168 Mr. D. D. Heath on the Dynamical Theory of 



Using the equations already found (A, B), we get 



or 



dv 



a? dy \ 



It~ 



k dco 



dv 

 dco 



a dy^ 

 k dco 



v = 



a 



(C) 



(assuming, in our integration, that there is no permanent cur- 

 rent, or that the average value of v is zero). 



We see, then, the necessary physical constitution of such a 

 wave. 



Contemplating the wave at a given instant, v is everywhere 

 proportional to y and of the same sign. At the ridge, or point 

 of greatest elevation, the particles of water are in their state of 

 greatest forward movement ; at the point of greatest depression 

 they have the same speed backward; at the mean level, inter- 

 mediate between the two other points, v = 0. 



And if we look to the change of velocity at a given place as 



dy 

 time goes on, we see that its rate is proportional to — —-, which 



represents the slope of the surface, estimated forward and down- 

 ward, as referred to the undisturbed level, or horizontal line, on 

 which co is measured. Assuming (what we shall presently jus- 

 tify) that the horizontal* velocity is uniform throughout each 

 vertical section, each particle in the canal has a definite oscilla- 

 tory motion on each side of a mean place which it occupies when 

 under the ridge or the lowest depression of the wave, reaching 

 its extreme distances eastward and westward when under the 

 forward and hinder shoulders respectively. 



The vertical motions and effective forces in such a wave as we 

 are considering may be neglected in comparison with the hori- 

 zontal ones. For the water displaced by the conversion of a 

 ridge between two level shoulders to a hollow is a large frac- 

 tion of the volume contained between the base (about 6000 miles) 

 and the whole vertical oscillation of a particle at the surface ; 

 and this has to be provided for by the lateral displacements 

 eastward and westward of the bounding sections, which yield a 

 space 2k x whole lateral oscillation ; and k is but a very few 

 miles. And the vertical motion of a particle on the surface is 

 very much larger than the average motion of the particles under 

 it ; for the water at the bottom remains there, and the rise and 

 fall in each horizontal stratum is proportional to the depth of 

 water under it, the reasoning by which the equation of conti- 

 nuity was obtained applying at every level* 



