777 



WAVES AND TIDES. 



WAVES AND TIDES. 



7/8 



Peculiarities in tides, arising from the interference of tide-waves, 

 occur in many different places. In the middle of the North Sea there 

 jj a considerable space within which the tide produced by the tide- 

 waves coming from the north and south takes places at one time. And 

 Dr. Whewell states, on the authority of the late Captain Hewett, that 

 about the Ower Shoal there is no sensible rise of the tide till 3 hours 

 after the time of low- water; but when the ebb stream has nearly 

 ceased, there ia a sudden rise of 6 or 6 feet ; so that nearly the whole 

 rise of the tide occurs in the last three hours. 



In 1740 the Acade'mie des Sciences offered a prize for the best 

 memoir on the theory of tides : and the paper by Daniel Bernoulli on 

 the flux and reflux of the sea shared it with those of Euler and Mae- 

 laurin. In that paper it is assumed lhat the water is kept in equilibrio 

 between the attractions of its particles towards the earth's centre of 

 gravity and the disturbing forces exercised by the sun and moon ; and 

 though the results of that theory are found to differ greatly from the 

 observed phenomena, the theory itself is deserving of attention, since 

 the analytical expressions which have been obtained by it first exhi- 

 bited the several phenomena distinctly from one another : those ex- 

 pressions consequently became guides to the observer or experimenter 

 in his efforts to ascertain the true values of the particular effects which 

 they represented. 



The attraction exercised by the solid nucleus of the earth on a 

 particle of water at any distance from its centre, being considered the 

 game as it would be if all the matter of the nucleus existed in that 



centre, is represented by , E being the mass of the earth and r 2 the 



square of the distance of a particle from the centre. But if x, >/, and z 

 are rectangular co-ordinates of a particle, the centre of the earth being 

 the origin, we have r ! =3J 1 + y 3 + s 1 ; and the partial differentials of the 



expression ? , relatively to y, y, and z, represent the effects of 



!.- + >/ + # 



that attraction upon a particle in the directions of the three axes. If 

 the attractions of the particles of water for each other are taken into 

 consideration, there must be determined the attraction exercised upon 

 a particle by all the water between the spherical nucleus and the ex- 

 terior surface (supposed to be spheroidal) of the surrounding fluid, and 

 the expression for this attraction must be added to that for the solid. 

 The disturbing force of the sun or moon upon a particle of water is 



represented by , s being the mass of the celestial body and R the 



distance of the particle of water from it ; and the partial differentials 

 of that expression relatively to x, y, and ; give the values of the attrac- 

 tion in the direction of the coordinate axes : but the disturbing force 

 exercised by the sun or moon on a particle of water being equal to the 

 difference between its attraction on the particle and its attraction 

 on the centre of the earth the latter, which ia represented by 



-; (D being supposed to be the distance between the centres of the 



earth and celestial body), is subtracted from the attraction exercised on 

 the particle in the direction of one of the coordinate axes, supposed to 

 be parallel to the line joining those centres, in order to have that 

 difference. The attracting forces of the earth in the directions of the 

 three axes being subtracted from the disturbing forces of the sun or 

 moon in the same directions, there remain three terms which are 

 usually represented by x, 1, and z. And since it has been demon- 

 strated by mathematicians that when a body is in equilibrio under the 

 action of attracting forces, the expression n:lx + \dy + zdz is an exact 

 differential; the form of the surface of equilibrium is determined by 

 making the integral of the expression constant. 



The resulting equation being found to correspond with the general 

 equation to a spheroid, a comparison of like terms in the two equations 

 gives the values of the constants which enter into the former. If r 

 represent the mean distance of the spheroidal surface of the water from 

 the centre of the earth, and + h represent the distance of any point 

 on that surface above or below the mean level ; then 3? + y* + $ = (>+ A)-' 

 at the surface ; and the determination of h for any place gives at that 

 place the height of the water above, or its depression below the mean 

 level 



Uniting the effects of the solar and lunar disturbances by simply 

 adding them together, since the disturbing forces are very small com- 

 pared with the force of gravity ; and introducing, in place of the 

 rectangular coordinates, angles which depend on the longitude and 

 latitude of a station, with the right ascension and declination of the 

 gun and moon, the value of the term j; h may be shown to consist of 

 three parts : one of these depends on the variation of the decimation 

 of the sun and rnoon, and indicates a slow tide which goes through its 

 change* in about fourteen days; the second depends on the hour 

 angles both of the sun and moon, and indicates two tides which go 

 through their changes in a solar and a lunar day respectively. These 

 being combined, there is produced a diurnal tide, the highest state of 

 which should precede, at a variable interval, the moon's culmination 

 between the times of passing from sy/ygy to quadrature, and should 

 f >llow it between the quadratures and syzygies. It has been found, 

 however, that the observed accelerations and retardations, and also the 

 absolute elevations of the water, in very few coses agree with the 

 result* of the theory. [ACCELERATION AXD HETAUDATIOX OF TIDES.] 



The third part depends upon the doubles of the hour angles just 

 mentioned, and consequently indicates two semi-diurnal tides, which 

 being combined constitute one such tide, whose highest state is 

 variable. The nature of the expression shows that the semi-diurnal 

 tide should be the greatest at the equator, and should diminish till it 

 vanishes at the poles : it denotes also that it is greatest at new or full 

 moon, and least at the quadratures. The theory moreover indicates 

 that the difference between two consecutive tides ought to be very 

 considerable in Europe ; whereas they are known to be nearly equal 

 to one another. Both Newton and Bernoulli endeavoured to explain 

 this circumstance by the hypothesis of a general oscillation of the sea, 

 in consequence of which the highest tide gives to the lowest a quantity 

 equal to the difference between them ; but the researches of La Place 

 have shown that, even with such oscillations, the two tides could not 

 (according to the theory) be equal unless the sea were everywhere 

 equally deep. 



Euler, departing from the hypothesis that the. sea is always in equi- 

 librio under the action of the sun and moon, endeavoured to introduce 

 the subject of fluid oscillations in his theory of the tides ; but the 

 laws of undulation were not then known, and Euler assumed that a 

 molecule of the sea in motion endeavours to regain the position which, 

 in a state of equilibrium, it would occupy in a vertical line with a 

 force proportional to its vertical distance from that position. 



The theory adopted by La Place, in which there are taken into con- 

 sideration the laws of the motion of fluid molecules when acted on by 

 attracting forces, was a great improvement on that of the mathema- 

 ticians before mentioned ; and it is found to produce a more near 

 agreement with the observed phenomena. The elaborate investigations 

 of La Place will be found in the ' Me'moires de 1'Academie des Sciences' 

 for the years 1775, 1776; and in the first and fourth books of the 

 ' Me'canique Celeste." They are also given, so far as contained in the 

 first book, in the late Dr. Thomas Young's ' Elementary Illustrations 

 of the Celestial Mechanics of Laplace ; ' Lond., 1821. As in the former 

 theory, the solid nucleus of the earth is supposed to be entirely 

 covered with water of uniform depth ; and the investigations com- 

 mence with the proof (' Mi5c. Ccl.', liv. i., ch. 8) that any portion of the 

 water, however its place may be changed, will always retain the same 

 volume. The equation expressing this law is called the equation of 

 continuity. 



A very small parallelopiped of water within that which covers the 

 solid nucleus of the earth is acted upon by accelerative forces arising from 

 pressures estimated in the directions of three rectangular coordinate 

 axes whose origin is at the centre of the earth : the first is supposed 

 to be parallel to the axis of rotation, and the others in the plane of the 

 equator : one being directed to the equinoctial point and the other at 

 right angles to that direction. The pressures are supposed to arise 

 from the attraction of the earth, from the angular velocity of its rota- 

 tion, and from the disturbing forces, and to tend towards the origin of 

 the coordinates. 



These pressures, which are expressed by partial differential coeffi- 

 cients relatively to x, y, and 2, in the coordinate axes, are subtracted 

 from the accelerative forces anting from the attraction of the earth, 

 and the perturbations exercised by the sun or moon, by which the 

 molecule would be made to recede from that origin ; and the differences 



in the directions of the axes are represented by rf "' r , ^X. and . 



dt- d(*' dP 



In these equations of motion the partial differential coefficients 

 representing the pressures are transformed into others depending on 

 the distance of the molecule from the centre of the earth, and on its 

 latitude and longitude; while the perturbations of the sun or moon in 

 the directions of the coordinate axes are expressed in terms of the right 

 ascension and declination of the disturbing body, and also of the 

 distances of the latter from the particle disturbed and from the centre 

 of the earth. The result is that the expression for the altitude of a 

 molecule of water above the mean level, in consequence of the perturba- 

 tion produced by the sun or moon, consists of three parts (' Mc"c. C^l.', 

 lib. iv., c. 1) ; the first does not depend on the rotation of the earth, 

 and indicates a tide which goes through its changes in a long period ; 

 it may consequently be disregarded. The second depends on that 

 rotation and on the hour angle of the disturbing body : it indicates the 

 diurnal tides, or those which take place when the celestial bodies are on 

 or near the meridian, above the horizon ; and which follow one another 

 at intervals of twenty-four hours for the sun, and about 24h. 50m. for 

 the moon. The third depends on an angle equal to the double of that 

 on which the second depends; and consequently it represents the semi- 

 diurnal tide. 



But the subject of waves and tides has been treated in conformity to 

 the theory of undulations by Mr. Airy, the astronomer royal, in a 

 valuable essay originally published in the ' Kncycloptcdia Metropolitan ;' 

 the investigations, though admitting of general application, are par 

 ticularly adapted to the phenomena of tides in rivers and arms of the 

 sea; and they are conducted by an analysis within the reach of persons 

 acquainted with the ordinary processes of the differential and integral 

 calculus. 



An in the theory of La Place, there is formed an " equation of con 

 tinmty," which is founded on the equality of a rectangular parallelo- 

 piped of water at rest, to the obli<jtu- panuldlopiped formed, when the 



