This expression is useful, because, in a long series of observations, it is possible to determine Tides a Wave 'M' S' the value of F with considerable accuracy, and that determination will immediately give us the value of the Sect. Equilibrium (56.) In the expressions for N and T, we have proceeded on the supposition, that it is most convenient to cos' to the values which they would have, if the respec Expres- All the fluctua tions may be ex cosines. 3 P' cos2μ and S' P μ or tive parallaxes had their mean values, and if the declinations were 0. With regard to the parallaxes, there m {1+ 3p' cos2 μ P cos2 A A similar change must be made in the value of T. that the expressions of (53.) and (54.) may still be retained, provided that we put M A Theory Then the greatest value Tides. and S', and that instead of sin2μ and sin2 we put (57.) With regard to the three principal parts into which we have resolved the lunisolar tide, we may make the following remarks: μ The first part treated in (45.) &c., is so slow in its period that it will generally be lost among the others whose recurrence is so much more frequent. If, however, we wished to examine its law, we should remark, that its variations depend chiefly on the variations of sinu and sino. And we see that sin u will be (for a few lunations) nearly proportional to the square of the sine of the moon's longitude from a given point, and therefore nearly proportional to the square of the sine of an angle increasing proportionally to the time (which we will call At+B, putting t for the time), or that sin uc sin (A+B) ∞ - cos (2At+2B). From this it appears that the fluctuation of the surface of the water, dependent on this cause, would be expressed (omitting the constant terms) by a multiple of cos (2At+2B). In like manner the fluctuation depending on cos2 would be expressed by cos (2Ct+2D). (58.) The fluctuations depending on the second and third parts, treated of from (46.) to (53.) are, as we have seen, expressed by cosines of multiples of l-m and l-s, which for a few tides can be expressed nearly enough by cosines of multiples of l-m. This angle is nearly proportional to the time, and thus these fluctuations can be expressed in the same form as those above. (59.) If we considered the variations which the factors of these different terms undergo, arising from the change of parallax, declination, &c., it would be found that they would all be expressed by series of cosines of multiples of the time; which combined as factors with the cosines of the former would produce cosines of new multiples of the time. (60.) Thus it appears that the fluctuation of the water will be expressed by a multitude of terms, each of which will be of the form C cos (2At+2B). If any one of these terms existed alone, the following curious law would be true. Suppose the lines of the extreme elevation and extreme depression of the surface of the water pressed by to be marked upon a wharf-wall; and suppose a circle to be described upon the wall, touching those two lines; then if the circumference of that circle be divided into equal parts, the fall of the water will expose the successive equal parts in successive equal times. For, since the whole fluctuation is 2C, C is the radius of our circle; and the elevation of any point of the circumference above the mean (or above the centre of the circle) is Cx cosine angle from the top; but the tidal term gives for that elevation Cx cosine (2A/+2B), and therefore 2At+2B must be the same thing as that angle from the top; and, therefore, that angle, and the circumference which is proportional to it, must have increased proportionally to the time. But this law does not hold for an assemblage of a multitude of such terms. (61.) We have now given a tolerably complete investigation of the equilibrium-theory. But before quitting this section we will point out roughly how far it agrees with observation. and (62.) The most conspicuous tide, on the coasts of Waves Europe at least, is the semidiurnal. The acceleration or retard of this tide on the moon's transit, does not at one port in a hundred agree in any measure with the result of this theory. The extreme differences of accery of leration or retard (F of article 55., &c.) agree better, but are not exactly the same at all ports. They do not occur on the days on which this theory predicts them, but always later. The absolute elevation of the tide is great at one port and small at another, without any relation to the quantity calculated from the theory. The proportions of the elevations however at the same port, in different stages of the lunation, agree pretty well with the theory (though not equally at all ports); yet the critical phænomena (spring and neap tides) occur later than the theory gives them, and that by a quantity which is not the same as the delay of extreme values of F, mentioned above. The peculiar phænomena of river tides are not touched by this theory. of (63.) The diurnal tide ought to be discovered, in observation, in one of the following ways. If the diurnal tide were much greater than the semidiurnal, there would appear, to common observation, to be only one tide in the day with some irregularities. If it were much smaller, its effect would be shown in either or both the following ways. If its high water occurred nearly at one high water of semidiurnal tide, its low water would occur nearly at another high water of semidiurnal tide; and one of the semidiurnal tides would be increased and the other would be diminished. If its high water occurred between two semidiurnal high waters, then at the first semidiurnal high water the surface would still be rising in consequence of diurnal tide, and the compound high water would be later; and at the second semidiurnal high water the surface would be falling in consequence of diurnal tide, and the compound high water would therefore be past; consequently the interval between these two high waters would be less than it ought to be on the usual laws of semidiurnal tide. The diurnal tide ought, in these latitudes, to be equal or nearly equal to the semidiurnal tide. Yet in the Thames it is absolutely insensible; and in other ports, as well of England as of other parts of Europe and America, though discoverable, it is not notorious, and has only been found from the observations made by men of science. It has been found to be very conspicuous at some places near the equator and some places hear the pole, where it ought not to be discoverable or scarcely discoverable. The Tides of longer period have scarcely been observed. (64.) Combining these remarks with those which we made at the introduction of this theory (14.), it must be allowed that it is one of the most contemptible theories that was ever applied to explain a collection of important physical facts. It is entirely false in its principles, and entirely inapplicable in its results. Yet, or strange as it may appear, this theory has been of very ty.great use. It has served to show that there are forces in nature following laws which bear a not very distant relation to some of the most conspicuous phænomena of the Tides; and, what is far more important, it has given an algebraic form to its own results, divided into separate parts analogous to the parts into which the tidal phænomena may be divided, admitting easily of calculation and of alteration, and thus at once suggesting the mode of separating the tidal movements, and affording numerical results of theory with which they Tides and are to be compared. The greatest mathematicians and Waves. the most laborious observers of the present age have agreed equally in rejecting the foundation of this theory and comparing all their observations with its results. And, till theories are perfect (a thing scarcely to be hoped for in any subject, and less in the Tides than in any other), this is one of the most important uses of theory. SECTION III.-LAPLACE'S THEORY OF TIDES. motion. (65.) In the theory which we are now about to Laplace's describe, a prodigious step was made towards a rational theory is a explanation, on mathematical principles, of the tidal theory of phænomena. The idea of a state of equilibrium was entirely laid aside, and the motion of the water was legitimately investigated, on the supposition that it is in motion, and subject to all the laws of fluids in motion. It was found necessary, however, in order to Supposimake the application of mathematics practicable, to tions by start with two suppositions, which are inapplicable to limited. the state of the earth. These are: that the earth is covered with water; and that the depth of this water is the same through the whole extent of any parallel of latitude. Under these suppositions it is evident that the theory is far from being one of practical application; though it clearly approaches much nearer to truth than the theory of equilibrium which we have already described. (66.) It would be useless to offer this theory in the same shape in which Laplace has given it; for the part of the Mécanique Celeste, which contains the Theory of Tides, is perhaps on the whole more obscure than any other part of the same extent in that work. We shall give the theory in a form equivalent to Laplace's, and, indeed, so nearly related to it, that a person familiar with the latter will perceive the parallelism of the successive steps. The results at which we shall arrive are the same as those of Laplace. (67.) We shall commence with a few considerations of a general nature, based upon the suppositions that we have already enunciated, and the additional supposition that the depth of the sea is small compared with the radius of the earth; and taking for granted a knowledge of the principal results of the equilibrium theory. which it is (68.) The motion of the water which forms the variable elevations of the Tides at different parts of the earth must be conceived to be principally a horizontal oscillation, the water on both sides of the highest point at any time having run towards that point in order to raise the surface there, and, consequently, (as the highest point occupies different positions at different times,) the water at any particular place running sometimes in one direction and sometimes in another. Com- A small bining this with the general result of the equilibrium- vertical theory as to semidiurnal Tides, (namely, that the water motion of is equally raised at two opposite points,) it will easily implies a be seen that, if a canal were traced through the water, large horiforming a great circle of the earth, it would (in certain zoutal mopositions at least of the sun and moon) be divided into tion. four parts, in two of which the water is running in one direction, and in the other two it is running in the opposite direction. Suppose that in one of these parts the water Waves. vertical forces de Partic origin be alt in a line. (70.) Now as the depth of the water is very small compared with the radius of the earth, the horizontal bottom of the water as at the top. Consequently we disturbing force will be very nearly the same at the such a kind that particles which were originally in a in av may conceive the whole motion of the water to be of vertical line remain in a vertical line, although that line n vertical line may have had a motion on the earth's suppo surface. The elevation of the water must be supposed to be produced merely by the approach of different vertical lines, (arising from their difference of horizontal velocities,) and the consequent forcing up of the water between them: the depression, by their separathe space between them. The reader will remark, that tion, and the consequent drop of the water to fill up this is not the most general supposition that we can make in regard to the motion of water, but it is one which is possible, and which is sufficient for our theory. Algebraically speaking, it is not our object to obtain a general solution of the equations applying to fluids, but a particular integral adapted to the case Tides and the length is 1000 times as great as the depth, and suppose that the water is depressed one foot through its whole extent. It is evident that the volume of the water (omitting the factor depending on the breadth of the canal), for which a new place is to be found, is = the length of the canal x1 foot, which = 1000 x depth of the canal xl foot, or depth of the canal x 1000 feet. Consequently the water at one end of the canal, if that at the other end remained unmoved in horizontal place, must have moved 1000 feet, or 1000 times as far Extraneous as the whole vertical motion of any part. The whole of the extraneous vertical forces then which act upon forces, and the particles of the water may be omitted in our investigations. For these forces are of two kinds. One pending on vertical is that which depends upon the acceleration or retardmotion, ation of the particles of water in their upward or may be downward direction: thus, if the water has been raised omitted. 1 foot in 6 hours, the force of which we speak is the pressure which must have acted, in order, by its action continued for 6 hours, to produce a motion of 1 foot. It is clear that this is insignificant in comparison with that force which in the same time has produced a motion of 1000 feet. The force of the other kind is the disturbing force of the sun or moon: now the pressure which this causes among the particles of the water depends not only upon the magnitude of the water depends not only upon the magnitude of the force, but also upon the depth of the column or the length of the canal through which it acts: and, therefore, though its resolved vertical part may be as great as its resolved horizontal part, yet as the vertical part acts only upon a column of water 5 or 6 miles deep (at the utmost), and the horizontal part acts along the whole length of a canal 5000 or 6000 miles long, the pressure among the particles of water which is caused by the former will be insignificant in comparison with that caused by the latter. As regards both these kinds of force then, the vertical force may be put out of consideration. But the same remark does not apply to the vertical force of gravity, nor to the difference in the pressure which it produces, depending on the small tidal difference of elevation of the water at different places. For the force of gravity, as we have Vertical seen in (16.), is nearly forty millions of times as great force of as the sun's disturbing force, and therefore the force of gravity on the small gravity acting on one foot of additional elevation of elevations water would cause as great an additional pressure among the particles of water as the sun's disturbing force acting along a canal whose length is 8000 miles. of water must not be omitted. Investiga (69.) From this we gather that the only forces which we shall have to consider are: the vertical force of gravity, the resolved disturbing forces of the sun and moon in the horizontal direction at each place, and under consideration. (71.) We shall now proceed to put the theory into a mathematical form. In figure 2, let P be the pole of the earth, PA a meridian fixed in space, PS a meridian fixed upon the earth's surface, and therefore travelling away from PA with a uniform angular velocity which we will call n; so that, at the end of the time t, PS will make with PA the angle nt: PT the meridian passing through the original place T of any point of the water whose motion is required. We shall suppose that in the quiescent state of the water the angle SPT would have been, but that in consequence of the tidal disturbance the particle of water is moved to T, and the angle is altered by the variable angle v, so that SPT☎ +v, and APT'= nt+w+v. shall conceive that the original polar distance of the water at T was 0; but that in consequence of the tidal disturbance it is now 0+u. Also we shall put y for the depth of the water at T, supposed quiescent, and w for its tidal elevation above the quiescent state when at T. Here y, in conformity with the supposition made in (65.), is to be considered a function of e only u, v, and w are all to be considered as functions of 0, 7, and t. We (72.) The equation which we shall first form is that which expresses that, however any part of the fluid is tion of the transported by the tidal and rotatory motions, it occupies still the same volume. Take another point U' corequation of responding to another particle of water U, which was originally on the same meridian with T, and whose polar continuity. du do distance was originally 0+de, and therefore is now 0+u+1+· do nearly. The angle APU' has the value which nt+a+v receives when 0+80 is used instead of in forming the value of v: it is therefore dv nt + w +v+ 20 nearly. Also take two points, V' and W', corresponding to two points V and W, whose polar disdo tances were originally 0, and 0+80, but which were upon a meridian making with PS the angle distances of these two points have now respectively the values 0+u+ S and 0+u+(1+· ᅡ( du do +ow. The polar du d, and du dw water) is sensibly equal to r(1+)58, and that from V' to W' is the same. 50. The distance, therefore, from T' to U' (if r be the radius of the spherical or spheroidal surface of the Also the difference of angular distances from PA, between T and V', is sensibly equal to r1+ W' is sensibly the same. and that between U' and Now the area of the surface upon which the water originally stood is seen without difficulty to be rex r sin( 0+: ( dv da: and, therefore, its original volume The area upon which it now stands is not estimated so readily, because its sides are inclined; namely, one pair dv dv Xr.sin 0. 80=sin 0.- and the other pair making with the du 1 du sin 0 do r sin odo do parallel the small angle sensible effect on the area. do These small inclinations do not, however, produce any The reader will perceive this most readily by estimating the area of a parallelogram abed, figure 3, whose sides are inclined to the sides of the rectangle acgf. For the area of abed =abx ad x sin bad-abx ad x sin (90°-bac-daf)=ab x ad x cos (bac+daf) =abx ad x(cos bac. cos daf-sin bac. sin daf) =(ab.cos bac)× (ad. cos daf) × (1-tan bac. tan daf which it is evident differs from ac. af only by the product of ac. af into a small quantity of the second order. The depth of the water, which at the original place was 7, is now altered, from two causes: first, because or, multiplying out these quantities, rejecting products of the small terms, and rejecting the insignificant terms depending on 80, Waves Tides and This is the equation expressing that the same water always occupies the same volume: it is frequently called Tides a the equation of continuity. Waves. fluids. Sect. I Explana- (73.) We now proceed to investigate the equations applying strictly to the motion of the water. We will Laplace tion of the first allude in a few words to the general equations of motion of incompressible fluids, referring to our article Theory pressure of HYDRODYNAMICS, page 276, or to other treatises, for a more detailed exposition. Let the places of the particles Tides. of a fluid at any instant of time be defined by three rectangular coordinates: suppose x, y, z, to be the coordinates of one point, and p the pressure among the particles at that point. The pressure may perhaps be most easily conceived in the following manner :-Suppose a plane to be inserted in the fluid, and suppose the fluid on one side of the plane to be removed; it will be necessary to apply a pressure to this plane, in order to maintain the remaining fluid in the same state (whether of repose, or of motion or change of motion); and the pressure for every square unit of surface on this plane is our quantity p. This pressure ought in strictness to be estimated as a statical pressure by the number of pounds and ounces under the action of gravity, at a given place on the earth's surface: but it will be preferable to take, instead, a quantity which bears a constant ratio to it, namely, the acceleration which this pressure will, by acting for one unit of time, cause in a cubic unit of the fluid. Investiga rectangular equations of the mo tion of (74.) Conceive, then, a small parallelopiped to be inclosed by planes corresponding to the coordinates x, tion of the xh; y, y+k; z, z+1; h, k, and being extremely small. The pressure per unit of surface on that end whose ordinate is x, is p; but the area of that end is kl; therefore the actual pressure is pkl. The actual dp pressure on the other end is kl × the value of p corresponding to x+h, or it is kl×(p+ h The former of dx these tends to push the parallelopiped forward in the direction of r: the latter tends to push it backward. dp The actual pressure, then, tending to push it backward, is hkl; and as the volume of the parallelopiped is dx fluids. dp dr hkl, the accelerating force in the direction opposite to that in which a is measured is If there is acting an dx der dp di dr Equality of pressure in moving fluids re (75.) It is to be remarked, that here we conceive the quantity p to be the same in the three equations which we have just found. That this is true when a fluid is in equilibrium (or that fluids press equally in all directions) there is no doubt: indeed it can be shown to be a necessary consequence of the possibility of quires in- division of the fluid by planes in all directions, and of the perpendicularity of the pressure of the fluid on any vestigation. such plane. It is not so self-evidently certain for fluids in variable motion. Without expressing any doubt of its truth, we wish at the same time to call the reader's attention to the difference of evidence for the principle in the different cases. motion of fluids. Investiga. (76.) To apply this to the movement of the sea: let x be parallel to the axis of revolution of the earth, y tion of the directed towards the first point of Aries, and z at right angles to these. And put 0' for 9+u, w' for w+v: polar equations of the r for the distance of any particle from the earth's centre. Then xr.cos e', y=r.sin 9′.cos nt + ', z=r. sin e′, sin nt +'. Our equations of motion, which depend on x, y, z, must be transformed into others depending on r, e', '. We must; then, instead of dp dr, dp do', dp da' + + dr dy do dy da' dy dr do' dp For then it is clear that we have &c. exactly in the same manner as if p was explicitly expressed in terms of x, y, and z. |