344 



Dr. E/ankine on Waves in Liquids, 



[May 7, 



III. On Waves in Liquids/^ By W. J. Macquorn Rankine, 

 C.E., LL,D., F.R.S. Received April 16, 1868. 

 (Abstract.) 



(1) Object of this Taper. — It has long been known that in an uniform 

 eanal filled with liquid, the speed of advance of a wave in which the horizontal 

 component of the disturbance is uniform from surface to bottom is equal to 

 the velocity acquired by a heavy body in falling through half the depth of 

 the canal. But, so far as I know, it has not hitherto been pointed out 

 that a similar law exists for waves transmitting a disturbance of any possible 

 kind in a liquid of limited or unlimited depth, provided only that the 

 upper surface of the liquid is a surface of uniform pressure. The object 

 of this paper is to demonstrate that law, and to show some of its applica- 

 tions. 



(2) Velocity of Advance defined. — Throughout this investigation the ve- 

 locity of advance of a wave will be defined to be the mean between the velo- 

 cities with w^hich the shape of the wave advances relatively to a surface- 

 particle at the crest, and to a surface-particle in the trough respectively. 

 In ordinary rolling waves the velocities of particles in those two positions 

 are equal and contrary, so that the speed of advance as above d-efined is 

 equal to the speed of advance of the wave relatively to the earth. A wave 

 of translation in which the velocities of particles at the crest and hollow 

 are not equal and contrary, may be regarded as produced by compounding 

 the motion of a rolling wave with that of a current whose velocity is half 

 the difference of the velocities of those particles. 



(3) Relation between height of wave and horizontal disturbance at the sur- 

 face. — The following relation between the height of a wave and the horizontal 

 disturbance of the surface-particles has already been proved and made use of 

 by various authors ; and it is demonstrated here for convenience only. Let 

 -f u-^ and —u^ be the velocities of a surface -par tide at the crest and trough of 

 a V. ave respectively. Let a be the velocity of advance of the wave as defined 

 in article 2. Conceive a horizontal current with the uniform velocity —a 

 to be combined with the actual wave-motion ; the resultant motion is that 

 of an undulating current, presenting stationary waves in its course ; and the 

 forces which act on the particles are not altered. The resultant velocity of 

 a particle at the crest becomes —a + u^; and the resultant velocity of a par- 

 ticle in the trough becomes —a — u^. Let the height from trough to 

 crest be denoted by b.z ; then, since the upper surface of the liquid is sup- 

 posed to be a surface of uniform pressure, the principle of the conservation 

 of energy gives the following equation : 



^A^=i{(«+w,)'— («— Wi)^} = 2«M,. ..... (1) 



(4) Virtual Depth of Uniform. Horizontal Disturbance. — By the phrase 

 ''virtual depth of uniform horizontal disturbance," or, for brevity's sake, 

 virtual depths I propose to denote the depth in the liquid to which an uni- 



