25 



coordinates Xo and i/o and including two constants m and C and a 

 parameter K which is a function of y only; then: 



x=Xo+K^\B.m{Xo—(Jt) (30a) 



y=yo-\-K cos m{Xo-Ct) (306) 



It will be easily seen that these equations, regarded merely as expressing the 

 geometrical motion of points, and apart from the physical possibility of the 

 motion, represent a wave disturbance of periodic character traveling in the direc- 

 tion Ox with a velocity of propagation C. 



Assuming infinite depth and applying the requirements of continuity 

 and constant pressure on the surface, Stokes shows that the displace- 

 ments must decrease exponentially with distance below the surface; 

 and that the wave velocity for all waves (small and finite height) is 

 given by 



Furthermore, if the motion is reduced to steady flow by superimposing 

 a velocity — C on the entire liquid mass, the stream lines, designated 

 by 2/o= constant, are also lines of constant pressure. 



This is undoubtedly no necessary property of wave motion converted into 

 steady motion, which only requires that the particular stream line at the surface 

 shall be one for which the pressure is constant, though Gerstner has expressed 

 himself as if he supposed it necessarily true; it is merely a character of the special 

 case investigated by Gerstner and Rankine. Nevertheless, in the case of deep 

 water it must be very approximately true. 



In any given case of wave motion, the flow which remains when the waves 

 have been caused to subside in the manner above explaiued is easily determined, 

 since we know that in the motions of a liquid in two dimensions the angular 

 velocity is not affected by forces applied to the surface. 



In Gerstner and Rankine's solution 



, bV bU 



vorticity = 2 CO = -:r ^ 





where ?7' = horizontal velocity remaining after wave motion has been destroyed. 



It appears then that in order that it should be possible to excite these waves in deep 

 water previously free from wave disturbance, by means of pressures applied to 

 the surface, a preparation must be laid in the shape of a horizontal velocity 

 decreasing from the surface downwards according to the value e"^'""', where y is 

 a function of the depth y' determined by the transcendental equation of y, and 

 moreover, a velocity decreasing downwards according to this law will serve for 

 waves of the present kind of only one particular height depending on the coeffi- 

 cient of the exponential in the expression for the flow. 



The oscillatory waves which most naturally present themselves to our attention 

 are those which are excited in the ocean or on a lake by the action of the wind, 

 or those which have been so excited and propagated into (practically, though not 

 in a rigorous mathematical sense) still water. Of the latter kind are the surf 



