•50 Dr. G. Green on Ship- Waves, and on Waves in 



<lr, can be obtained to represent the motion due to any 

 system of applied pressure. 



In working out the application of this process to ship- 

 waves, we may, without loss of generality, take the case of 

 -a pressure symmetrical about a vertical line, represented by 



p{x, y)=f(yr), where vr 2 = x 2 +y 2 . ... (6) 



Let this pressure be applied to the surface at time t = 0, with its 

 mid point at the origin, and let it move with uniform velocity 

 t>, in the positive direction of the x axis. At time t from the 

 commencement of its motion, the moving pressure has reached 

 the point (vr, 0, 0), and in the ensuing interval dr it applies 

 the impulsive pressure /{(#— vr, y\dr to the surface. The 

 corresponding velocity-potential at any point in the fluid, at 

 any time t from the commencement of motion, is represented 



d<j> = --dTf](x-VT),t/,z,t-r}, . . (7) 



where the complete function f(x, y 9 z, t) is determined to 

 satisfy equations (1) and (3). The velocity-potential of the 

 resultant motion due to all the impulses delivered up to 

 time t is therefore 



<f>(x,y,z,'t) = - ( drf{x—vT, y, z, t— t}. . (8) 

 "Jo 

 In an exactly similar way we can make a summation of the 

 vertical component velocities, or of the vertical component 

 displacements, corresponding to each increment d<j> of velo- 

 city-potential appearing in the above summation. From (7) 

 above, with dlj used to denote the vertical component velo- 

 city corresponding to d<f>, we have 



d£±-h T ^-f\(x-vT),y,z, (t-r)\, . . (9) 



p 0~ 

 or in virtue of (3) above : 



4 = - -<*r|J /{(«-«*•), y, z, t-r\, . (10) 



and <*f = - -dr- f{{.i—vT), y, z, t-r}, . (11) 



99 & 



provided the function /(#, y, z, t) satisfies the equation 



|£=0, when * = (12) 



at 



