Deep Water due to Motion of Submerged Bodies. 51 



Accordingly, in the resultant motion, the vertical velocity 

 and vertical displacement at any point in the fluid are given 

 by 



and 



?--ifV"U*-w),y,«,(t-rT)}, . (13) 

 f=-^r^/'K*-7«r),y,r,(*-T)K . (14) 



9PJo 



From the equations (8), (13), and (11), it appears that the 

 solution of our problem is reduced to the determination of 

 /{as, y> z, t) to satisfy equations (1), (2), (3), and (12): and 

 this determination is easily made by means o£ the theorem 

 analogous to Fourier's double integral theorem, according to 

 which 



far) = if J {k*r)kdkC f{«)J (kx)ad*, . . (15) 



-'vo Jo 



and 



/ (-BT, S, t) = ^ f ^ J (**) ' * COS fofe*)*<tt 



/»oo 



I /(a)J (&*W«. . • . (16) 



The complete solution of: our problem to determine the 

 velocity-potential and vertical component displacement, at 

 any point in the fluid, due to moving pressure, /(-st), which 

 is applied*at the origin at time t = 0, is thus contained in the 

 integrals 



<£(*,*/, z, t) = - 5— \ dr e~" ! J (k^)k cos {gk{t-rYfdk 



-"Wo Jo 



f /{a)J {ka)uda, (17) 



X(x,y,z,t) = + j^-i \'dr) e~ k < J (fe')**sin {^(*-t)^ 



f /(«)J (^«)«^ (18) 



fo 



v2i„ 



in which -sr' 2 = (x — vr) 2 +y 2 . 



When v is put equal to zero in these formulas we obtain 

 the solution corresponding to the case where the pressure 

 /(«■) is applied at the origin at t = and kept applied without 

 change of position till time t. If we indicate the velocity- 

 potential and vertical-component-displacement for the case 



E 2 



