37 



ACCELERATION OF TIDES. 



ACCELERATION OF TIDES. 



38 



since A x M is half of A P N M, and the latter contains a square foot for 

 every foot of length which would hare been described if M N had been 

 the velocity from the beginning, we must infer that the length described 

 by a uniformly accelerated motion from a state of rest, is half that which 

 would have been described, if the body had had its last velocity from 

 the beginning. 



If the body begin with some velocity, instead of being at rest, the 

 space which it would have described from that velocity must be added 

 to that which, by the last rule, it describes by the acceleration. 

 Suppose that it sets out with a velocity of 10 feet per second, and 

 moves for 3 seconds uniformly accelerated in such a manner as to gain 

 6 feet of velocity per second. Hence it will gain 18 feet of velocity, 

 which, had it had at the beginning, would have moved it through 

 18 x 3 or 54 feet of length, and the half of this is 27 feet. This is 

 what it would have described had it had no velocity at the beginning ; 

 but it has 10 feet of velocity per second, which, in 3 seconds, would 

 move it through 30 feet. Hence 30 feet and 27 feet, or 57 feet, is the 

 length really moved through in the 3 seconds. 



Similarly we can calculate the effects of a uniform retardation of 

 velocity. This we can imagine to take place in the following way. 

 While the body moves uniformly from left to right of the paper, let 

 the paper itself move with a uniformly accelerated velocity from right 

 to left of the table. Let the body at the beginning of the motion be at 

 the left edge of the paper, and let that edge of the paper be placed on 

 the middle line of the table. Let the body begin to move on the 



ll:r uniformly 10 inches per second, and let the paper, which at 

 the beginning ia at rest, be uniformly accelerated towards the left, so 

 as to acquire 2 inches of velocity in every second. At the end of 3 

 seconds, the body will be at B, 30 inches from A, but the paper itself 

 will then have acquired the velocity of 6 inches per second, and will 

 have moved through the half of 18 inches or 9 inches ; that is, A c will 

 be 9 inches. Hence the distance of the body from the middle line will 

 be C B, or 21 inches. Relatively to the paper, the velocity of the body 

 is uniform, but relatively to the table, it has a uniformly retarded 

 velocity. At the end of the fourth second, it will have advanced 40 

 inches on the paper, and the paper itself will have receded 16 inches, 

 giving 21 inches for c B. At the end of the fifth second, A B will be 60 

 inchec, A C 25 inches, and c B 25 inches. At the end of the sixth 

 econd, A B will be 60 inches, A c 36 inches, and B c 24 inches, so that 

 the body, with respect to the table, stops in the sixth second, and then 

 begins to move back again. We can easily find when this takes place, 

 for, -iuce the velocity on the paper is 10 niches per second, and that of 

 the paper gains 2 inches hi every second, at the end of the fifth second 

 the body will cease to move forward on the table. At the end of 10 

 second* it will have returned to the middle line again, and afterwards 

 will begin to move away from the middle line towards the left. At the 

 the twelfth second, it will have advanced 120 inches on the 

 paper, and the paper will have receded 144 inches, BO that the body will 

 be 24 inches on the left of the middle line. 



The general algebraical formula; which represent these results are as 

 Let a be the velocity with which the body begins to move, t 

 the number of seconds elapsed from the beginning of the motion, g the 

 velocity acquired or lost during each second. Then the space described 

 in a uniformly accelerated motion from rest is ^yt i ; when the initial 

 velocity is a, the space described in an accelerated motion is at + i gt 1 , 

 and in a retarded motion the body will have moved through a t - \ g C 

 in the direction of it* initial velocity if a t be greater than 4</< 8 , or 

 will have cutne back and passed its first position on the other side by 

 ^yl--at, if at be less than 4 jr < '. In the last case it continues to 



in the direction of ita initial velocity for seconds and proceeds 

 in that direction through the space 4 



a 



iurther explanation as to velocities which ore accelerated or 

 retarded, but not uniformly, see VELOCITY. 



ACCELERATION AND RETARDATION OF TIDES are certain 

 deviations of the timtvi of consecutive high-water at any place from 

 those which would be observed if the tides occurred after the lapse of 

 a mean interval. The interval between the culmination of the moon, 

 or the occurrence of her principal phases, and the nearest time of high- 

 water, U al*o called the retardation of the tide. 



The tide* are caused by the attractions exercised by both the sun 

 and moon on the water* of the earth ; but the effect produced by the 

 moon exceeds that which is produced by the sun, and the difference is 

 uch, that the phenomena of the tides depend principally on the 

 former. The mean interval between two consecutive returns of the 

 moon, above and below the pole, to the meridian of any place, 



50m. 28'32s. ; and since, neglecting all causes of irregularity, two lunar 

 high-tides occur in that time, the mean interval between two con- 

 secutive lunar tides should be 12h. 25m. 14'16s. ; while the mean 

 interval between two consecutive solar tides should be 12k. Hence, if 

 at the time of a conjunction or opposition of the sun and moon, the 

 high tides which are produced by the actions of the luminaries sepa- 

 rately were coincident, the next lunar tide would be retarded with 

 respect to the next solar tide, by 25m. 14'16s., that is, by the excess of 

 half a lunar day above half a solar day. These retardations continuing 

 daily, the lunar high-water would coincide, at the time of quadrature, 

 with the solar low-water, and thus produce the neap or diminished 

 tides; after which, the like retardation continuing, the solar and lunar 

 high-waters would again become coincident at the times of syzygy, and 

 so on. The observed daily retardation of the lunar high-tides varies 

 however according to the position of the moon with respect to the sun, 

 to the moon's declination, and to the distance of that luminary from the 

 earth. At Brest, when the sun and moon are in conjunction or in 

 opposition, at the summer or winter solstice, the retardation is equal to 

 40m. 51 '69s., and at the time of the equinoxes 37m. 38'15s. Again, 

 when the sun and moon are in quadrature at either solstice, the retar- 

 dation is Ih. 7m. 27'49s., and at the time of the equinoxes Ih. 

 23m. 16'34s. 



If the earth were a solid of revolution, and were covered by the sea, 

 the high tides produced by the sun and moon separately would, at any 

 place, occur at the instants when those celestial bodies are on the 

 meridian of the place ; but such is not the fact in the actual condition 

 of the earth ; and local circumstances produce, at different ports, great 

 differences in the intervals between the culmination of the sun or moon 

 at the time of high-water, even on the days when the luminaries are in 

 conjunction or opposition. The interval between the instant that the 

 sun passes the meridian of a place and the occurrence of the solar high- 

 tide, is found to be greater than the interval between the transit of the 

 moon and the occurrence of the lunar high-tide ; and this acceleration, 

 as it is called, of the lunar tide, is with much probability ascribed by 

 Dr. Young to a difference in the resistances experienced by the waters 

 on account of the different velocities which are communicated to them 

 by the separate actions of the sun and moon. 



It should be observed however that at Ipswich the time of high- 

 water is nearly coincident with the time at which the moon passes the 

 meridian of that port ; and both at Glasgow and Greenock, the high- 

 tide generally precedes the transit (Mr. .Mackie's ' Report,' at the 

 seventh meeting of the British Association) ; but such phenomena are 

 of rare occurrence, and at almost every place the high-tide occurs some 

 time after the moon has culminated. 



From a series of observations continued during sixteen years, at 

 Brest, La Place, taking the excesses of the height of the evening tide 

 above that of the morning for the day of syzygy, for the day preceding 

 it, and for four days following it, has ascertained that at the syzygies 

 which occur about the vernal and autumnal equinox the highest tides 

 at that port take place V48013 days after the instant of the conjunction 

 or opposition ; and at the syzygies which occur about the summer and 

 winter solstices they take place 1'54684 days after conjunction or oppo- 

 sition. Again, taking the excesses of the height of the morning tide 

 above that of the evening for six days, as above, he ascertained that 

 at the quadratures which occur about the equinoxes the highest tides 

 take place 1 '50964 days after the instant of quadrature, and at the 

 solsticial quadratures 1 '51269 days after such instant. 



Mr. Airy (' Tides and Waves,' Encycl. Metrop.) observes that these 

 retardations cannot be accounted for by delays in the trausmission of 

 the tide-waves, since no cause for such delay can be imagined to exist 

 in the Southern Ocean, where the waves are formed ; and it is known 

 that the time of high-water at Brest is only fifteen hours later than at 

 the Cape of Good Hope : he conceives, therefore, that the retardation 

 must be ascribed to friction. By taking the means of the daily retar- 

 dations of the morning and evening tides at Brest, La Place found that 

 at the equinoctial syzygies such mean retardation was equal to 37m. 

 38s.; at the solsticial syzygies, 40m. 52s; at the equinoctial quadra- 

 tures, 83m. 16s. ; and at the solsticial quadratures, 67m. 27s. 



From a series of observed heights of the tides, Sir John Lubbock 

 has detei-miued that the highest tides occur at London 2'013 days after 

 the conjunction or opposition of the sun and moon; and at Liverpool, 

 1'68 days. ('Phil. Trans.' 1831, 1836.) Also, from the observed 

 heights, Dr. Whewell has found that the highest tides occur at Bristol 

 1-667 days after the syzygies; and at Dundee 1'639 days. ('Phil. 

 Trans.' 1838, 1839.) On the supposition that the mean retarda- 

 tion of the tide at London at the times of syzygy is 2'459 days, Mr. 

 Airy has computed the moon's true hour-angle west of the meridian, 

 at the time of high- water, for every half hour's difference in the time 

 of her transit ; and from the table it appears, that when the moon 

 passes the meridian of London at noon (that is, at the time of 

 conjunction), that angle, in time, is Ih. 57m. 17s. ; when it passes 

 at 3 P.M., the angle is Ih. 10m. 45s.; at 6 P.M., or at quadrature, 

 Oh. 41m. 17s.; and at 9 P.M., Ih. 55m. 29s. The hour angle is the 

 greatest at 104 p - M - when it is equal to 2h. 9m. 65s. ; and at 114 p - M -> 

 or nearly at the time of opposition, it is 2h. 3m. 9s. : all these times are 

 found to agree very nearly with the results of observation. From 

 such results it is ascertained that, on the days following the times of 

 ayzygy and quadrature, the intervals between the time of the moon s 



