Dr. llankiue on Waves in Liquids. 53 



ceive a horizontal current with the uniform velocity —a to be com- 

 bined 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 1 ; and the resultant 

 velocity of a particle in the trough becomes — a — u x . Let the height 

 from trough to crest be denoted by Az ; then, since the upper surface 

 of the liquid is supposed to be a surface of uniform pressure, the 

 principle of the conservation of energy gives the following equation : 



ffAz=i{(a + u i y-(a-u i y} = 2au l (1) 



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

 phrase " virtual depth of uniform horizontal disturbance," or, for 

 brevity's sake, virtual depth, I propose to denote the depth in the 

 liquid to which a uniform horizontal disturbance would have to 

 extend, in order to make the amount of horizontal disturbance equal 

 to the actual amount. That is to say, conceive that a pair of vertical 

 planes normal to the direction of advance, 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 

 passing backwards through the plane at the trough, and -f u f the ve- 

 locity with which particles are passing forwards through the plane at 

 the crest ; then the rate by volume at which liquid is passing into 

 the space between those two planes is 



$u'dz+ £u"dz,— 



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

 denote the virtual depth ; then 



Ju'dz+£u n dz 

 J (2) 



2u x 



(5) Relation between Virtual Depth and Speed of Advance. — In 

 an indefinitely short interval of time dt, the volume of liquid which 

 passes into the space between the two vertical planes mentioned in 

 article 4 is 



2ku x dt ; 



and in order to make room for that volume of liquid, 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 



adtAz ; 



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

 equation : 



aAz=2ku l ; 



