Encyclopædia Britannica, Ninth Edition/Tides/Chapter 3
III. Dynamical Theory of Tides.
§ 10. Historical Explanation.
Dynamical theory.The problem of tidal oscillation is essentially a dynamical one. Even when the ocean is taken as covering the whole earth, it presents formidable difficulties, and this is the only case in which it has been hitherto solved.[1] Laplace gives the solution in bks. i. and iv. of the Mécanique Céleste; but his work is unnecessarily complicated by the inappropriate introduction of spherical harmonic analysis, and it is generally admitted that his investigation is difficult. Airy, in his "Tides and Waves" (in Ency. Metrop.) presents the solution free from that complication, but he has made a criticism of Laplace's method which we believe to be wrong. Sir W. Thomson has written some interesting papers (in Phil. Mag., 1875) in justification of Laplace, and on these we base the following paragraphs. This portion of the article is given more fully than others, because there exists no complete presentment of the theory free from objections of some kind.
§ 11. Equations of Motion.
Equaations of motion.Let r, 6, tj> be the radius vector, co-latitude, 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, 3 be the radial, co-latitudinal, and longitudinal accelerations of the point, we have
R=(15).
Now suppose that the point never moves far from a zero position and that its displacements, 77 sin 6 co-latitudinally and longi tudinally are very large compared with its radial displacement p, and that the velocities are so small that their squares and products are negligible compared with nV 2; then we have
dr dp -r. = -f., a very small quantity; at at
Hence (15) is approximately li= -?iVsin 2
- =^-2ntrin0co605Z at " at ff=su10^, + 2ncos0 ( *l dt* dt (16).
Component accelerations.With regard to the first equation of (16), we observe that the time has disappeared, and that has exactly the same form as if the system were rendered statical by introducing a potential tyfir-siu-O and annulling the rotation of the axes. Since inertia plays no sensible part radially, it follows that, if we apply these expressions to the formation of equations of motion for the ocean, the radial motion need not be considered. We are left, therefore, with only the last two equations of (16).
Component forces.We now have to consider the forces by which an element of the ocean is urged in the direction of co-latitude and longitude. These forces are those due to the external disturbing forces and to the pressure of the surrounding fluid, the attraction of the fluid on itself being supposed negligible. We have seen in (9) that, if fluid on a sphere of radius a be under the action of disturbing forces whose potential is Ur 3, and if r=a + ty be the equation to the sur face, then must gfy= Ua 2 . Hence, if e be the equilibrium height of tide, the potential of the disturbing force is gtr~/a 2 . But, if the elevation DC }j, the potential under which it would be in equili brium is (jh^/a 2 . Therefore, if fy be the elevation of the tide in our dynamical problem, the forces due to hydrostatic pressure on an element of the ocean are the same as would be caused by a potential - gtyr~/a Hence it follows that the whole forces on the element are those due to a potential - <?(fj - t)r 2 / 2 . Therefore from (16) we see that the equations of motion are dt add sin + 2 cos (!-) (17)dt a sin d d<j>
Equations of motion.It remains to find the equation of continuity. This maybe 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 V be a velocity potential, the equation of con tinuity for incompressible fluid is
The element referred to in this equation is defined by r, 0, <f>, r + 8r, + d0, $ + 50. The co-latitudinal and longitudinal veloci ties are the same for all the elementary prism defined by 6, <j>, + Se, d> + 8d>, send the sea bottom. Then -3s=-4,, . rdO dt rsin sin 0-j-; and, since the radial velocity is d}}fdt at the surface of the (it ocean, where r = a + y, and is zero at the sea bottom, where r = a, we have = = - -s?. Hence, integratin with respect to r from dr 7 dt r=a + y to r=a, and again with respect to t from the time t to the time when i),, 77 all vanish, and treating y and fj as small com pared with a, we have lasin0 + ^ a (y^sme)+~(yr,sm6) = Q ......... (18). Equa* * tionof
Equation of continuity.This is the equation of continuity, and, together with (17), 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 7, the depth of the ocean, is only a function of latitude. In this case (18) becomes
sin 6 .(19).
§ 12. Adaptation to Forced Oscillations.
Adaptaption toforced 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 (13) that t (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 terms vary slowly, and the whole of the third varies slowly. Hence c 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 approxi mately semi-diurnal, diurnal, and slowly varying terms, each of which is a strictly harmonic function of the time. Thus we may assume for t a form e cos (2nft + k<j> + a), where/ and k are numbers, and where e is only a function of co-latitude and of the elements of the orbit of the disturbing body. According to the usual method of treating oscillating systems, we may therefore make the follow ing assumption for the form of solution
r = e cos (2nft + k<j> + a)
I) = h cos (2nft + k<(> + a)
| = x cos (2nft + k<f> + a)
7? = y sin (271/5 + k<f> + a)
where e, h, x, y are functions of co-latitude only. Substituting these values in (19), we have
- -; ^(-yxsin 0) + kyy + ha = (21).
Then, if we write u for h -e, andjput m= n-a/g, substitution from
(20) in (17) leads at once to x/ + y/siu0cos0 = i Lg I ^ y^siu0 + x/cos0=-A s JLj Solving (22) for x and y, we have 1 /dn k cos 0 ^ 4md0 / sin 0} - 1 /cos dn kn y sin (f 2 - cos 2 0) = -( -r- - + -r ) 4wi / do sin OJ Then substituting from (23) in (21), we have cos di ...(20), ...(23). fru Final 1 equation. sin + 47na(u + e) = (24).
This is Laplace's equation for tidal oscillations in an ocean whose depth is only a function of latitude. When u is found from this equation, its" value substituted in (23) will give x and y.
§ 13. Preparation for Solution.
Preparation for solution.The ocean which is considered in this case is not like that on the earth's surface, and therefore it does not seem desirable to pursue the integration of (24) except in certain typical cases.
In (13) we have the expansion of the disturbing potential and implicitly of the disturbing forces in three terms, the first of which is variable in half a day, the second in a day, and the third in half the period of revolution of the tide-raising body. Forestal ling the results of chapter iv. each of these terms may be expressed as the sum of a series of strictly harmonic functions of the time; the first set of these have all approximately semi-diurnal periods, the second approximately diurnal periods, and the third vary slowly in dependence on the periodic time of the tide-generating body. The first set involve twice the terrestrial longitude, the second the longitude, and the third set are independent of the longitude of the place of observation. From these statements compared with (13) we see that in the semi-diurnal terms / is approximately unity, k=2, and c = E sin 2; in the diurnal terms / is approximately, k=l, and e = E sin cos; in the terms of long period /is a small fraction (for the fortnightly tide about ^V). = 0, c = E(-cos 2 0). The departure from exactness in the rela tion /= 1 for the semi-diurnal, and /= | for the diurnal terms is generally (except for certain critical depths of ocean) not such as to greatly change the nature of the results from those obtained when Laplace's/=l and rigorously. Laplace's three kinds of oscillation.Hence the integration of (24) will be pursued on these three hypotheses, giving Laplace's three kinds of oscillation. The hypothesis which will be made with regard to -y is that 7 = 7(1 q cos- 0), and in the case of the semi-diurnal tides we shall be compelled by mathematical difficulties to suppose q to be either unity or zero. The tides of zonal seas may be worked out, and more complex laws of depth may be assumed; but for the discussion of such cases the reader is referred to Thomson's papers in Phil. Mag., 1875.
There might be reason to conjecture that the form of u would be similar to that of e, and this is in fact the case for the diurnal tides for any value of q and for the semi-diurnal tides when q is unity. Preliminary transformations.Before proceeding further it will be convenient to exhibit two purely analytical transformations of the first two terms of (24) which hold true for certain values of k and/, and when u has such a form as that suggested. If we put &=!,/=, y = l(l -q cos 2 0), then, if v=A sin cos 0, it will be found on substitution that
sin0d0 i-cos 2 7 4 Again, if we put k=2,f=l, j=l, y=l(I-cos* 0) = l sin 2 0, and if v=A sin 2 0, / . n dv dv 2v
- y I sin -j-r + 2v cos j cot | -. -2? /*...f g =-8to ...(26). 1 d sin0d0 l-cos 2 " l-cos 2 Another general property of (24) is derived from the supposition that u is expressed in a series proceeding by powers of I; thus! + <. + (27)Let v, v v v.2, &c., be so chosen that, when u is substituted in (24), the coefficient of each power of I vanishes independently; then the term independent of I obviously gives v = -e, and the connex ion between successive v's is /cos0 dv m kv m, / dO sin Q) sin (f 2 - cos 2 0) (28).
We shall suppose below that u is expansible in the form. (27), and shall use (28) in conjunction with (25) or (26) for finding the successive values of the v's.
§ 14. Diurnal Tide.
Let us first consider the diurnal tides. We have e = E sin cos 0, Diurnal k=l, and /=; then v = - E sin cos 0. Hence by (28) and (25) tide.
- 8?gr + 47/ifoj = (29), and therefore t j = v^ Applying the same theorem a second time, i- 2 = (2qfm}v 1, and so on; therefore u = vj[l + 2lq/ma + (2lq/ma) 2 + ...] V 6 l-2lq/ma~ l-2lq/ma * *" But u = h - e; hence, _ 2lqjma .(31). 1 - 2lq/ma
It appears, therefore, that the tide is " inverted," giving low water where the equilibrium tide gives high water. If q = 0, so that the ocean is of uniform depth, the tide vanishes.
§ 15. Semi-Diurnal Tide, with Variable Depth.
Semi-diurnal tide.Next let us consider the semi-diurnal tide in the case where q=l, so that 7 = I sin 2 0. Then e = E sin 2 0, k = 2, f= 1; also t- = - e = - E sin 2 6. Hence by (28) and (26) - 8lv + 4mlv^ = 0, whence v, = 2/?n . Applying the same theorem a second time, i < 2 =(2/m) 2 t7 OJ and so on; therefore u = v [l + 2ljma + (21/ma) 2 + . . .]
Hence 1 - 2l/ma 1 - 2l/ma, _ 21/ma, ~l-2l/ma C "
If 2l/ma = &, the height of tide is equal to the equilibrium height; but it is inverted, giving low water where the equilibrium theory gives high water. In the case of the earth m= 1/289, and therefore this relation is satisfied if 1= a/1 156. Hence in a sea 3000 fathoms deep at the equator, and shallowing to the poles, we have inverted semi-diurnal tides of the equilibrium height.
§ 16. Semi-Diurnal Tide, with Uniform Depth.
The method of development used above, where we proceed by powers of the depth of the ocean, is not applicable where the depth is uniform, because it leads to a divergent series. We have there fore to resume equation (24). In the case of the semi-diurnal tides we have for the depth y=l (a constant by hypothesis), k=2,f=l
approximately, and e = Esin 2 0. Now for brevity let fi=4ma/l, / = sin 0, so that e = Ev 2 . Then we find that on development (24) _,{* (8 _. dv* dv " 6 (33).
Let us now assume as the solution of this equation
u = (A" 2 -E> 2 + J fir 4 / 4 + ^ 6 +...+ J K 2i v 2 + (34).
Substituting from (34) in (33), and equating to zero the coefficients of the successive powers of v, we find K^ E, K apparently inde terminate, and
2i(2i + 6)^>i+4- 2i(2i + 3)^ai+2 + /3Jr 2i = (35). Since A = 0, this equation of condition may be held to apply for all positive integral values of i, beginning with i = 0. It is obvious that K 6 is determinable in terms of K and K z, K 8 in terms of L~ and A" 4, &c., so that all the A"s are to be found in terms of K z, which is known, and of K 4, which is apparently indeterminate. The condition for the convergency of the series (34) for u and for the series du/dv is that K K+ ilK-x shall tend to a limit less than unity. The equation (35) may be written AT2144_2i-f3 _____ K&,gg> AVh2~2i + 6 2i(2i + 6) AW" Now A^+2/ A*2i tends to be either infinitely small or not infinitely small. If it be not infinitely small in the limit, the second term on the right of (36) becomes evanescent when i is very great, and we have in the limit when i is very large V=flBut the ratio of successive terms of /(l ~ " ) tends to become (l-f/i> 2 . Hence, if K^^Ky. does not tend to become infinitely small, u = A + BVl -v 2, where A and B are finite for all values of v. Again, under the same circumstances we have in the limit when i is very large .A 1 V, 3 A 1 4 + l But the ratio of successive terms of (I-? 2 ) ~* tends to (1 -/i) 2 Hence, if Kx+n/Ka does not tend to become infinitely small, duldv = C + D(l - 1- 2 )"*, where C and D are finite for all values of v. XT da ^ u i o~ /~( /i i T Now -j3=-r Vl- 2 = Cvl-i 2 + D. dO dv Therefore at the equator, where v = l, dn/d6 = D, a. finite quantity. Hence the hypothesis that AVi-2/^2 tends to be not infinitely small leads to the conclusion that u and dn/d& are finite at the equator. But on account of the symmetry of the system the co-latitudinal displacement must vanish at the equator, and therefore x also. By (23), when/=l, k=2, y = sin 0, But we have just seen that this hypothesis makes u finite when / =! or 0=90, and therefore at the equator 1 du _ ., ... x = -;—,-r, a finite quantity. 4m d& Now symmetry necessitates a vanishing value of du/dO at the equator. Thus the hypothesis that A 2,-+2/^2i tends to be not in finitely small is negatived, and we conclude that, on account of the symmetry of the motion, it is infinitely small for infinitely great values of i. This being established, let us write (36) in the form Hence by repeated application of (36a) we have And we know that this is a continuous approximation to A" L _ which must hold in order that the latitudinal velocity may vanish at the equator. Writing N, = AT^+a/ K^, all the N s may be com puted from the continued fraction (37). Then We cannot compute K 6 from AT 4, A g from AT 6, and so on; for, if we do, then, short of infinite accuracy in the numerical values, we shall be gradually led to successive values of the K s which tend to equality. 1 This process was followed by Laplace without explanation. It was attacked by Airy in his " Tides and Waves " (in Ency. Metrop. ) and by Ferrel in his Tidal Researches (U.S. Coast Survey, 1873), but was justified by Sir W. Thomson in the Phil. Mag. (1875, p. 230). The investigation given here is substantially Thomson's. Solu- Laplace gives numerical solutions for three different depths of lions. the sea, -3-5^, T^*S-, sa 1 ts f the earth's radius. Since tn~^^, these correspond respectively to the cases of =40, 10, 5, and the solutions are =40, h = E{v 2 + 20-1862v 4 + 10 1164i 6 - 13 1047 8 -15-4488y 10 -7-4581i/ 12 - 2-1975? 14 - 4501i 16 - O-OeS?? 18 -0-0082V 20 - 0-0008? 22 - O OOOlc 24 ...} =10, h = E{i> 2 + e-igeOi 4 + 3-2474c 6 + 07238 8 + Q-QVlQv 10 + 0-0076p 12 + 0004v 14 ...} = 5, h = E{v 2 + 0-7504J/ 4 + 0-1566i/ 6 i Thomson calls this a dissipation of accuracy. It may t>e illustrated thus. Consider the equation 2-31+2=0, which may be written either z=j+Jx 2 or x= 3 - 2/x. Now let x tt+l = + Jj: 2,^ and suppose we start with any value XQ, less than unity, and compute xj, 3%, . . . x n . Then, starting with x n in the equation x n _ l =3-Z/x n, if we work backwards, we ought to come to the original value zo. In fact, however, we shall only do so if there is infinite accuracy in all the numerical values. For, start with &o=i. then Zi = -75, a%2= 8542, z 3 =-9099, a; 4 = -9527, 5 = 9692; and the values go on approximating to 1, which is a root of the equation. Next start backwards with x 5 = -97, and we find z 4 = -938, 3= -868, z. 2 = -696, x l = -l-27,x = - 1275, x_ l =3~157, x_ 2 =2 367, z_ 3 =2-155, x_ 4 = 2-072; and the values go on approximating to 2, the other root of the equation. Since h vanishes when v = 0, there is no rise and fall of water at the poles. When v = 1 at the equator, we find =40, h=-7 434E = 10, h = 11-267E = 5, h= 1-924E. The negative sign in the first case shows that the tide is inverted at the equator, giving low water when the disturbing body is on the meridian. Near the pole, however, that is, for small values of v, the tides are direct. In latitude 18 (approximately) there is a nodal line of evanescent semi-diurnal tide. In the second and third cases the tides are everywhere direct, increasing in magnitude from pole to equator. As diminishes the tides tend to assume their equilibrium value, because all the terms, save the first, become evanescent When =1 (depth 7 of radius) the tide at the equator still exceeds its equilibrium value by 11 per cent. As diminishes from 40 to 10 the nodal line of evanescent tide contracts round the pole, and when it is infinitely small the tides are infinitely great. The particular value of for which this occurs is that where the free oscillation of the ocean has the same period as the forced oscilla tion. The values chosen by Laplace were not well adapted for the illustration of the results, because in the cases of =40 and = 10 the depth of the ocean is not much different from that value which would give infinite semi-diurnal tide. For values of greater than 40 we should find other nodal lines dividing the sphere into regions of direct and inverted tides. We refer the reader to Sir W. Thomson's papers for further details on this interesting point.
§ 17. Tides of Long Period; Laplace's Argument from Friction.
Laplace's argument from friction unsound.In treating these oscillations i^aplace remarks that a very small amount of friction will be sufficient to cause the surface of the ocean to assume at each instant its form of equilibrium, and he adduces in proof of his conclusion the considerations given below. The friction here contemplated is such that the integral effect is represented by a retarding force proportional to the velocity of the water relatively to the bottom. Although proportionality to the square of the velocity would probably be nearer to the truth, yet Laplace's hypothesis suffices for the present discussion. In oscillations of this class the water moves for half a period north, and then for half a period south. In oscillating systems, where the resistances are proportional to the velocities, it is usual to specify the resistance by a modulus of decay, namely, that period in which a velocity is reduced to c" 1 of its initial value by friction. Now the friction contemplated by Laplace is such that the modulus of decay is short compared with the semi-period of oscillation. The quickest of the important tides of long period is the fortnightly (see chapter iv.); hence, for the applicability of Laplace's conclusion, the modulus of decay must be short compared with a week. Now it seems prac tically certain that the friction of the bed of the ocean would not materially affect the velocity of a slow ocean current in a day or two. Hence we cannot accept Laplace's discussion as satisfactory. How ever this may be, we now give what is substantially his argument.
Let us write 6 for the reciprocal of the modulus of decay. Then the frictional forces introduced on the left-hand side of (17) are + Gd%/dt in the first and sin OSdij/dt in the second. Laplace's hypothesis with regard to the magnitude of the frictional forces enables us to neglect the terms d-/dt- and sinQd^y/dt- compared with the frictional forces. Then, if we observe that in oscillations of this class the motion is entirely latitudinal, equations (17) and (19) become
dt +2n cos 0=0 dt dt sin + j(y sin 6)
From the first two of these we easily obtain
q d . (, 4?r g+ ^ (38). .(39).
As a first approximation we treat d/dt as zero, and obtain I)=r, or the height of water satisfies the equilibrium theory. In these tides (see chap, iv.) f = I (J-cos 2 0) cos it, so that from the third equation of (38) we can obtain a first approximation to |; then, sub stituting in (39), we obtain on integration a second approximation to h. Laplace, however, considers as adequate the first approxima tion, which is simply the conclusion of the equilibrium theory.
§ 18. Tides of Long Period in an Ocean of Uniform Depth.
Tides of long period without friction.As it seems certain that these tides do not satisfy even approximately the equilibrium law, we now proceed to find the solution where there is no friction. In the case of these tides k= O,/ a small fraction, and e = E (^-cos 2 0). The equation (24) then becomes
or, writing fj. for cos 6 and e = E ( - fj?}, (40) We shall confine the investigation to the case where y=l, a con stant, and where the sea covers the whole surface of the globe. The symmetry of the motion in this case demands that u when ex panded in a series of powers of n shall only involve even powers. Let us assume, therefore, that
(41). Again, (42). ...... (43), y+ (44),
where C is a constant. Then, writing /3 for 4ma/l, as in the case of the semi-diurnal tide, substituting fiom (42), (43), and (44) in (40), and equating to zero the successive coefficients of the powers of /*, we find
C=-|E + B 1 //3-s
Thus the constant C and B 3, B B, &c., are all expressible in terms of Bj, and BX is apparently indeterminate. We may remark that, if
the equation of condition (45) may be held to apply for all values of i, from one to infinity. Let us write (45) in the form
When i is large Boi+i/Bsi-i either tends to become infinitely small or it does not do so. Let us suppose that it does not tend to become infinitely small. Then it is obvious that the successive B s tend to become equal to one another, and so also do the values of I>j;-2-/ 2 Boi_i) /2t and the coefficients of du/dfj.. Hence we have du/dfj. = L + M/(1- fjf), for all values of /i, where L and M are finite. Hence this hypothesis gives infinite velocity to the fluid at the pole, where /* =!. But with a water-covered globe this infinite velocity is impossible, and therefore the hypothesis is negatived, and B 2l -+i/B 2i _i must tend to become infinitely small. This being fraction, established, let us write (46) in the form
P B_i_ 2i(2i + l) 12i(2i
By repeated applications of (47), we have in the form of a continued fraction 12t(2i 1(2i+2)(2t And we know that this is a continuous approximation, which must hold in order to satisfy the condition that the water covers the whole globe. Let us denote this continued fraction by - JVj. Then, if we remember that B_ x = - 2E, we have
B 1 = 2FJVi, B 3 /B 1 = - A * B 5 /B 3 = - A"* E./E^ -A* &c., so that 63= -2EAyV 7 2, B 8 =2EAiAV^s, B 7 = -2EA T .jr a A r 8 A 4, &a, and C= Then n=u + e (49).
Now we find that, when = 40, which makes the depth of the sea 3000 fathoms or ^^ of the radius of the earth, and with /= -0365012, which is the value for the fortnightly tide (see chap, iv.), A T 1 = 3-040692, A r ., = l-20137, A~ 3 = -66744, A" 4 = -42819, ^, = -29819 A~ 6 = -21950, JV 7 =-16814, A 7 ^ -13287, JV 9 =-107, A T 10 = -1. These values give 2AV/3 =15203, 1+/ 2 A 7! = 1-0041, J.V 1 (1+/ 2 A:! ) = 1*5228, JA*iAVl +/%) = 1-2187, iAjAyVVl +/ 2 A 7 4 ) = -60988, iA*! . . . AVI +/ 8 A T 5 ) = -20888, JA^ . . . AVI +/ 2 A* 8 ) = 05190, iA 7! . . . AVI +/*A" 7 ) = -00976, | A*! . . . A* 7 (l +/*A 8 ) = 0014, |A*j . . . AVI +/ 2 A 7 9 ) = -00017. So that h/e= (-1 520-1 -0041 Ai 2 + l-5228/i 4 -l -21 87M 6 + 6099/i 8 - -2089M 10 Solutions. + -0519A1 12 - -0098/x 14 + -0014/* 18 - -0002/t 18 } -f-( j - M 2 ) (50). At the pole, where /i=l,h= -E x -1037 = ex -1556,, n and at the equator, where M = 0, h= +Ex -1520 = ex -4561 / Now let us take a second case, where /3=10, which was also one of those solved for the case of the semi-diurnal tide by Laplace, and we find h/E = -2363 - 1 0016/x*+ -5910/* 4 - -1627/x 6 + 0258/* 8 - -0026/x 10 + -0002/x 12 .
At the pole, where /t=l, we find h= -Ex -3137 = ex -471, and at the equator h=+Ex -2363 = ex 709. With a deeper ocean we should soon arrive at the equilibrium value for the tide, for A^, A" 2, &c., become very small, and 2A T 1 /j8 becomes equal to J. In this case, with such oceans as those with which we have to deal, the tides of long period are considerably smaller than the equilibrium value.
§ 19. Stability of the Ocean.
Imagine a globe of density 5, surrounded by a spherical layer of Stability water of density cr. Then, still maintaining the spherical figure, and of the with water still covering the nucleus, let the layer be displaced ocean, sideways. The force on any part of the water distant / from the centre of the water and r from the centre of the nucleus is tcai 1 towards the centre of the fluid sphere and $r(5 - <r)r towards the centre of the nucleus. If S be greater than <r there is a force tend ing to carry the water from places where it is deeper to places where it is shallower; and therefore the equilibrium, thus arbitrarily dis turbed, is stable. If, however, S is less than y (or the nucleus lighter than water) the force is such that it tends to carry the water from where it is shallower to where it is deeper, and therefore the equilibrium of a layer of fluid distributed over a nucleus lighter than itself is unstable. As Sir William Thomson has remarked, 1
if the nucleus is lighter than the ocean, it will float in the ocean Stabilities with part of its surface dry. Suppose, again, that the fluid layer of various be disturbed, so that its equation is r=a(l+s t ), where Sj is a sur- orders, face harmonic of degree i; then the potential due to this deforma 4iro- ai+3 tion is - -v | s i} and the whole potential is
4?r5rt 3 47T(r ai+3 3r 2t + l rt+l *
If, therefore, o-/(2t + l) is greater than J5, the potential of the forces due to deformation is greater than that due to the nucleus. But we have seen that a deformation tends to increase itself by mutual attraction, and therefore the forces are such as to increase the deformation. If, therefore, <r=^(2t + l)3, all the deformations up to the z th are unstable, but the t + lth is stable. 2 If, however, <r be less than 5, then all the deformations of any order are such that there are positive forces of restitution. For our present purpose it suffices that this equilibrium is stable when the fluid is lighter than the nucleus.
§ 20. Precession and Nutation.
Suppose we have a planet covered with a shallow ocean, and that Precesthe ocean is set into oscillation. Then, if there are no external dis- sion and turbing forces, so that the oscillations are "free," not " forced, " nutation, the resultant moment of momentum of the planet and ocean remains constant. And, since each particle of the ocean executes periodic oscillations about a mean position, it follows that the oscillation of the ocean imparts to the solid earth oscillations such that the re sultant moment of momentum of the whole system remains constant. But the mass of the ocean being very small compared with that of the planet, the component angular velocities of the planet necessary to counterbalance the moment of momentum of the oscillations of the sea are very small compared with the component angular velocities of the sea, and therefore the disturbance of planetary rotation due to oceanic reaction is negligible. If now an external disturbing force, such as that of the moon, acts on the system, the resultant moment of momentum of sea and earth is unaffected by the interaction between them, and the processional and nutational couples are the same as if sea and earth were rigidly connected together. Therefore the additions to these couples on account of tidal oscillation are the couples due to the attraction of the moon on the excess or deficiency of water above or below mean sea-level. The tidal oscillations are very small in height compared with the equatorial protuberance of the earth, and the density of water is T * T ths of that of surface rock; hence the additional couples are very small compared with the couples due to the moon's action on the
[3] Corrections to precession and nutation insensible.solid equatorial protuberance. Therefore precession and nutation take place sensibly as though the sea were congealed in its mean position. If the ocean be regarded as frictionless, the principles of energy show us that these insensible additional couples must be periodic in time, and thus the corrections to nutation must consist of semi-diurnal, diurnal, and fortnightly nutations of absolutely insensible magnitude. We shall have much to say below on the results of the introduction of friction into the conception of tidal oscillations as a branch of speculative astronomy.
§ 21. Some Phenomena of Tides in Rivers.
Tides in rivers.In § 2 we have given a description of some of the phenomena of the tide-wave in rivers. As a considerable part of our practical knowledge of tides is derived from observations in estuaries and rivers, we give an investigation of two of the most important features of the tide-wave in these cases. It must be premised that when the profile of a wave does not present the simple harmonic form it is convenient to analyse its shape into a series of partial waves superposed on a fundamental wave ; and generally the prin ciple of harmonic analysis is adopted, in which the actual wave is regarded as the sum of a number of simple harmonic waves.
The tide-wave in a river is a " long " wave in which the vertical motion of the water is very small compared with the horizontal, the river very shallow compared with the wave-length, and the water which is at any moment in a vertical plane always remains so throughout the oscillation.
Suppose that the water is contained in a straight and shallow canal of uniform depth ; then take an origin of coordinates at the bottom, with the x axis horizontal in the direction of the canal, and the y axis vertical ; let h be the undisturbed depth of water ; let h + 77 be the ordinate of the surface corresponding to that fluid whose undisturbed abscissa is x and disturbed abscissa x + ; and let g be gravity. The equations of motion and continuity 1 are
This represents the oceanic tide, and n is that which we call below ( 23) the speed of the tide. Then obviously m=n/v, so that at any point x up the river " =//sinw O-^) (56) - (56) gives the first approximation to the forced tide-wave, and it is clear that any number of oscillations may be propagated inde pendently up the river with the velocity fgh due to the depth of Over the river. In passing to the second approximation we must separate tides, the investigation into two branches. (i.) Over-Tides (see 24). We now suppose that the tide at the river mouth is simply (55). On substituting the approximate values (54) in (53) our equations become (57) _ ?7 ~ For brevity we shall write v- = gh and u = vt - x. Since for " long " waves d/dx is small, the equations (52) become approximately
.(53). 27_ h dx For finding a first approximation we neglect the second term on the right of each of (53). The solution is obviously
=a cosm(vt-x) = acosmu ,.< 77= -mahsin mu j **P (54) gives the height of the water whose undisturbed abscissa is x, and since is small this is approximately the height at the point on the bank whose abscissa is x. But now suppose that at the origin (the mouth of the river) the canal communicates with a basin in which there is a forced oscillation of water-level given by
--- ...(55).
We have now to assume an appropriate form for the solution of (57), such as = a cos mu + Ax cos 2mu + B sin 2mu ......... (58). We have here in effect assumed that the second and third terms of (58) are small compared with the first. It is clear, however, that at a distance from the origin the term in A will become large. This difficulty may be eluded by taking the canal of finite length, and supposing that, where the canal debouches into a second basin, a second appropriate forced oscillation is maintained. The length of the canal remains arbitrary, save that the second term of (58) shall still be small compared with the first. On substituting from (58) in (57) we have B indeterminate and A= - | 2 m 2 ; hence This gives the elevation of the water whose undisturbed abscissa is x, that is to say, at the point whose abscissa along the bank is X=x + . If we put x=X- in the largest term of (59), and treat as small, and put x=X in the small terms, (59) becomes tj/h = - ma sin m(vt - X) + f ra s a 2 Jf sin 2m(vt - X) + (2mB - |mV-) cos 2m(vt - X-}. But at the origin (55) holds true, therefore B = T ma 2 , -mah = H, and mv=n. Thus the solution is TT ( . x Z&n ( ^ 77=^8111 n^- ^ + /rT =..Ysm24- ) (60). From (60) we can see what the proper forced oscillation at the Solution further end of the canal must be ; but this matter has no present giving interest. The first term of (60) being called the fundamental, the first second gives what is called the first over -tide; and by further over-tide. approximation we can get the second, third, &c. The over-tide travels up the river at the same rate as the fundamental, but it has double frequency or "speed," and the ratio of its amplitude to that o rr y of the fundamental is -,- =. As a numerical example, let the range of tide at the river mouth be 20 feet and the depth of river 50 feet. The "speed" of the semi-diurnal tide is about 1/1 9 radians per hour ; /gh=27 miles per hour ; hence j_ -7=7 =379 ^ Therefore 34 miles up the river Fig. 1 the over-tide is ^th of the fundamental and has a range of 2 feet. If the river shallows very gradually, the formula will still hold, and we see that the height of over-tide varies as (depth )~~i. Fig. 1 2 read from left to right exhibits the progressive change of shape. The steepness of the advancing crest shows that it is a shorter time from low to high water than vice versa. The law of the ebb and flow of currents mentioned in 2 may also be easily determined from the above investigation. We leave the reader to determine the effect of friction, which is given by inserting a term - (id/dt on the right-hand side of (57). Inter- (ii.) Compound Tides (see 24). We shall now consider the ference mutual influence of two waves of different periods travelling up of waves the river together. In the first approximation they are quite inde nt shal- pendent, and we may assume low % = acosm(vt-x) + bcos[n(vt-x) + e] (61). In proceeding to the second approximation, we only take notice of those terms which result from the interaction of the two, and omit all others, writing for the sake of brevity {m - n =(m - ri)(vt -x)-e, {m + n = (m + n}(vt - x) + e. With the value of assumed in (61), we find, on substituting in (53) and only retaining terms depending on mutual influence, that the equations for the second approximation are 1 See, for example, Lamb s Hydrodynamics, chap. vii. 2 From Airy, "Tides and Waves." i> 2 -v-o sin {m + n} - (m - n} sin {m - n} ] y/h = - abmn[cos {m + n} - cos {m - n} ] - d^jdx Now let us assume as the solution
+Bsin{m+}
, , and let us elude the difficulty about the increasing magnitude of the second term in the same way as before. Substituting in the equation of motion, we have for all time, 2(m + w) A sin {m + n} + 2(m - n}C sin {m - n} + %abmn[(m + n) sin {m + n} -(m-n) sin {m - n} ] = 0. This gives A= - %abmn and C= +%abmn. B and D remain arbi trary as before, and will be dropped, because they are to be deter mined by the condition that at the origin the terms of d/dx in cos {m + n}, cos {m-n} are to vanish, whence rj/h = am sin m(vt x) - bn sin [n(vt x) + e] + %abmn[(m + n)x sin {m + n} - (m - n]x sin {m - n} ] + terms in cos {m + n} and cos {m - n} . Then we pass from x to X as in the last section, and make the terms in cos {m + n} and cos {m - n} vanish by proper values of B and D, and we have i} = amksinm(vt - X) - bnhs [n(vt - x) + e] + $abmnX[(m + ri)&in {m + n} - (m - n} sin {m - n} ] (64). Now at the river s mouth, where x = Q, suppose that the oceanic tide is represented by 77 = H^ sin n^t + H. 2 sin (n.J, + e). Then so that (64) becomes habmn = H^ V0*
As a numerical example, suppose at the mouth of a river 50 feet deep that the solar semi-diurnal tide has a range 27/ 1 =4 feet, and the lunar 2//" 2 = 12 feet; then n + n 2 =j$ radians per hour,?^ - n z =-f radians per hour, and as before Jgh= c il miles per hour. With these figures
3//!/7 2 n l + n 2 _ 1 4h ^gh ~170
Thus 15 miles up the river the quater-diurnal tide (in 24 below, called MS) has a semi-range of an inch. But the luni-soiar fort nightly tide (called MSf in 24) would have a semi-range of -faila of an inch. Where the two interacting tides are of nearly equal speed, the summation compound tide is very large compared with the differential tide. As before, when the river shallows gradually this formula will still hold.
It is interesting to note the kind of effect produced by these compound tides. When the primary tides are in the same phase
n^t = nj, + e. Then (i + "o)^- X ^ and
Hence the front slope of the tide-wave is steeper at spring than at neap tide, and the compound tide shows itself in the form of an augmentation of the first over-tide; and the converse statements hold of neap tide. Also mean-water mark is lower and higher alternately up the river at spring tide, and higher and lower alter nately at neap tide, by a small amount which depends on the dif ferential tide. With the river which we were considering, the alternation would be so long that it would in actuality be either all lower or all higher.
- ↑ Sir W. Thomson's paper "On the Gravitational Oscillations of Rotating Water," in Phil. Mag., August 1880, bears on the same subject. It is the only attempt which has hitherto been made to consider the effects of the earth's rotation on the oscillations of land-locked seas.
- ↑ 1 Thomson and Tait, Nat. Phil., 816.
- ↑ 2 Compare an important paper by Poincare, in Acta Math. (1SSG), 7; 3, 4.