64 



6t HrDRAULICS. 



crements from the contiguous increments on the same side, 

 and the mean of the sagittas is half of the mean of the se- 

 cond fluxions: but the second fluxion of the space may 

 always be expressed by twice its second increment ; the se- 

 cond fluxion of the space is therefore equal to the mean of 

 the second fluxions of thesagittas corresponding to the places 

 of the moveable points, and the space to the mean of tlie 

 sagittas themselves, since the same mode of reasoning may 

 be extended step by step throughout the length of the 

 surface. 



The actions of any two or more forces being always ex- 

 pressed by the addition or subtraction of the results pro- 

 duced by their single operations, it may easily be under- 

 stood that any two or more impressions may be propagated 

 in a similar manner through the canal, without impeding 

 each other, the inclination of the surface, which is the ori- 

 ginal cause of the acting force, being the joint effect of the 

 inclinations produced by the separate impressions, and pro- 

 ducing singly the same force as would have resulted from 

 the combination of the two separate inclinations ; and the 

 elevation or depression becoming always the sum or differ- 

 ence of those which belong to the separate impressions. 

 If then we suppose two similar impulses, waves, or scries 

 of waves, to meet each other in directions precisely oppo- 

 site, they will still pursue their course, but the point, in 

 which their similar parts meet must be free from all horizon- 

 tal motion, since tlie motions peculiar to each, destroy each 

 other : consequently a solid obstacle fixed in a vertical di- 

 rection would produce precisely the same effect on either 

 series, as is produced by the opposition of a similar scries, 

 •and would refleci it in a form similar to that of the opposite 

 series. 



Scholium. The limited elasticity of liquids actually 

 existing produces some variations in the phenomena of 

 waves, which have not yet been investigated; but its effect 

 may be in some degree estimated by approximation. For 

 a finite time is actually required in order for the propagation 

 of any effect to the parts of the fluid situated at any given 

 depth below the surface, and for the return of the impulse 

 or pressure to the superficial pans : so that the summit of 

 every wave must have^ravelled through a certain portion of 

 its track before the neighbouring parts of the fluid can have 

 partaken in the whole effects which its pressure would pro- 

 duce by means of th« displacement of the lower part of the 

 fluid. This cause probably cooperates with the cohesion of 

 the liquid in rounding off any sharp angles which may ori- 

 ginally have existed ; it limits the effect that an increase of 

 depth can produce in the velocity of the transmission of 

 •waves of a finite magnitude, and diminishes the velocity of 



all waves the more u the depth approaches more to Jhfa 

 limit. If the surface was originally in the form of the harmo- 

 nic curve, it may be shown that the force acting at any time 

 on a given point in consequence of the sum of (he results of 

 the forces derived from the effect of a given portion of a wave 

 which has already passed by, will still follow the law of the 

 same curve : but the force will be diminished in the ratio 

 of the arc corresponding to half the space described by the 

 wave while the impulse returns from the bottom, to its sine, 

 the whole distance of the wave being considered as the cir- 

 cumference; and the velocity will be diminished in the sub- 

 duplicate ratio ; but the arc which, when diminished in the 

 subduplicate ratio that it bears to the sine, is the greatest, 

 is that of which the length is equal to the tangent of its ex- 

 cess above a right angle, or an arc of about 70°^, its sine Is 

 .94 and its lengtli a. 8, the subduplicate ratio that of 1 to 

 .57, andthe velocity will be so much less than that which is 

 due to the height : but v^-ith this velocity the wave will de- 

 scribe a portion equal to |^ of its breadth, while the effect 

 descends and reasccnds to the depth concerned ; and sup- 

 posing the velocity with which the impulse is transmitted 

 through the fluid to be equal to that which is acquired by 

 a body falling through a space equal to \m, and calling the 

 depth h, and the breadth of the wave o, while JiJo is- de- 

 scribed by V, 2/1 is described by that which is due to im, or 



hy bv' ( — j i andv being .57l'^/{~J, as .57iv(-) 



'° sis'^> «o '5 *v^ ( —V° ^Aj and 1 .14A ' :i:2^fl ^ m, whence 



hz:.5{a:'m)'. For water, according to Mr. Canton's ex- 

 riments, m is not more than 7 50,000 feet, but we may ven- 

 ture to call it a million ; then if a, the breadth of the wave, 

 were 1 foot, h would be 50, and the velocity nearly 23 feet 

 in a second. If a were 1000 feet, h would be 5000 ; and 

 the addition of a greater depth could not increase the velo- 

 city. Where the depth is given, the correction may be 

 made in a similar manner. For h being in this case given, 

 we must find the arc which is to its sine in the duplicate 

 ratio of the velocity due to the height to the diminished ve- 

 locity, represented by that arc, while that of the impulse 

 propagated in the medium is expressed by twice the depth- 

 Thus if h were 8 feet, and o a foot, the velocity being u, 

 the arc must be to its sine as 256 to rv, and v to 5660 a^ 

 twice the arc to twice the depth and the arc ^3, or in 

 degrees .blv ; but this arc is somewhat more than 6°, and 

 exceeds its sine so little that the velocity is scarcely dimi- 

 nished one thousandth by the compressibility of the water. 

 The ftiction and tenacity of the water must also tend ia 

 some degree to lessen the velocity of the waves. 



