56 Dr. G. Green on Ship- Waves, and on Waves in 



The first integration gives 



- * Wf ^t * *-t ) 2 • -^3 cos /^=^; - » i 



2 y Jo {x-vr)i [4(*-w 



(31) 



_g{t-Tyjh+z) 



IT 



vt) 4 



. . . (32) 



each of these calculations being subject to the condition that 

 (t — t)I{x—vt) is large, and that (x — vt) is large in com- 

 parison with the space over which the applied pressure is a 

 first order effect. These conditions, which are explained in 

 ■a former paper*, are fulfilled in the present case where we 

 are considering places and times at which the motion has 

 become steady. On proceeding to the final integration, we 

 find that, corresponding to each value of t, only one value of 

 r fulfils the condition for stationary phase — that given by 



(ff-rr) =(«*—*) (33) 



In this, values of x greater than vt are evidently inadmissible 

 as they require that t should exceed t, hence we may con- 

 clude that the effective part of the wave-disturbance is 

 behind the mid line of the applied pressure at each instant, 

 the forward part being negligible in comparison. The final 

 evaluations of (31) and (32), obtained by means of the 

 stationary phase principle, are 



(34) 

 (35) 



■*,(*, z,t)=- (iffa^/v) e «» ( * + * ) cos J | («*_«) 1, 



— 9 (h+z) r i 



? 2 0,V)= (W"> •* smU {vt - x )[. 



These results are valid at all points %vell behind the moving 

 pressure at which steady motion has been established. In 

 the immediate neighbourhood of the mid line of the npplied 

 surface-pressure the complete solutions of (29) and (30) are 

 required. The problem of the moving line pressure has 



* See short note at end of this paper, or Proc. R.S.E. vol. xxx. 

 p. 247. 



