Encyclopædia Britannica, Ninth Edition/Tides/Chapter 2

II. Tide-generating Forces.

§ 5. Investigation of Tide-generating Potential and Forces.

Tide-generating 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 satellite, the motion of any forces. 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 accelera tion 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 the 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* 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+m. 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^, M 2 r, M 3 r- be the coordinates of the satellite referred thereto; and let |p, rip, fy be the coordinates of the particle P whose radius is p. Then the component accelerations for reducing the planet's centre to rest are - mMj/r 2, - mM 2 /r 2, - m M 3 /r 2; and since these are the differential coefficients with respect to p, py, pf of the function

"ip

and since cos z = Mj -t- M 2 rj + M 3 f; it follows that the potential of the forces by which the planet's centre is to be reduced to rest is

mp - fcos z. 1*

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 of its body, then, the planet contributes forces of which the poten tial 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

V {r* + P* ~ 2

To determine the forces from this potential we regard p and z as the variables for differentiation, ana 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

• J {r 2 + p* - 2rp cos z} r

It is obvious that r is very large compared with p, and we may therefore expand this in powers of p/r. This expansion gives us

$vgt$

where P = cos z, P 2 = f cos2 z - i. PS = I cos 3 z - f cos z, &c. The reader familiar with spherical harmonic analysis of course recog nizes the Legendre's 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 . -P l -~ cos z, and is therefore equal and oppo site 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 coordinates are p|, py, p

Potential.-+ m (% cos * z -ft + n ri -(t cos 3 s-f cos2)-f ...... (1).

The first term of (1) is the potential of gravity, and the terms of the series, of which two only are written, constitute the tide-gener ating potential. In all practical applications this series converges so rapidly that the first term is amply sufficient, and thus we shall generally denote

as the tide-generating potential.

Moon and anti-moon.In many mathematical works the tide-generating force is presented as being due to an artificial statical system, which produces nearly the same force as the dynamical system considered above. This statical system is as follows. Stopping all the rotations, we divide the satellite into two equal parts, and place them diametrically opposite to one another in the orbit. Then it is clear that, instead of the term

we have

V {r* + p* - 2rp cos z} r

And this reduces to

The first term is the same as before; hence the statical system produces approximately the same tide-generating force as the true system. The "moon" and "anti-moon," however, produce rigor ously the same force on each side of the planet, whereas the true system only satisfies this condition approximately.^[1]

§ 6. Form of Equilibrium.

Let us consider the shape assumed by a layer of fluid of density a, lying on a globe of mass M, when acted on by disturbing forces whose potential is

3?n . 2 .. y fy3P ~( cos z ~ s) (*)

Suppose the layer to be very thin, and that the mean radius of the layer is a, and let the equation to the boundary of the fluid be

p=o[l + e(cos 2 s- )] (4).

We assume this form, because the theory of harmonic analysis tells us that the departure from sphericity must be represented by a function of the form cos 2 z - ^. That theory also gives us as the potential of a layer of matter of depth ta(cos 2 z- ), and density a, at an external point the value

EQUATION

Hence the whole potential, outside of and up to the fluid layer, is

a's - I E(COS 2 Z- i) (5).

The first term of (5) is the potential of the globe, the second that of the disturbing force, and the third the potential due to departure from sphericity.

Now the fluid must stand in a level surface; hence, if we equate this potential to a constant, we must get back to the equation (4), which was assumed to be that of the surface. In other words, if we put p = a[l + t(cos 2 z- J)] in (5), the result must be constant, provided the departure from sphericity is small In effecting the substitution for p, we may put p=a in the small terms, but in the first term of (5) we put

|"=^[l-r(cos 2 z-4)].

The whole potential (5) can only be constant if, after this substitution, the coefficient of cos 2 z - vanishes. Thus we must have AI Bma 2 1 + ^ o + *7r<ra-c = 0. a 2r 3 But if 5 be the mean density of the planet J/=fjra 3 5, and gravity g=3I/a~. Then we easily find that " t= * l (6] Sgr 3 1 - 4(7/5 Form of Thus the equation to the surface is equili- ( _ 3ma 1 If <r be small compared with 5, the coefficient is Sma/Zgr 3; thus we see that 1/(1 - f<r/5) is the coefficient by which the mutual attrac tion of the fluid augments the deformation of the fluid under the action of the disturbing force. If the density of the fluid be the same as that of the sphere, the augmenting factor becomes, and we have t ^-ma/gr 3, which gives the form of equilibrium of a fluid sphere under the action of these forces. Since H-3 = -( 1 fiT CL yd it follows that, when the form of equilibrium is p a[l + c(cos 2 z - )], the potential of the forces is %)vK*+-V (8).

More generally, if we neglect the attraction of the fluid on itself, so that ff/S is treated as small, and if p = a(l + j) be the equation to the surface of the fluid, where's is a function of latitude and longi tude, then the potential of the forces under which this is an equi librium form is

(9).

Tide-generating force specified by equilibrium form.It thus appears that we may specify any tide-generating forces by means of the figure of equilibrium which the fluid would assume under them, and in the theory of the tides it has been found practically convenient to specify the forces in this way.

By means of the principle of "forced vibrations" referred to in the historical sketch, we shall pass from the equilibrium form to the actual oscillations of the sea.

§ 7. Development of Tide-generating Potential in Terms of Hour-Angle and Declination.

Development of tide-generating potential.We now proceed to develop the tide-generating potential, and shall of course implicitly (§ 6) determine the equation to the equilibrium figure.

We have already seen that, if 2 be the moon's zenith distance at the point P on the earth's surface, whose coordinates referred to A, B, C, axes fixed in the earth, are a%, arj, af, then cos z = Mj + 7?M 2 + f M 3, where Mj, M s, M 3 are the moon's direction cosines referred to the same axes. Then with this value of cos z cos 2 2 - = S8 The axis of C is taken as the polar axis, and AB is the equatorial plane, so that the functions of, 77, f are functions of the latitude and longitude of the point P, at which we wish to find the potential The functions of Mj, M 2, JI 3 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 ( 23)jui 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 II be the moon, the inclination of the plane MC to the plane CA is the moon's easterly local hour-angle. Let h local hour-angle of moon and 5= moon's declination: we have M I = cos5cosA, M 2 =cos5sin&, M 3 =sin5, whence 2MjM 2 = cos 2 5 sin 2k, M^ - M 2 2 = cos 2 S cos 2k, = 2 sin 5 cos 5 sin k, 2MiM 3 =2 sin 5 cos 5 cos h, = - sin 2 5. O Also, if X be the latitude of P, = cos X, 17 = 0, f=sin, and Hence (10) becomes cos 2 z - 4 = cos 2 X cos 2 5 cos 2k + sin 2X sin 5 cos 5 cos k (11). The angle h, as defined at present, is the eastward local hour-angle, and therefore diminishes with the time. As, however, this function does not change sign with h, it will be more convenient to regard it as the westward local hour-angle. Also, if h be the Greenwich westward hour-angle at the moment under consideration! and I be the west longitude of the place of observation P, we have k = k Q -l (12). Hence we have at the point P, whose radius vector is a, V= -9-5— (k cos 2 X cos 2 5 cos 2(h -l) + sin 2X sin 5 cos 5 cos (k - I) +f (4-sin 2 5)(&-sin 2 X)} (13). Potential The tide-generating forces are found by the rates of variation of pdevelfor latitude and longitude, and also for radius a, if we care to find UP**! in the radial disturbing force. hourangle and

§ 8. Evaluation of Tide-generating Forces, and Lunar Deflexion of Gravity.

The westward component of the tide-generating force at the earth's surface, where p=a, is dVja cos dl, and the northward component is dVjadX; the change of apparent level is the ratio of these to gravity g. Therefore, differentiating (13), changing signs, and Zm/a 3, 3ma, writing ^-jJ - ) lor ^3, we have component change of level southward j, = 7T/-") i sin 2X cos2 5 cos 2 (^o - ^ - 2 cos 2X sin 25 cos (h - Z) Lunar

• V/ 4- sin 5>Xn - a sin 2 XV, + sin2X(l-3sin 2 5)}; component change of level westward deflexion ofgravity. _ - {cos X cos 2 5 sin 2(k - 1) + sinXsin25sin(A -Z)} (14).

The westward component is made up of two periodic terms, one going through its variations twice and the other once a day. The southward component has also two similar terms; but it has a third term, which does not oscillate about a zero value. If A be a de clination such that the mean value of sin 2 8 is equal to sin 2 A, then, to determine the southward component so that it shall be a truly periodic function, we must subtract from the above sin 2X(1 - 3 sin 2 A), and the last term then becomes 3sin2X(sin 2 A-sin 2 S).

In the case of the moon, A varies a little according to the position of the moon's node, but its mean value is about 16 31 .

The constant portion of the southward component of force has its effect in causing 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 in order that the meaning to be attributed to lunar deflexion of gravity may be clearly defined.

If we consider the motion of a pendulum-bob 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, with ratio of axes equal to the sine of the latitude, and with linear dimensions proportional to cos 2 5, and it once describes an ellipse whose north and south axis is proportional to sin 25 cos 2X and whose east and west axis is proportional to sin 28 sin X. 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

Sm/a 3 . ^-A - I cos X.

At Cambridge in latitude 52 43 this angle is 0"-0216.

An attempt, made by George and Horace Darwin,^[2] to measure the lunar deflexion of a pendulum failed on account of incessant variability of level occurring in the supports of the pendulum and arising from unknown terrestrial changes. The work done, therefore, was of no avail for the purposes for which it was instituted, but remained as a contribution to an interesting subject now be ginning to be studied, viz., the small changes which are always taking place on the upper strata of the earth.

§ 9. Correction to Equilibrium Theory for Continents.

Effect of land in equilibrium theory.In the equilibrium theory as worked out by Newton and Bernoulli 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. Sir W. Thomson has, however, reasserted^[3] a point which was known to Bernoulli, but has since been overlooked, namely, that this law 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 law would still hold if water were appropriately supplied to and exhausted from the ocean; and, if in any configuration of the tide-generating body we imagine water to be instantaneously so supplied or exhausted, the level will every where rise or fall by the same height Now the amount of that rise or fall depends on the position of the tide-generating body with reference to the continents, and is different for each such position. Conversely, when the volume of the ocean remains con stant, we have to correct Bernoulli's simple equilibrium theory by an amount which is constant all over the globe at any instant, but which changes in time. Thomson's solution of this problem has since been reduced to a form which is easier to grasp intelligently than in the shape in which he gave it, and the results have also been reduced to numbers.^[4] It appears that there are four points on the earth's surface at which in the corrected theory the semi diurnal tide is evanescent, and four others where it is doubled. A similar statement holds for the diurnal tide. As to the tides of long period, there are two parallels of latitude of evanescent and two of doubled tide.

Now in Bernoulli's theory the semi-diurnal tide vanishes at the poles, the diurnal tide at the poles and the equator, and the tides of long period in latitudes 35 16 north and south. The numerical solution of the corrected theory shows that the points and lines of doubling and evanescence in every case fall close to the points and lines where in the unconnected theory there is evanescence. When in passing from the unconnected to the corrected theory we speak of a doubled tide, the tide doubled may be itself nil, so that the result may still be nil. The conclusion, therefore, is that Thomson's correction, although theoretically interesting, is practically so small that it may be left out of consideration.

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1. The reader may refer to Thomson and Tait's Natural Philosophy (1883), part ii. 798-821, for further considerations ou this and analogous subjects, together with some interesting examples.
2. Reports to the British Assoc., 1881 (York) and 1882 (Southampton).
3. Thomson and Tait, Nat. Phil.,§807.
4. Darwin and Turner, Proc. Roy. Soc., 1886.