better to determine the mean establishment, which is the mean interval from the moon's transit to high-water at spring tide, and the age of the tide, which is the mean period from full moon and change of moon to spring tide.
For these purposes the observations ma y be conveniently treated graphically. 2 An equally divided horizontal scale is taken to represent the twelve hours of the clock of civil time, Graphical Det ermina- regulated to the time of the port, or - more accurately - Det r arranged always to show apparent time by being fast tionEstablish- or slow by the equation of time; this time-scale represents ment, &c. the time-of-clock of the moon's transit, either upper or lower. The scale is perhaps most conveniently arranged in the order V, VI, ...XII, I ... IIII. Then each interval of time from transit to high-water is set off as an ordinate above the corresponding time-of-clock of the moon's transit. A sweeping curve is drawn nearly through the tops of the ordinates, so as to cut off minor irregularities. Next along the same ordinates are set off lengths corresponding to the height of water at each high-water. A second similar figure may be made for the interval and height at low-water. In the curve of high-water intervals the ordinate corresponding to XII. is the establishment, since it gives the time of high-water at full moon and change of moon. That ordinate of high-water intervals which is coincident with the greatest ordinate of high-water heights gives the mean establishment. Since the moon's transit falls about fifty minutes later on each day, in setting off a fortnight's observations there will be about five days for each four times-of-clock of the upper transit. Hence in these figures we may regard each division of the time-scale I to II, II to III, &c., as representing twenty-five hours instead of one hour. Then the distance from the greatest ordinate of high-water heights to XII is called the age of the tide. From these two figures the times and heights of highand low-water may in general be predicted with fair approximation. We find the time-of-clock of the moon's upper or lower transit on the day, correct by the equation of time, read off the corresponding heights of highand low-water from the figures, and the intervals being also read off are added to the time of the moon's transit and give the times of highand low-water. At all ports there is, however, an irregularity of heights and intervals between successive tides, and in consequence of this the curves present more or less of a zigzag appearance. Where the zigzag is perceptible to the eye, the curves must be smoothed by drawing them so as to bisect the zigzags, because these diurnal inequalities will not present themselves similarly in the future. When, as in some equatorial ports, the diurnal tides are large, this method of tidal prediction fails in the simple form explained above. It may however be rendered applicable by greater elaboration.3 This method of working out observations of highand low-water was not the earliest. In the Mecanique Celeste, bks. i. and v., Laplace treats a large mass of tidal observations by dividing them into classes depending on the configurations of the tide-generating bodies. Thus he separates the two syzygial tides at full moon and change of moon and divides them into equinoctial and solstitial tides. He takes into consideration the tides of several days 1 Founded on Whewell's article " Tides," in Admiralty Manual (ed. 1841), and on Airy's " Tides and Waves," in Ency. Metrop. 2 For a numerical treatment, see Directions for Reducing Tidal Observations, by Commander Burdwood, R.N. (London, 1876).
3 G. H. Darwin " On Tidal Prediction," Phil. Trans. (1891), vol. 189 A.
embracing these configurations. He goes through the tides at quadratures on the same general plan. The effects of declination and parallax and the diurnal inequalities are similarly Methods of treated. Lubbock (Phil. Trans., 1831, seq.) improved the LLaplace, method of Laplace by taking into account all the obu bbock, served tides, and not merely those appertaining to certain Wh ewell, configurations. He divided the observations into a num ber of classes. First, the tides are separated into parcels, one for each month; then each parcel is sorted according to the hour of the moon's transit. Another classification is made according to declination; another according to parallax; and a last for the diurnal inequalities. This plan was followed in treating the tides of London, Brest, St Helena, Plymouth, Portsmouth and Sheerness. Whewell (Phil. Trans., 1834, seq.) did much to reduce Lubbock's results to a mathematical form, and made a highly important advance by the introduction of graphical methods by means of curves. The method explained above is due to him. Airy remarks of Whewell's papers that they appear to be " the best specimens of reduction of new observations that we have ever seen." VI. - Tidal Deformation Of The Solid Earth § 34. Elastic Tides. - The tide-generating potential varies as the square of the distance from the earth's centre, and the corresponding forces act at every point throughout its mass. No Elastic matter possesses the property of absolute rigidity, Tides. and we must therefore admit the probable existence of tidal elastic deformation of the solid earth. The problem of finding the state of strain of an elastic sphere under given stresses was first solved by G. Lame; 4 he made, however, but few physical deductions from his solution. An independent solution was found by Lord Kelvin, 5 who drew some interesting conclusions concerning the earth.
His problem, in as far as it is now material, is as follows. Let a sphere of radius a and density w be made of elastic material whose bulk and rigidity moduli are k and n, and let it be subjected to forces due to a potential per unit volume, equal to Twr 2 (sin 2 X - 3), where X is latitude. Then it is required to find the strain of the sphere. We refer the reader to the original sources for the methods of solution applicable to spherical shells and to solid spheres. The investigation applies either to tidal or to rotational stresses. In the case of tides T = gm/c 3 , m and c being the moon's mass and distance, and in the case of rotation - 10) 2, being the angular velocity about the polar axis. The equation to the surface is found to be (2 r =a i + I i?na I + '16r 4?n /k ] T (sin 2 X - 2) .
In most solids the bulk modulus is considerably larger than the rididity modulus, and in this discussion it is sufficient to neglect n compared with k. With this approximation, the ellipticity e of the surface becomes 5wa2 = e - 19n Now suppose the sphere to be endued with the power of gravitation, and write 19n 2s r= 5 w a2, g= 5a' where g is gravity at the surface of the globe. Then, if there were no elasticity, the ellipticity would be given by e = r/g, and without gravitation by e = r/r. And it may be proved in several ways that, gravity and elasticity co-operating, e - r +g g' i+r/g.
If n be the rigidity of steel, and if the globe have the size and mean density of the earth, r/g =2, and with the rigidity of glass r/g = 3. Hence the ellipticity of an earth of steel under tide-generating force would be a of that of a fluid earth, and the fraction for glass would be 6. If an ocean be superposed on the globe, the visible tide will be the excess of the fluid tide above the solid tide. Hence for steel the oceanic tides would be reduced to 1, and for glass to of the tides on a rigid earth.
It is not possible in general to compute the tides of an ocean lying on an unyielding nucleus. But Laplace argued that friction would cause the tides of long period (§ 17) to conform to the equilibrium law, and thus be amenable to calculation. Acting on this belief, G. H. Darwin discussed the tides of long period as observed during 33 years at various ports, and found them to be as great as on an unyielding globe, indicating an elasticity equal to that of steel.° Subsequently W. Schweydar repeated the calculation from 194 years of observation with nearly the same result.' But as Laplace's argument appears to be unsound (§ 17), the conclusion seems to become of doubtful validity. Yet subsequently Lord Rayleigh showed 4 Theorie math. de l'elasticite (1866), p. 213.
5 Thomson and Tait, Nat. Phil. §§ 73 2 -737 and 833-842, or Phil. Trans. (1863), pt. ii., p. 583. Compare, however, J. H. Jeans, Phil. Trans. (1903), 201 A, p. 157.
° Thomson & Tait, Nat. Phil. § 843.
7 Beitreige zur Geophysik (3907) ix. 41.
that the existence in the ocean of continental barriers would have the same effect as that attributed by Laplace to friction, and thus he re-established the soundness of the result.' A wholly independent estimate derived from what is called the variation of latitude also leads to the same conclusion, namely that the earth is about as stiff as steel.' The theory of the tides of an elastic planet gives, mutatis mutandis, that of the tides of a viscous spheroid. The reader who desires to know more of this subject and to obtain references to original memoirs may refer to G. H. Darwin's Tides. VII. - Tidal Friction § 35. Investigation of the Secular Effects of Tidal Friction. - We have indicated in general terns in § 9 that the theory of tidal friction leads to an interesting speculation as to the origin of the moon. We shall therefore investigate the theory mathematically in the case where a planet is attended by a single satellite moving in a circular orbit, and rotates about an axis perpendicular to that orbit. In order, however, to abridge the investigation we shall only consider the case where the planetary rotation is more rapid than the satellite's orbital motion.
Suppose an attractive particle or satellite of mass m to be moving in a circular orbit, with an angular velocity co, round a planet of mass M and suppose the planet to be rotating about an axis perpendicular to the plane of the orbit, with an angular velocity n; suppose, also, the mass of the planet to be partially or wholly imperfectly elastic or viscous, or that there are oceans on the surface of the planet; then the attraction of the satellite must produce a relative motion in the parts of the planet, and that motion must be subject to friction, or, in other words, there must be frictional Energy tides of some sort or other. The system must accordingly Diminished be losing energy by friction, and its configuration must Di inishe change in such a way that its whole energy diminishes. by Such a system does not differ much from those of actual planets and satellites, and, therefore, the results deduced in this hypothetical case must agree pretty closely with the actual course of evolution, provided that time enough has been and will be given for such changes. Let C be the moment of inertia of the planet about its axis of rotation, r the distance of the satellite from the centre of the planet, h the resultant moment of momentum of the whole system, e the whole energy, both kinetic and potential, of the system. It is assumed that the figure of the planet and the distribution of its internal density are such that the attraction of the satellite causes no couple about any axis perpendicular to that of rotation. A special system of units of mass, length and time will now be adopted such that the analytical results may be reduced to their simplest forms. Let the unit of mass be Mm/(M+m). Let the unit of length -y be such a distance that the moment of inertia of the planet about its axis of rotation may be equal to the moment of inertia of the planet and satellite, treated as particles, about their centre of inertia, when distant y apart from one another. This condition gives M(my My +m) -?-m M+m =C; Let the unit of time T be the time in which the satellite revolves through 57.3° about the planet, when the satellite's radius vector is equal to y. This system of units will be found to make the three following functions each equal to unity, viz.
µiMm(M+m)-, µMm, and C, where is the attrac tional constant. The units are in fact derived from the consideration that these functions shall each be unity. In the case of the earth and moon, if we take the moon's mass as a - of the earth's and the earth's moment of inertia as sMa 2 (as is very nearly the case), it may easily be shown that the unit of mass is s i rs of the earth's mass, the unit of length 5.26 earth's radii or 33,506 kilometres (20,807 miles), and the unit of time 2 hrs. 41 mins.
In these units the present angular velocity of the Moment of earth's diurnal rotation is expressed by 0.7044, and the Momentum. moon's present radius vector by 11.454. The two bodies being supposed to revolve in circles about their common centre of inertia with an angular velocity w, the moment of momentum of orbital motion is (m r r Mm r2 w. w M M +m) - M r Then, by the law of periodic times in a circular orbit, Or' = (M +m); whence wr2 = µ 2 (M + m) Zr=. Thus the moment of momentum of orbital motion =µ"Mm(M+m)-?r2, and in the special units this is equal to r i. The moment of momentum of the planet's rotation is Cn, and C= i in the special units. Therefore h = n + r 2 . (62) Since the moon's present radius vector is 11.454, it follows that the orbital momentum of the moon is 3.384. Adding to this the ' Phil. Mag. (1903), V. 136.
2 Hough, Phil. Trans. (1897), 187 A, p. 319.
rotational momentum of the earth, which is 0.704, we obtain 4.088 for the total moment of momentum of the moon and earth. The ratio of the orbital to the rotational momentum is 4.80, so that the total moment of momentum of the system would, but for the obliquity of the ecliptic, be 5.80 times that of the earths rotation. Again, the kinetic energy of orbital motion is 1 m r l r 1 2 2 _ 1 Mm 2 2_ 1 p Mm M M +m/ W -I- M +m l co - 'M+m - 2 The kinetic energy the planet's rotation is z Cn 2. The potential energy of the system is -µMm/r. Adding the three energies together, and transforming into the special units, we have 2e = n 2 - 1/r. (63) Now let x = r 2, = n, Y = 2e.
It will be noticed that x, the moment of momentum of orbital motion is equal to the square root of the satellite's distance from the planet. Then equations (62) and (63) become h = y + x (64) Y=y2 - I /x2= (h- - i/x2 (65) (64) is the equation of conservation of moment of momentum, or, shortly, the equation of momentum; (65) is the equation of energy. Now consider a system started with given positive moment of momentum h; and we have all sorts of ways in which it may be started. If the two rotations be of opposite maximum kinds, it is clear that we may start the system with and any amount of energy, however great, but the true Energy. maxima and minima of energy compatible with the given moment of momentum are supplied by d Y/dx = o, or x - h+ i/x3=o, that is to say, x4 - hx 3 = o. (66) We shall presently see that this quartic has either two real roots and two imaginary, or all imaginary roots. The quartic may be derived from quite a different considera- No Relative tion, viz. by finding the condition under which the Motion satellite may move round the planet so that the planet between shall always show the same face to the satellite - in fact, Satellite and so that they move as parts of one rigid body. The Planet when condition is simply that the satellite's orbital angular velocity w = n, the planet's angular velocity of rotation, Minimum. or y = 1 /x 3, since n =y and r' = w = x. By substituting this value of y in the equation of momentum (64), we get as before x 4 -hx 3 +i = 0.
At present we have only obtained one result, viz. that, if with given moment of momentum it is possible to set the satellite and planet moving as a rigid body, it is possible to do so in two ways, and one of these ways requires a maximum amount of energy and the other a minimum; from this it is clear that one must be a rapid rotation with the satellite near the planet and the other a slow one with the satellite remote from the planet. Of the three equations h=y+x, (67) Y= (h-x)'-I/x2, (68) x3y 1, (69) (67) is the equation of momentum, (68) that of energy, and (69) may be called the equation of rigidity, since it indicates that the two bodies move as though parts of one rigid body.
To illustrate these equations geometrically, we may Equations of take as abscissa x, which is the moment of momentum Momentum, of orbital motion, so that the axis of x may be called y, and the axis of orbital momentum. Also, for equations (67) n o tion r e and (69) we may take as ordinate y, which is the moment n. of momentum of the planet's rotation, so that the axis of y may be called the axis of rotational momentum. For (68) we may take as ordinate Y, which is twice the energy of the system, so that the axis of Y may be called the axis of energy. Then, as it will be convenient to exhibit all three curves in the same figure, with a parallel axis of x, we must have the axis of energy identical with that of rotational momentum. It will not be necessary to consider the case where the resultant moment of momentum h is negative, because this would only be equivalent to reversing all the rotations; h is therefore to be taken as essentially positive. The line of momentum whose equation is (67) is a straight line inclined at 45° to either axis, having positive intercepts on both axes. The curve of rigidity whose equation is (69) is clearly of the same nature as a rectangular hyperbola, but it has a much more rapid rate of approach to the axis of orbital momentum than to that of rotational momentum. The intersections (if any) of the curve of rigidity with the line of momentum have abscissae which are the two roots of the quartic 4 -hx 3 +i = o. The quartic has, therefore, two real roots or all imaginary roots. Then, since x = r', the intersection which is more remote from the origin indicates a configuration where the satellite is remote from the planet; the other gives the configuration where the satellite is closer to the planet. We have already learnt that these two correspond respectively to minimum and maximum energy. When x is very large the equation to the curve of energy is Y= (h-x) 2 , which is the equation to a parabola with a vertical axis parallel to Y and distant h from the origin, so that the axis of the parabola passes through the intersection of the line of momentum Special Units. with the axis of orbital momentum. When x is very small, the equation becomes Y = - I /x 2. Hence the axis of Y is asymptotic on both sides to the curve of energy. If the line of momentum intersects the curve of rigidity, the curve of energy has a maximum vertically underneath the point of intersection nearer the origin and a minimum underneath the point more remote. But if there are no intersections, it has no maximum or minimum.
Fig. 8 shows these curves when drawn to scale for the case of the earth and moon, that is to say, with h=4. The points a and b, which are the maximum and minimum of the curve of energy, are supposed to be on the same ordinates as A and B, the intersections of the curve of rigidity with the line of momentum. The intersection of the line of momentum with the axis of orbital momentum is FIG. 8.
denoted by D, but in a figure of this size it necessarily remains indistinguishable from B. As the zero of energy is quite arbitrary the origin for the energy curve is displaced downwards, and this prevents the two curves from crossing one another in a confusing manner. On account of the limitation imposed we neglect the case where the quartic has no real roots. Every point of the line of momentum gives by its abscissa and ordinate the square root of the satellite's distance and the rotation of the planet, and the ordinate of the energy curve gives the energy corresponding to each distance of the satellite. Part of the figure has no physical meaning, for it is impossible for the satellite to move round the planet at a distance less than the sum of the radii of the planet and satellite. For example, the moon's diameter being about 2200 m. and the earth's about 8000, the moon's distance cannot be less than 5100 miles. Accordingly a strip is marked off and shaded on each side of the vertical axis within which the figure has no physical meaning. The point P indicates the present configuration of the earth and moon. The curve of rigidity 3 y = I is the same for all values of h, and by moving the line of momentum parallel to itself Least Mo- nearer to or further from the origin, we may represent all possible moments of momentum of the whole system. The smallest amount of moment of momentum with for which which it is possible to set the system moving as a rigid Motion body, with centrifugal force enough to balance the. mutual attraction, is when the line of momentum touches Possible the curve of rigidity. The condition for this is clearly that the equation x 4 - hx 3 + I =0 should have equal roots. If it has equal roots, each root must be 4h, and therefore (lh) 3 - h(lh) 3 -1-I =o, whence h4= 44/33, or h= 4/3 I =1.75. The actual value of h for the moon and earth is about 4; hence, if the moon-earth system were Maximum started with less than i'z of its actual moment of momen- Maxi of tum, it would not be possible for the two bodies to move so that the earth should always show the same Days . face to the moon. Again, if we travel along the line Month of momentum, there must be some point for which yx 3 is a maximum, and since yx 3 =n/w there must be some point for which the number of planetary rotations is greatest during one revolution of the satellite; or, shortly, there must be some configuration for which there is a maximum number of days in the month. Now yx 3 is equal to x 3 (h - x), and this is a maximum when x = fh and the maximum number of days in the month is (lh) 3 (h-1h) or 33h4/44; if h is equal to 4, as is nearly the case for the earth and moon, this becomes 27. Hence it follows that we now have very nearly the maximum number of days in the month. A more accurate investigation in a paper on the " Precession of a Viscous Spheroid " in Phil. Trans. (1879), pt. i., showed that, taking account of solar tidal friction and of the obliquity to the ecliptic, the maximum number of days is about 29, and that we have already passed through the phase of maximum.
We will now consider the physical meaning of the figure. It is assumed that the resultant moment of momentum of the whole system corresponds to a positive rotation. Now Figure. imagine two points with the same abscissa, one on the of momentum line and the other on the energy curve, and suppose the one on the energy curve to guide that on the momentum line. Since we are supposing frictional tides to be raised on the planet, the energy must degrade, and however the two points are set initially the point on the energy curve must always slide down a slope, carrying with it the other point. Looking at the figure, we see that there are four slopes in the energy curve, two running down to the planet and two down to the minimum. There are therefore four ways in which the system may degrade, according to the way it was started; but we shall only consider one, that corresponding to the portion ABba of the figure. For the part of the line of momentum AB the month is History of longer than the day, and this is the case with all known Sa tellite satellites except the nearer one of Mars. Now, if a satellite as Energy be placed in the conditionA - that is to say, moving rapidly round a planet which always shows the same face to the satellite - the condition is clearly dynamically unstable, for the least disturbance will determine whether the system shall degrade down the slopes ac or ab - that is to say, whether it falls into or recedes frc,m the planet. If the equilibrium breaks down by the satellite receding, the recession will go on until the system has reached the state corresponding to B. It is clear that, if the intersection of the edge of the shaded strip with the line of momentum be identical with the point A, which indicates that the satellite is just touching the planet, then the two bodies are in effect parts of a single body in an unstable configuration. If, therefore, the moon was originally part of the earth, we should expect to find this identity. Now in fig. 9, drawn to scale to represent the earth and moon, there is so close an approach between the edge of the shaded band and the intersection of the line of momentum and curve of rigidity that it would be scarcely possible to distinguish them. Hence, there seems a probability that the two bodies once formed parts of a single one, which broke up in consequence of some kind of instability. This view is confirmed by the more detailed consideration of the case in the paper on the " Precession of a Viscous Spheroid," already referred to, and subsequent papers, in the Phil. Trans.' § 36. Effects of Tidal Friction on the Elements of the Moon's Orbit and on the Earth's Rotation. - It would be impossible within the limits of the present article to discuss completely the effects of tidal friction; we therefore confine ourselves to certain general considerations which throw light on the nature of those effects. We have in the preceding section supposed that the planet's axis is perpendicular to the orbit of the satellite, and that the latter is circular; we shall now suppose the orbit to be oblique to the equator and eccentric. For the sake of brevity the planet will be called the earth, and the satellite the moon. The complete investigation was carried out on the hypothesis that the planet was a viscous spheroid, because this was the only theory of frictionally resisted tides which had been worked out. Although the results would be practically the same for any system of frictionally resisted tides, we shall speak below of the planet or earth as a viscous body.' We shall show that if the tidal retardation be small the obliquity of the ecliptic increases, the earth's rotation is retarded, and the moon's distance and peri- Obliquity of odic time are increased. the Ecliptic Fig. 9 represents the earth as seen from above the south pole, so that S is the pole and the outer circle the equator. The earth's rotation is in the direction of the curved arrow at S. The half of the inner circle which is drawn with a full line is a semi-small-circle of south latitude, and the dotted semicircle is a semi-small-circle in the same north latitude. Generally FIG. 9.
dotted lines indicate parts of the figure which are below the plane of the paper. If the moon were cut in two and one half retained at the place of the moon and the other half transported to a point diametrically opposite to the first half with reference to the earth, there would be no material change in the tide-generating forces. It is easy to verify this statement by reference to § II. These two halves may be described as moon and anti-moon, and such a substitution will facilitate the explanation. Let M and M' be the projections of the moon and anti-moon on to the terrestrial sphere. If the fluid in which the tides are raised were perfectly frictionless,' or if the earth were a perfect fluid or perfectly elastic, the apices of the tidal spheroid would be at M and M'. If, however, there is internal friction, due to any sort of viscosity, the tides will lag, and we may suppose the tidal apices to be at T and T'. Now suppose the tidal protuberances to be replaced by two equal heavy particles at T and T', which are instantaneously rigidly connected with the earth. Then the attraction of the moon on T is greater ' For further consideration of this subject see a series of papers by G. H. Darwin in Proceed. and Trans. of the Royal Society from 1878 to 1881, and app. G. (b) t. pt. ii. vol. i. of Thomson and Tait's Nat. Phil. (1883); or Scientific Papers, vol. ii.
2 These explanations, together with other remarks, are to be found in the abstracts of G. H. Darwin's memoirs in Proc. Roy. Soc., 1878 to 1881.
We here suppose the tides not to be inverted. If they are inverted the conclusion is precisely the same.
than on T', and that of the anti-moon on T' is greater than on T. The resultant of these forces is clearly a pair of forces acting on the earth in the direction TM, T'M'. These forces cause a couple about the axis in the equator, which lies in the same meridian as the moon and anti-moon. The direction of the couple is shown by the curved arrows at L,L'. If the effects of this couple be compounded with the existing rotation of the earth according to the principle of the gyroscope, the south pole S will tend to approach M and the north pole to approach M'. Hence, supposing the moon to move in the ecliptic, the inclination of the earth's axis to the ecliptic diminishes, or the obliquity increases. Next the forces TM, T'M' clearly produce, as in the simpler case considered in § 9, a couple about the earth's polar axis, which tends to retard the diurnal rotation.
This general explanation remains a fair representation of the state of the case so long as the different harmonic constituents of the aggregate tide-wave do not suffer very different amounts of retardation; and this is the case so long as the viscosity is not great. The rigorous result for a viscous planet shows that in general the obliquity will increase, and it appears that, with small viscosity of the planet, if the period of the satellite be longer than two periods of rotation of the planet, the obliquity increases, and vice versa. Hence, zero obliquity is only dynamically stable when the period of the satellite is less than two periods of the planet's rotation.
It is possible, by similar considerations, to obtain some insight into the effect which tidal friction must have on the plane of the lunar orbit, but as the subject is somewhat complex Inclination we shall not proceed to a detailed examination of the of Plane question. It must suffice to say that in general the inclin of Orbit ation of the lunar orbit must diminish. Now let us con- Genera/ly sider a satellite revolving about a planet in an elliptic Decreases. orbit, with a periodic time which is long compared with the period of rotation of the planet; and suppose that frictional tides are raised in the planet. The major axis of the tidal spheroid Eccentricity always points in advance of the satellite, and exercises of en r s y on it a force which tends to accelerate its linear velocity the satellite is in perigee the tides are higher, and Generally . th i s d i sturbing force is greater than when the satellite is in apogee. The disturbing force may therefore be represented as a constant force, always tending to accelerate the motion of the satellite, to which is added a periodic force accelerating in perigee and retarding in apogee. The constant force causes a secular increase of the satellite's mean distance and a retardation of its mean motion. The accelerating force in perigee causes the satellite to swing out farther than it would otherwise have done, so that when it comes round to apogee it is more remote from the planet. The retarding force in apogee acts exactly inversely, and diminishes the perigean distance. Thus, the apogean distance increases and the perigean distance diminishes, or in other words, the eccentricity of the orbit increases. Now consider another case, and suppose the satellite's periodic time to be identical with that of the planet's rotation. Then, when the satellite is in perigee, its angular motion is faster than that of the planet's rotation, and when in apogee it is slower; hence at apogee the tides lag, and at perigee they are accelerated. Now the lagging apogean tides give rise to an But it M accelerating force on the satellite, and increase the peri ButitMa Decrease. gean distance, whilst the accelerated perigean tides give rise to a retarding force, and decrease the apogean distance. Hence in this case the eccentricity of the orbit will diminish. It follows from these two results that there must be some intermediate periodic time of the satellite for which the eccentricity does not tend to vary.
But the preceding general explanation is in reality somewhat less satisfactory than it seems, because it does not make clear the existence of certain antagonistic influences, to which, however, we shall not refer. The full investigation for a viscous planet shows that in general the eccentricity of the orbit will increase. When the viscosity is small the law of variation of eccentricity is very simple: if eleven periods of the satellite occupy a longer time than eighteen rotations of the planet, the eccentricity increases, and vice versa. Hence in the case of small viscosity a circular orbit is only dynamically stable if the eleven periods are shorter than the eighteen rotations.
Viii. - Cosmogonic Speculations Founded On Tidal Friction § 37. History of the Earth and Moon. - We shall not attempt to discuss the mathematical methods by which the complete history of a planet, attended by one or more satellites, is to be traced. The laws indicated in the preceding sections show that there is such a problem, and that it may be solved, and we refer to G. H. Darwin's papers for details (Phil. Trans., 1879-1881). It may be interesting, however, to give the various results of the investigation in the form of a sketch of the possible evolution of the earth and moon, followed by remarks on the other planetary systems and on the solar system as a whole.
We begin with a planet not very much more than 8000 m. in diameter, and probably partly solid, partly fluid, and partly gaseous. It is rotating about an axis inclined at about II° or 12° to the normal to the ecliptic, with a period of from two to four hours, and is revolving about the sun with a period not much shorter than our present year. The rapidity of the planet's rotation causes so great a compression of its figure that it cannot continue to exist in an ellipsoidal form with stability; or else it is so nearly unstable that complete instability is induced by the solar tides. Conjectural The planet then separates into two masses, the larger being the earth and the smaller the moon. It is Moon from not attempted to define the mode of separation, or Earth. to say whether the moon was initially a chain of meteorites. At any rate it must be assumed that the smaller mass became more or less conglomerated and finally fused into a spheroid, perhaps in consequence of impacts between its constituent meteorites, which were once part of the primeval planet. Up to this point the history is largely speculative, for the investigation of the conditions of instability in such a case surpasses the powers of the mathematician. We have now the earth and moon nearly in contact with one another, and rotating nearly as though they were parts of one rigid body. This is the system which was the subject of dynamical investigation. As Moon Sub- the two masses are not rigid, the attraction of each /ect of In- distorts the other; and, if they do not move rigorously with the same periodic time, each raises a tide in the other. Also the sun raises tides in both. In consequence of the frictional resistance to these tidal motions, such a system is dynamically unstable. If the moon had moved orbitally a little faster than the earth rotated, she must have fallen back into the earth; thus the existence of the moon compels us to believe that the equilibrium broke down by the moon revolving orbitally a little slower than the earth rotates. In consequence of the tidal friction the periodic times both of the moon (or the month) and of the earth's rotation (or the day) increase; but the month increases in length at a much greater rate than the day. At some early stage in the history of the system the moon was conglomerated into a spheroidal form, and acquired a rotation about an axis nearly parallel to that of the earth.
The axial rotation of the moon is retarded by the attraction of the earth on the tides raised in the moon, and this retardation takes place at a far greater rate than the similar retardation The Moon. of the earth's rotation. As soon as the moon rotates round her axis with twice the angular velocity with which she revolves in her orbit, the position of her axis of rotation (parallel with the earth's axis) becomes dynamically unstable. The obliquity of the lunar equator to the plane of the orbit increases, attains a maximum, and then diminishes. Meanwhile the lunar axial. rotation is being reduced towards identity with the orbital motion. Finally, her equator is nearly coincident with the plane of the orbit, and the attraction of the earth on a tide, which degenerates into a permanent ellipticity of the lunar equator, causes her always to show the same face to the earth.
All this must have taken place early in the history of the earth, to which we now return. At first the month is identical with the day, and as both these increase in length the lunar orbit The Earth will retain its circular form until the month is equal to I i'i days. From that time the orbit begins to be eccen- and Lunar tric, and the eccentricity increases thereafter up to its present magnitude. The plane of the lunar orbit is at first practically identical with the earth's equator, but as the moon recedes from the earth the sun's attraction begins to make itself felt. We shall not attempt to trace the complex changes by which the plane of the lunar orbit is affected. It must suffice to say that the present small inclination of the lunar orbit to the ecliptic accords with the theory.
As soon as the earth rotates with twice the angular velocity with which the moon revolves in her orbit, a new instability sets in. The month is then about twelve of our present hours, and the day about six such hours in length. The inclination of the equator to the ecliptic now begins to increase and continues to do so until finally it reached its present value of 231°. All these changes continue and no new phase now supervenes, and at length we have the system in its present configuration. The minimum time in which the changes from first to last can have taken place is 54,000,000 years.
There are other collateral results which must arise from a supposed primitive viscosity or plasticity of the earth's mass. For during this course of evolution the earth's mass must have Distortion suffered a screwing motion, so that the polar regions of Fusion have travelled a little from west to east relatively to the Plane t. equator. This affords a possible explanation of the north and south trend of our great continents. The whole of this argument reposes on the imperfect rigidity of solids and on the internal friction of semi-solids and fluids; these are ver g e causae. Thus changes of the kind here discussed must be going The Theory on, and must have gone on in the past. And for this history of the earth and moon to be true throughout, Sufficient it is only necessary to postulate a sufficient lapse of time, Lapse of and that there is not enough matter diffused through Time. space materially to resist the motions of the moon and earth in perhaps 200,000,000 years. It seems hardly too much to say that, granting these two postulates, and the existence of a See criticism, by Nolan, Genesis of Moon (Melbourne, 1885); also Nature (Feb. 18, 1886).
primeval planet such as that above described, a system would necessarily be developed which would bear a strong resemblance to our own. A theory, reposing on ver g e causae which brings into quantitative correlation the lengths of the present day and month, the obliquity of the ecliptic, and the inclination and eccentricity of the lunar orbit should have claims to acceptance.
§ 38. The Influence of Tidal Friction on the Evolution of the Solar System and of the Planetary Sub-systems. 1 - According to the nebular hypothesis of Kant and Laplace the planets and satellites are portions detached from contracting nebulous masses, and other theories have been advanced subsequently in explanation of the present configuration of the solar system. We shall here only examine what changes are called for by the present theory of tidal friction. It may be shown that the reaction of the tides raised in the sun by the planets 'must have had a very small influence in changing the dimensions of the planetary orbits round the sun, and it appears improbable that the planetary orbits have been sensibly enlarged by tidal friction since the origin of the several planets.
Similarly it appears unlikely that the satellites of Mars, Jupiter and Saturn originated very much nearer the present surfaces of the planets that we now observe them. But, the data being Planetary insufficient, we cannot feel sure that the alteration Sub-systems. in the dimensions of the orbits of these satellites has not been considerable. It remains, however, nearly certain that they cannot have first originated almost in contact with the present surfaces of the planets, in the same way as in the preceding sketch has been shown to be probable with regard to the moon and earth. Numerical data concerning the distribution of moment of momentum in the several planetary sub-systems exhibit so striking a difference between the terrestrial system and those of the other planets that we should from this alone have grounds for believing that the modes of evolution have been considerably different. The difference appears to lie in the genesis of the moon close to the present surface of the planet, and we shall see below that solar tidal friction may be assigned as a reason to explain how it has happened that the terrestrial planet had contracted to nearly its present dimensions before the genesis of a satellite, but that this was not the case with the exterior planets. The efficiency of solar tidal friction is very much greater in its action on the nearer planets than on the farther ones. The time, however, during which solar tidal friction has been operating on the external planets is probably much longer than the period of its efficiency for the interior ones, and a series of numbers proportional to the total amount of rotation destroyed in the several planets would present a far less rapid decrease as we recede from the sun than numbers simply expressive of the efficiency of tidal friction at the several planets. Nevertheless it must be admitted that the effect produced by solar tidal friction on Jupiter and Saturn has not been nearly so great as on the interior planets. And, as already stated, it is very improbable that so large an amount of momentum should have been destroyed as materially to affect the orbits of the planets round the sun.
We will now examine how the difference of distances from the sun may have affected the histories of the several planets. According to the nebula hypothesis, as a planetary nebula contracts, Distribution the increasing rapidity of the rotation causes it to of Satellites become unstable, and an equatorial portion of matter Amongst detaches itself. The separation of that part of the mass the Planets. which before the change had the greatest angular momentum permits the central portion to resume a planetary shape. The contraction and the increase of rotation proceed continually until another portion is detached, and so on. There thus recur at intervals epochs of instability, and something of the same kind must have occurred according to other rival theories. Now tidal friction must diminish the rate of increase of rotation due to contraction, and therefore if tidal friction and contraction are at work together the epochs of instability must recur more rarely than if contraction alone acted. If the tidal retardation is sufficiently great, the increase of rotation due to contraction will be so far counteracted as never to permit an epoch of instability to occur. Since the rate of retardation due to solar tidal friction decreases rapidly as we recede from the sun, these considerations accord with what we observe in the solar system. For Mercury and Venus have no satellites, and there is progressive increase in the number of satellites as we recede from the sun. Whether this be the true cause of the observed distribution of satellites amongst the planets or not, it is remarkable that the same cause also affords an explanation, as we shall now show, of that difference between the earth with the moon and the other planets with their satellites which has caused tidal friction to be the principal agent of change with the former, but not with the latter. In the case of the contracting terrestrial mass we may Case of suppose that there was for a long time nearly a balance Earth and between the retardation due to solar tidal friction and Earth ffer-- the acceleration due to contraction, and that it was not Mo until the planetary mass had contracted to nearly its entfrom . present dimensions that an epoch of instability could others occur. It may also be noted that if there be two equal planetary masses which generate satellites, but under very different conditions as to the degree of condensation of the masses, the 1 A review of this and of cognate subjects is contained in G. H. Darwin's presidential address to the Brit. Assoc. in 1905.
XXVI. 31 two satellites will be likely to differ in mass; we cannot, of course, tell which of the two planets would generate the larger satellite. Thus, if the genesis of the moon was deferred until a late epoch in the history of the terrestrial mass, the mass of the moon relatively to the earth would be likely to differ from the mass of other satellites relatively to their planets. If the contraction of the planetary mass be almost completed before the genesis of the satellite, tidal friction will thereafter be the great cause of change in the system; and thus the hypothesis that it is the sole cause of change will give an approximately accurate explanation of the motion of the planet and satellite at any subsequent time. We have already seen that the theory that tidal friction has been the ruling power in the evolution of the earth and moon co-ordinates the present motions of the two bodies and carries us back to an initial state when the moon first had a separate existence as a satellite; and the initial configuration of the two bodies is such that we are led to believe that the moon is a portion of the primitive earth detached by rapid rotation or by other causes.
Let us now turn to the other planetary sub-systems. The satellites of the larger planets revolve with short periodic times; for the smallness of their masses would have prevented tidal friction from being a very efficient cause of change in the dimensions of their orbits, and the largeness of the planet's masses would have caused them to proceed slowly in their evolution. The satellites of Mars present one of the most remarkable features in the solar system, for, whereas Mars rotates in 24h. 37m., Deimos has a period of Soh. 18m. and Phobos of only 7h. 39m. The minuteness of these satellites precludes us from supposing that they have had much influence on the rotation of the planet, or that the dimensions of their own orbits have been much changed.
The theory of tidal friction would explain the shortness of the periodic time of Phobos by the solar retardation of the planet's rotation, which would operate without directly affecting S atellites the satellites' orbital motion. We may see that, given o f Mars. sufficient time, this must be the ultimate fate of all satellites. Numerical comparison shows that the efficiency of solar tidal friction in retarding the terrestrial and martian rotations is of about the same degree of importance, notwithstanding the much greater distance of the planet Mars. In the above discussion it will have been apparent that the earth and moon do actually differ from the other planets to such an extent as to permit tidal friction to have been the most important factor in their history.
By an examination of the probable effects of solar tidal friction on a contracting planetary mass, we have been led to assign a cause for the observed distribution of satellites in the solar Summary. system, and this again has itself afforded an explanation of how it happened that the moon so originated that the tidal friction of the lunar tides in the earth should have been able to exercise so large an influence. We have endeavoured not only to set forth the influence which tidal friction may have, and probably has had in the history of the system, if sufficient time be granted, but also to point out what effects it cannot have produced. These investigations afford no grounds for the rejection of theories more or less akin to the nebular hypothesis; but they introduce modifications of considerable importance. Tidal friction is a cause of change of which Laplace's theory took no account; and, although the activity of that cause may be regarded as mainly belonging to a later period than the events described in the nebular hypothesis, yet it seems that its influence has been of great, and in one instance of even paramount, importance in determining the present condition of the planets and their satellites. Throughout the whole of this discussion it has been supposed that sufficient time is at our disposal. Yet arguments have been adduced which Limitation seemed to show that this supposition is not justifiable, of Time. for Helmholtz, Lord Kelvin and others have attempted to prove that the history of the solar system must be comprised within a period considerably less than a hundred million years.' But the discovery of radio-activity and the consequent remarkable advances in physics throw grave doubt on all such arguments, and we believe that it is still beyond our powers to assign definite numerical limits to the age of the solar system.
Dr T. J. J. See (Researches on the Evolution of Stellar Systems; vol. ii. (1910) Capture Theory) rejects the applicability of tidal friction to the cosmogony of the solar system, and argues that the satellites were primitively wandering bodies and were captured by the gravitational attraction of the planets. Such captures are considered by Dr See to be a necessary result of the presence in space of a resisting medium; but the present writer does not feel convinced by the arguments adduced. (G. H. D.)
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Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Tide'. 1911 Encyclopedia Britanica. https://www.studylight.org/​encyclopedias/​eng/​bri/​t/tide.html. 1910.