218 LECTURE XXIII. 



or depression is to the whole length of the tube. Hence it follows, that if 

 two such tubes were united below, so as to form a single bent tube, the' 

 vibrations might take place in the whole compound tube, in the same 

 manner, and in the same time, as in each of the separate tubes ; nor would 

 the effects be materially altered if any part of the middle of the tube were 

 in a horizontal or in an oblique direction, provided that the whole length 

 remained unaltered. In such a tube, also, all vibrations, even if of con- 

 siderable extent, would be performed in the same time, and would long 

 remain nearly of the same magnitude ; but in a single tube, open below, 

 the vibrations would continually become less extensive, and their duration 

 would also be altered as well as their extent; besides the unavoidable 

 resistances, which would in both cases interfere with the regularity of the 

 effects. 



But it does not appear that the laws of the vibrations of fluids in pipes 

 will at all serve to elucidate the phenomena of waves. Sir Isaac Newton * 

 has supposed that each wave may be compared with the fluid oscillating in 

 a bent pipe ; but the analogy is by far too distant to allow us to found any 

 demonstration on it. The motions of waves have been investigated in a 

 new and improved manner by Mr. Lagrange ;t and I have given a concise 

 demonstration of a theorem similar to his, but perhaps still more general 

 and explicit. It appears from these determinations, that supposing the 

 fluids concerned to be infinitely elastic, that is, absolutely incompressible, 

 and free from friction of all kinds, any small impulse communicated to a 

 fluid, would be transmitted every way along its surface with a velocity 

 equal to that which a heavy body would acquire in falling through half 

 the depth of the fluid ; and I have reason to believe, from observation and 

 experiment, that where the elevation or depression of the surface is con- 

 siderably extensive in proportion to the depth, the velocity approaches 

 nearly to that which is thus determined, being frequently deficient one 

 eighth or one tenth only of the whole ; in other cases, where a number of 

 small waves follow each other at intervals considerably less than the depth, 

 I have endeavoured to calculate the retardation which must be occasioned 

 by the imperfect elasticity or compressibility of the fluid ; but it seems 

 probable that the motion of small waves is still much slower than this 

 calculation appears to indicate. 



Whatever corrections these determinations of the velocity of waves may 

 be found to require, the laws of their propagation may still be safely 

 inferred from the investigation. Thus, it may be shown, supposing the 

 waves to flow in a narrow canal of equable depth, that, whatever the 

 initial figure of the waves may be, every part of the surface of the fluid 

 will assume in succession the same form, except that the original elevations 

 and depressions, extending their influence in both directions, will produce 

 effects only half as great on each side, and those effects will then be con- 

 tinued until they are destroyed by resistances of various kinds. It may 

 also be inferred that the surface of a fluid thus agitated by any series of 

 impressions, will receive the effects of another series, in the same manner 



* Principia, Lib. II. Prop. 46. 



f Mecanique Analytique, 2de Partie, xi. 



