

781 



WAVES AND TIDES. 



WAVES AND TIDES. 



762 



into a rise, to which another rapid descent succeeds; so that there 

 seem to be two tides, or elevations of the water, in the upper part of 

 the canal, corresponding to one elevation at the mouth. 



Tho value of , or the velocity of the particles of water, is found 



ill 



also to contain the sines and cosines of the angles above mentioned ; 

 and, substituting in these the greatest positive and the greatest 

 negative values of the elevation, it is found that the velocity corre- 

 sponding to the first of these values that is, at the top of the wave 

 is less than the velocity corresponding to the other ; but the motion, 

 in the first case, is up the canal, and in the other down it, and these 

 are nearly the same as the greatest velocities of the water : conse- 

 quently, the velocity of the flow of the wave in the canal is less than 

 that of the ebb. The preceding conclusions relate to the case in which 

 the water was at rest in the canal previously to the formation of the 

 wave : in the event of the water having a general movement towards 

 the sea, the time in which the wave rises, or the time from low-water 

 to high-water, is still less than the time of the descent; but the 

 difference between the two times is greater than in the former case. 



If a section of the bed of the canal, instead of being rectangular, has 

 the form of an isosceles triangle, the investigations show that the 

 velocity of tho wave would be equal to that of a wave in a rectangular 

 bed whose depth is equal to half the perpendicular of the triangle. If 

 the section were a parabola, the velocity would be two-thirds of that 

 which the waves would have in a rectangular bed of equal breadth 

 and depth. 



When the water, still supposed to be in a canal of uniform breadth 

 and depth, is disturbed by extraneous forces, as the attraction of the 

 sun or moon, the term P in the equation of equal pressure is conceived 

 to consist of two, one represented by H sin. (itmx) for the horizontal 

 intensity of such force in the direction of x, and the other by 



d?x 

 cos. (it mx) for the vertical intensity; and the equation for 



being then satisfied by the equation x = <J>" (y) sin. (it mx), in which 

 <l>" (y) represents the second differential coefficient of a function of y, 

 there is obtained a value of X at the surface of the fluid in terms of 

 sin. (it mx), and a value of the height above the level of still water 

 in terms of cos. (itmx). The wave thus indicated depends upon the 

 continuance of the actions of the extraneous disturbing forces, and is 

 designated by Mr. Airy the forced tide-wave. This wave, he observes, 

 would cease to exist if those forces were to cease ; but other waves 

 resulting from the previous action would still continue to exist, and 

 these he distinguishes by the name of free tide-waves. If the canal be 

 supposed to surround the earth at the equator, the length of the forced 

 tide-wave is equal to half the circumference of that great circle ; and 

 from the expressions for x and Y, it appears that the effect of the 

 vertical disturbing forces on the phenomena of the tides is insignificant, 

 almost the whole sensible effect being due to the horizontal force. 



Taking into account the effects of friction, which may be considered 

 as a horizontal retarding force proportional to the velocity, and which 



may consequently be represented by / ~: ; the Talue o{ x contains 



terms involving the sines and cosines of angles represented by it mx 

 and i ( + q *, and the expression for the vertical elevation contains the 

 sine anoTcosine of itmx. The analytical expression arising from the 

 introduction of this additional perturbation indicates the fact that 

 the highest tides take place later than the times at which the disturb- 

 ing forces arising from the action of the sun or moon are the greatest; 

 and this circumstance gives to the wave theory an important advantage 

 over those of Newton and La Place ; for in both these theories the 

 greatest tides take place when the force is the greatest. 



In the case of a canal bounded at both extremities, the expression 

 for x, the horizontal disturbance of a particle, is found to consist of 

 two parts, ono of which is the horizontal movement due to the disturb- 

 ing forces, and the other a combination of free tide-waves, probably 

 caused by reflexions of the forced tide-waves from the opposite ends of 

 the canal. When a canal so bounded is of small extent, the horizontal 

 u of the particles is found to be the greatest in the middle of its 

 length, and to diminish gradually to the ends, where it vanishes. 

 There is proved to be no variation of level in the middle of the length, 

 and the variation in other parts is proportional to the distance from the 

 middle, the elevation at one end taking place at the same time as the 

 depression at the other. It results, also, that the greatest horizontal 

 and vertical displacements of the particles take place at the same time ; 

 whereas in other circumstances, from the circular or elliptical motions 

 of the particles, the greatest horizontal displacements take place when 

 the vertical displacements are zero, and vice versd. 



In a deep gulf open to the sea at one end and closed at the other, 

 and in which the waters have a tidal fluctuation, the termination of 

 the flow upwards takes place at the mouth a considerable time after 

 high- water ; but near the bottom of the gulf the difference between 

 the times is very small. When a tide-wave is propagated up a river, 

 the analysis shows that the vertical elevations of the wave, and also the 

 horizontal motion of the particles of water, diminish continually as the 

 wave advances : also the direction of the tide-current changes sooner 

 after the instant of high-water than it would if friction were not con- 



sidered. When a rivtr runs on a declivity towards the sea, the latter 

 being affected by tides, it is shown that the low-water at certain points 

 up the river may be higher than the level of high-water on the sea. 



The theory, of which a brief outline has just been stated, applies to 

 what are called negative waves by a mere change in the sign of tha 

 coefficients of the trigonometrical factors. These waves are depressions 

 below the general surface of the water, and, like the others, they have 

 a progressive motion. Such waves, for example, are those which are 

 formed by the paddles of a steam-boat. 



All the theories concur in showing that the difference between the 

 diurnal and semidiurnal tides is great in middle latitudes, and small at 

 the equator and poles ; and in this respect they are at variance with 

 the actual phenomena. From observations it is found that this differ- 

 ence is aa great at certain places near the equator as near the latitude 

 of either tropic : it has also been found to be great at Petropaulowski 

 and in Norfolk Sound, while in Europe it is small. It has been 

 attempted to account for the latter circumstance by assuming that 

 each tide-wave in this part of the world is composed of two, which 

 flow towards the same place in opposite directions at intervals of about 

 twelve hours. It is supposed that the semidiurnal waves of these 

 tides, being in the same state or phase, produce together a like effect, 

 but that the diurnal waves are in opposite states ; so that the superior 

 high tide of one wave coinciding with the inferior high tide of the other, 

 they together produce a mean height of water differing but little from 

 that of the united semidiurnal tides. 



We cannot here enter into the details of the investigations relating 

 to the theories of the oscillations of water, or the discussion of the 

 experiments which have been mado on waves in artincial canals, the 

 methods of making observations on tides, and accounts of the par- 

 ticular tides in rivers and seas; but the experimental researches of 

 Mr. Scott Russell have made so important an accession to our know- 

 ledge of waves, in its relation to practical as well as theoretical science, 

 that this article would be defective without a summary of their results. 

 The details of his experiments will be found chiefly in the ' Trans- 

 actions of the Royal Society of Edinburgh,' vol. xiv., and in the 

 ' Report of the Seventh Meeting of the British Association for the 

 Advancement of Science." 



At the time when Mr. Russell's hydrodynaniical researches were 

 commenced, the celestial mechanics of the tides, as we have seen in 

 the preceding portion of this article, had been analysed and explained 

 in a manner satisfactory both to astronomers and mathematical 

 physicists, but a great variety of considerations relating to the pro- 

 pagation of the tides along the surface of the globe constituting their 

 terrestrial mechanism remained without explanation. The solar 

 and lunar attraction having generated the tides, exercise little or no 

 influence over the subsequent propagation of them. It is not until 

 50 or 60 hours after their creation that the tides reach our shores, 

 having moved in the interval in every possible direction, and with 

 every velocity from 10U down to 10 miles an hour. " This moving 

 elevation of fluid," in the words of the committee on waves, appointed 

 by the British Association in 1836, " may be conveniently designated 

 a wave, and ite history will be the history of the tidal it-arc ; but to 

 confer upon it the name of wave does not imply that its laws are 

 those which belong to any other similar elevation with which we are 

 acquainted. It was necessary to investigate the nature of this tide- 

 wave to examine the hydrodynamical mechanism by which it ig 

 transferred from one place to another to determine the laws which 

 regulate its form and its velocity to ascertain if any relations exist 

 between the form and dimensions of its bed, and its own form and 

 rate of transference. These and many similar points," including also 

 the effect of the wind upon the tide-wave, " were still unknown." 

 Laplace, Lubbock, and Whewell had severally pointed out how much 

 was required to be known, and the last had shown that a great number 

 of curious facts in fluid motion had been established by the tide 

 researches, some of which had been discussed and others instituted 

 and pursued by himself, of which he expressed a hope that the theory 

 of hydrodynamics would one day be able to render a reason. 



Such having been the condition of science on the subject when Mr. 

 Russell began his inquiries, the following is a condensed statement, 

 but nearly in the words of the Committee, of the " General Results" 

 he obtained, and which have eventually been found to possess much 

 more than the value which had been anticipated. 



1. The existence of a great primary wave of fluid, differing in its 

 origin, its phenomena, and its laws, from the undulatury and oscillatory 

 waves which alone had been investigated previous to the researches of 

 Mr. Russell, have been confirmed and established. This wave was first 

 observed by him in 1834. 



2. The velocity of this wave in channels of uniform depth is inde- 

 pendent of the breadth of the fluid, and equal to the velocity acquired 

 by a heavy body falling freely by gravity through a height equal to 

 half the depth of the fluid, reckoned from the top of the wave to the 

 bottom of the channel. 



3. The velocity of this primary wave is not affected by the velocity 

 of impulse with which the wave has been originally generated, neither 

 does its form or velocity appear to be derived in any way from the form 

 of the generating body. 



4. This wave has been found to differ from every other species of 

 wave hi the motion which is given to the individual particles of the 



