418 



H V D R O D Y N A M I C S. 



HUtory. t ne latter increased with the slowness of their motion. 

 S "~"Y~~"' The valuable labours of Bossut were recompensed by 

 M. Turgot, who established for him in the Louvre a 

 professorship of Hydrodynamics, to which he was ap- 

 pointed in 1775. 



Labours of We llave already seen, that Newton was the first 

 La Place, philosopher who investigated the laws of the motion of 

 Born 1749. waves. His theory was, however, only an approxima- 

 tion to the truth, and, as he himself was aware, was 

 suited only to the hypothesis, that the particles of the 

 fluid ascended and descended vertically in the course of 

 their vibrations. When the ascent and descent is made 

 in curve lines, the velocity of the waves cannot be ac- 

 curately determined by Newton's method. It is only 

 by means of the general laws of the motion of fluids 

 that this subject can be properly treated. M. De La 

 Place was the first who applied this mode of investiga- 

 tion to rectilineal undulations, in the Memoirs of the 

 Academy of Sciences for 1776. This investigation is 

 contained in a separate section, Sur let ondes, publish- 

 ed in his paper entitled Suite des Recherches stir plu- 

 sieurs points du Systeme du Monde. He supposes the 

 water to be shut up in a canal infinitely narrow, and 

 of an indefinite length, but of a constant depth and 

 breadth. He imagines that the wave is produced by 

 immersing a curve in the fluid to a very small depth. 

 The curve being kept in its place till the water has re- 

 covered its equilibrium, it is then drawn out, and 

 waves are formed by the water while it recovers its 

 equilibrium. When the curve is plunged more or less 

 deep into the fluid, the time of the propagation of the 

 waves to a given distance will be always the same, as 

 the oscillations of a pendulum are constant, whatever 

 be the length of the arcs which they describe, provided 

 they are very small. If the depth of the canal is very 

 great, in proportion to the radius of curvature of the 

 curve at its lowest point, the times of the propagation 

 of waves generated by different curves, or by the same 

 curves in different situations, are reciprocally as the 

 square roots of the radii of curvature ; and the veloci- 

 ties of the waves are directly as the same roots. Hence 

 La Place concludes, that the velocity of waves is not 

 like that of sound, independent of the primitive agita- 

 tion of the air. 



Fluugergnes The subject of the oscillation of waves was now ex- 

 un Wave*, amined experimentally by M. Flaugergues, who endea- 

 A.n. 1769. voured to overturn the opinions of Sir Issac Newton. 

 In a memoir on the motion and figure of waves, of which 

 an abstract is given in the Journal des Sfavans for Octo- 

 ber 1789, Flaugergues gives an account of a series of ex- 

 periments which he made upon this subject. He com- 

 bats the opinion of Newton, that waves arise from a mo- 

 tion of the particles of the fluid, in virtue of which they 

 ascend and descend alternately in a serpentine line, 

 while they move from their common centre ; and he 

 attempts to prove, that they are a kind of intumescence 

 raised round the common centre, by the depression 

 which- the impulse has occasioned ; and that this intu- 

 mescence is afterwards propagated circularly from the 

 centre of impulse. A portion of the intumescence, or 

 elevated water, flows, as he conceives, from all sides 

 into the cavity formed at the centre of impulse ; and 

 this water being, as it were, heaped up, produces an- 

 other intumescence, which occasions a new wave, that 

 is propagated circularly as before. M. Flaugergues 

 proceeds to determine the figure of a wave, and gives 

 the equation of it, and also the equation of the curve 

 which the centre of gravity of a vessel describes from 

 the motion of waves. From tlu's theory he deduces 



the conclusion, that all waves, whether great or small/ History, 

 have the same velocity ; whereas Newton made their ve-c S ""Y"*" > ' 

 locity proportional to the square roots of their breadth. 

 In order to examine this result, our author made the 

 following experiment on a branch of the Rhone, shut 

 up at one end to make the water stagnant. Having 

 measured a distance or base of thirty feet, he threw 

 into the water small stones at the end of this base, and 

 he found that the waves which they produced, whe- 

 ther they were great or small, occupied exactly twen- 

 ty-one seconds in moving over the space of thirty 

 feet. 



In the Memoirs of the Academy of Berlin for 1781 La Grange 

 and 1786, and also in his Mecanique Anali/tiquc, M. De Bom J7:;6, 

 La Grange, one of the most distinguished mathemati- Die( * 181 ~ 

 cians of the last century, has endeavoured to determine 

 the oscillation of waves in a canal. He found that it is 

 the same as that which a heavy body would acquire by 

 falling through a height equal to half the depth of the 

 water in the canal. Hence, if the depth of the canal is 

 1 foot, the velocity of the wave will be 5A()5 feet, and, 

 at greater or less depths, the velocity will be as the 

 square roots of the depth, provided it is not very great. 

 If it is admitted, that when waves are formed, the wa- 

 ter is affected only to a small depth, the theory of. La 

 Grange will give tolerably correct results whatever be 

 the depth of the water in the canal, and the figure of 

 its bottom ; but although this supposition is counte- 

 nanced by experience, and derives probability from the 

 viscidity of water, yet La Grange's theory does not har- 

 monize with experiment. Dr U'ollaston observed, that 

 in a place where the depth of the water was said to be 

 50 fathoms, a bore, or large wave, moved at the rate of 

 one mile in a minute ; whereas La Grange's theory 

 gives only 40 fathoms as the depth which corresponds 

 with the velocity. Dr Thomas Young has also obser- 

 ved, that the waves/ or oscillations of water in a cistern, 

 always move with a velocity smaller than that of a 

 body falling through half the depth, but nearly in the 

 same proportion. 



The first engineer who examined experimentally the Experi- 

 motion of water in canals, in reference to the resist- men ts and 

 ances arising from the cohesion of water, and to that formula; of 

 kind of friction of which fluids are capable, was M. ^ Q 7 j 7 

 Chezy, the predecessor of M. Prony, in the direction of 

 the School of Roads and Bridges. Towards the year 

 1775, when he was working with Perronet on the sub- 

 ject of the canal of Yvette, he was anxious to deter- 

 mine from observation and calculation, the relation 

 which subsisted between the declivity and length of a 

 canal, the width and figure of its transverse section, 

 and the velocity of the water which it conveyed. In 

 the course of this investigation, he obtained a very sim- 

 ple expression of the velocity, involving these dif- 

 ferent variable quantities, and capable, by means of a 

 single experiment, of being applied to all currents what- 

 ever. He assimilates the resistance of the sides and 

 bottom of the canal to known resistances, which follow 

 the law of the square of the velocity, and he obtains the 

 following very simple formula, 



V = A-, where g is =16.087 feet, the velocity acqui- 



~ v /3 * 



red by a heavy body after falling one second ; d, the 

 hydraulic mean depth, which is equal to the area of the 

 section divided by the perimeter of the part of the 

 canal in contact with the water ; *, the slope or declivi- 

 ty of the pipe ; and /), an abstract number to be deter- 

 mined by experiment. 





