Growth of a Train of Waves. 19 



and by integration (175) becomes 



SC*, s> i)=i[v(*,r,0-V(*,*,0)] . (177). 



Putting now in this 3 = 1, and using- (176), we find 



K*, l,l)=irv(*, l.t)-V(.<; l,0)]=|[v(*,l,0 + n] (178). 



The interpretation o£ this, as t increases from to go , is that 

 the sudden application and continued maintenance of a 

 pressure — Y(x, I, 0) over the whole fluid surface, initially- 

 plane and level, produces a depression, ?, which gradually 

 increases from 0. at t=0, to its hydrostatic value U/g, at 

 £ = co. The gradual subsidence of the difference from the 

 static condition, as time advances from to go , is illustrated 

 by the diagrams of fig. 34, for the case in which we choose 

 for Y(x, 1, 0) the ty(x, 1, 0) of §§ 100-104 above. 



§ 13]. To understand thoroughly the meaning of Y(x, z, q) 

 as defined in § 127 ; remark first that it is the velocity- 

 potentiil of a possible motion of water, under the influence 

 of gravity, with no surface-pressure, or with merely a pressure 

 uniform over its infi ite free surface. This is equivalent to 

 saying that V [x 3 3, q) fulfils the equations 



d 2 Y d 2 V A . dY d 2 Y „„ n . 



c7? + -p-=0, -and <^ = ^ . . (179). 



Secondly, remark that at the instant (7 = 0, there is no surface 

 displacement; hence Y(x,z,q) is the velocity-potential at 

 time q, due to an instantaneous impulsive pressure, — Y(x, 1, 0), 

 applied to the surface of the fluid at rest and in equilibrium, 

 at time q = 0. Now, allowing negative values of q, think of 

 a state of motion from which our actual condition of no 

 displacement, and of velocity-potential equal to Y(x } 2,0), 

 would be reached and passed through when q passes from 

 negative to positive. It is clear that the values of Y(x, z, q) 

 are equal for equal positive and negative values of q. Hence, 



when q=0, we have j Y(x, 2,0)^0 . (180). 



§ 132. Consideration of the Y(x, z, q), defined in § 127, 

 which allows Y(x, 1,0) to be any arbitrary function of x, 

 but requires dY/dq to be zero when q = 0, suggests an allied 

 hydrokinetic problem : — to find W fulfilling (179) with W 

 in place of Y : and, at time </=0, having W=0 and dW/dg 



C 2 



