WAVES. 



as that with wliich they ftruck agakift the obftacle. From 

 this motion, and the motion above-mentioned along E F, 

 arifes a motion along E H, whofe celerity is exprefled by 

 the line E H, which is equal to the line C E ; and by the 

 refleftion, the celerity of the ware is not changed, but it 

 returns along E H, in the fame manner as if, taking away 

 the obftacle, it had moved along E h. 



If from the point C, C D be drawn perpendicular to the 

 obftacle, and then produced, fo that D c (hall be equal to 

 C D, the line H E continued will go through c ; and as 

 this demonftration holds good in all points of the obftacle, 

 it follows, that the reflefted wave has the fame figure on 

 that fide of the obftacle, as it would have had beyond the 

 line A B, if it had not ftruck againft the obftacle. If the 

 obftacle be inclined to the horizon, the water rifes and de- 

 fcends upon it, and fuffers a fritlion, by which the refleftion 

 of the wave is difturbed, and often wholly deftroyed ; and 

 this is the reafon why very often the banks of rivers do not 

 refleft the waves. 



If there be a hole, as I, in the obftacle B L, the part of 

 the wave which goes through the hole, continues its motion 

 direftly, and expands itfelf towards Q Q ; and there is a 

 new wave formed, which moves in a femicircle, whofe 

 centre is the hole. For the raifed part of the wave, which 

 firft goes through the hole, immediately flows down a little 

 at the fides ; and, by defcending, makes a cavity which is 

 furrounded with an elevation on every part beyond the hole, 

 wliich moves every way in the fame manner as was laid down 

 in the generation of the firft wave. 



In the fame manner, a wave to which an obftacle, as A O, 

 is oppofed, continues to move between O and N, but ex- 

 pands itfelf towards R, in a part of a circle, whofe centre 

 is not very far from O. Hence, we may eafdy deduce 

 what muft be the motion of a wave behind an obftacle, 

 as MN. 



Waves are often produced by the motion of a tremulous 

 body, which alfo expand themfelvos circularly, though the 

 body goes and returns in a right line ; for the water which 

 is raifed by the agitation, defcending, forms a cavity, which 

 is every where furrounded with a rifing. 



Different waves do not difturb one another, when they 

 niove according to different direftions. The reafon is, that 

 whatever figure the furface of the water has acquired by the 

 motion of the waves, there may in that be an elevation and 

 depreffion ; as alfo fuch a motion as is required in the motion 

 of a wave. 



To determine the celerity of the waves, anotlier motion, 

 analogous to their's, muft be examined. Suppofe a fluid 

 in the bent cylindric tube E H [Jig. 13.) ; and let the fluid 

 in the leg E F be liigher than in the other leg by the dif- 

 tance / E ; which diftance is to be divided into two equal 

 parts at i. The fluid, by its gravity, delcends in the leg 

 E F, while it ..fcends equally m the leg G H ; f o that when 

 the furf.ice of the tiuid is arrived at ;, it is at the fame height 

 in both legs ; which is the only pofition in which the liquid 

 can be at r'?ft : but by the celerity acquired in defcending, 

 it continues its motion, and afcends higher in the tube G H ; 

 and in E F is depreffed quite to /, except fo much as it is 

 hindered by the friftion againft the fides of the tube. The 

 fluid in the tube G H, which is higher, alfo defcends by its 

 gratvity, and fo the fluid in the tube rifes and falls, till it has 

 loft all its motion by the friftion. 



The quantity of matter to be moved is the whole fluid in 

 the tube ; the moving force is the weight of the column / E, 

 whofe height is always double the diltance E i ; which dif- 

 tance, therefore, increafes and diminiftics in the fame ratio 

 with the moving force. But the diftance E ;' is the fpacc to 



be run through by the fluid, in order to its moving from thi- 

 pofition E H, to the pofition of reft ; which fpacc, theru- 

 fore, is always as the force continually afting upon the 

 fluid : but it is demonftrated, that it is on this account thai 

 all the vibrations of a pendulum, ofcilkting in a cycloid, are 

 ifochronal ; and, therefore, here alfo, whatever be the ine- 

 quality of the agitations, the fluid always goes and returns 

 in the fame time. The time in which a fluid thus agitated 

 afcends, or defcends, is the time in which a pendulum vi- 

 brates, whofe lengLh is equal to half the length of the fluid 

 in the tube, or to half the fum of the hues E F, F G, E K. 

 This length is to be meafured in the axis of the tube. Sec 

 Pendulum. 



From thefe principles, to determine the celerity of tlie 

 waves, we muft confider feveral equal waves following oi.t 

 another immediately ; as A, B, C, D, E, F, {Jig. 14. 

 which move from A towards F : the wave A has run ii.» 

 breadth, when the cavity A is come to C ; which cannot 

 be, unlefs the water at C afcends to the height of the top 

 of the wave, and again delcends to the depth C ; in which 

 motion, the water is not agitated fenfibly below the line hi ; 

 therefore, this motion agrees with the motion in the tube 

 above-mentioned ; and the water afcends and defcends, that 

 is, the wave goes through its breadth, while a pendulum of 

 the length of half B C performs two oicillations, or while a 

 pendulum of the length BCD, that is, four times as long 

 as the firft, performs one vibration ; fince the times in wliich 

 pendulums of different lengths perform their vibrations are 

 as the fquares of their lengths. [ See Vibration. ) There- 

 fore, the celerity of the wave depends upon the length of 

 the line BCD; which is greater, as the breadth of the 

 wave is greater, and as the water defcends deeper in the 

 motion of the waves. In the broadeft waves, which do not 

 rife high, fuch a line as B C D do^•s not much differ from 

 the breadth of the wave ; and in that cafe a wave moves its 

 breadth, while a pendulum, equal to that wave, ofcillates 

 once. Hence, if the breadth of a wave be 39.1 196 inches, 

 ( this being the length of a pendulum which vibrates feconds, ) 

 then that wave will move on at the rate of 39.1196 inches 

 per fecond of time; that is, at the rate of 195 feet per 

 minute, nearly. 



In every equable motion, the fpace gone through in- 

 creafes with the time and the celerity ; wherefore, multiply- 

 ing the time by the celerity, you have the fpace gone 

 through ; whence it follows, that the celerities of the waves 

 are as the fquare roots of their breadths : for as the times in 

 which they go through their breadths are in that ratio, the 

 fame ratio is required in their celerities, that the produfts 

 of the times, by their celerities, may be as the breadths of 

 the waves, which are the fpaces gone through. 



Dr. Young is of opinion, that fir Ifaac Newton's ana- 

 logy, refultiiig from a comparifon of a wave with the ofcil- 

 lation of a fluid in a bent tube, is too diftant to admit our 

 founding any demonftration upon it. Legrange, he fays, 

 has inveftigated the motions of waves in a new and improved 

 manner ; and Dr. Young has alfo demonilrat. d a theorem 

 fimilar to his, but, as he apprehends, more general and ex- 

 plicit. From thefe premifcs it appears, that, fuppoCng the 

 fluids concerned to be infinitely elaftic, thai is, abfolutely 

 incompreflible, and free from friftion of all kinds, any fmall 

 impulfe communicated to a fluid would be trarfmittcd every 

 way along its furface, with a velocity equal to t.Sat which a 

 heavy body would acquire in faUing through half the depth 

 of the fluid ; and he concludes, from oblervaiion and ex- 

 periment, that where the elevation or depreflion of the fur- 

 face is confiderably estenfive in proportion to the depth, 

 the velocity approKchcs nearly to that which is thus deter- 

 mined, 



