1868.] 



Dr. Eankiue on Waves in Liquids. 



345 



form horizontal disturhance would have to extend^ in order to maTce the 

 amount of horizontal disturhance equal to the actual amount. That is to 

 say, conceive that a pair of vertical planes normal to the direction of ad- 

 vance, and each of the breadth unity, coincide at a given instant, one with 

 the trough-line or furrow, and the other with the crest-line or ridge, which 

 bound one of the slopes of a wave. We will suppose this to be the front 

 slope, merely to fix the ideas ; for similar reasoning applied to the back 

 slope leads to the same results. At a given depth z below the surface, let 

 — w" be the horizontal velocity with which particles are in the act of pass- 

 ing backwards through the plane of the trough, and the velocity with 

 which particles are passing forwards through the plane at the crest ; then 

 the rate by volume at which hquid is passing into the space between those 

 two planes is 



the integrations extending from the surface to the bottom. Let k denote 

 the virtual depth ; then 



^ Jxidz+lu"dx ^2) 



2m, 



(5) Relation hetiveen Virtual Depth and Speed of Advance. — In an in- 

 definitely short interval of time dt, the volume of liquid which passes into 

 the space between the two vertical planes mentioned in article 4, is 



21cu^dt ; 



and in order to make room for that volume of hquid, the front slope of the 

 wave must sweep in the same interval of time through an equal volume. 

 But the volume swept through by the front of the wave is 



adt^z ; _ 



so that, cancelling the common factor dt, we have the following equation : 



a^z=2kii^ ; 



but, according to equation (1), ^z=^^ ; which value being substituted in 

 the above equation, gives 



and therefore 



i = 2Jm,, 



9 



— = ^, and a = ^gk', .... (3) 

 ff 



so that the velocitij of advance of a wave (defined as in article 2) is equal 

 to that acquired by a body in falling through half the virtual depth ; and 

 this is true for all possible waves in which the upper surface is a surface of 

 uniform pressure. 



(In article 6 of the paper, the speed of advance of a wave of translation 

 is expressed by combining the speed of a rolling wave, ^/ gk, with that of 

 a supposed current, as stated in article 2. 



