the Breaking of Waves, 357 



unity. This motion is inconsistent with the equation of con- 

 tinuity of an incompressible fluid, and if set up must lead to 

 instant rupture of the continuity of the liquid, which will 

 break up into drops. But liquids appear to be capable of 

 resisting a small amount of internal tension, and this, 

 together with the surface-tension, may suffice, when the 

 motion is not too violent, to prevent the rupture, modifying 

 the motion in such a way as to render it consistent with the 

 equation of continuity. If m does not differ much from unity, 

 the difference between this actual motion and the motion 

 given by the ordinary theory on the one hand, and that 

 obtained on the hypothesis of compressibility on the other 

 hand, will be small compared with the motion itself. If then 

 we substitute the velocities — u, mv in the equation of con- 

 tinuity for a compressible fluid, the variation of density may 

 be taken as an indication of the tendency to rupture, and we 

 may say that rupture will take place if the maximum density 

 exceeds, or the minimum density falls short of the actual 

 density of the liquid by more than a certain small amount 

 depending on the nature of the liquid. 



Deep Water Waves. 



Taking the origin at the obstacle, and measuring y vertically 

 downwards, let the incident waves have a profile given by 



77 = sin Tc(jS"—ct) 



indicating a train of waves of altitude rj and length 27r/k 

 advancing parallel to the axis of a. The velocity potential 

 for such a train is 



(f) = ce~ k,J cos k (x — ct ) , 



making _-£ =„ when ?/ = 0, and ~— = when y — co . For an 



ordinary reflected wave we should have 



<p' = ce~ ky cos k{x + ct), 



so that the velocity system at any point of the reflected 

 wave is 



u'=— cke~ k y sin k (x -f ct) ; v'=— cke~ ky cos k {x + ct) , 



In accordance with our hypothesis, therefore, we investi- 

 gate the system 



u' = — eke'** sin k{x + ct) ; i J =— mcke~^ cos k [x + ct) , 



and the motion of any element is supposed to be compounded 

 of this and the motion due to the incident wave. The latter 

 Phil. Mag. S. 6. Vol. 2. No. 10. Oct. 1901. 2 B 



