Deep Water due to Motion of Submerged Bodies. 55 



an origin in the free surface vertically above the initial 

 position of the axis of the cylinder in expressing the motion 

 due to the pressure equal and opposite to that given in (26)_ 

 If we use 0i — 4>i to represent the solution corresponding to 

 the motion of the solid and its negative image, and cj> 2 to 

 represent the motion due to the moving surface-pressure, 

 equal and opposite to that given by {26) above, then, 

 according to (17) and (18), 



<J> 2 (x, z,t)=-{2gaVi/ir) 



Jo Jo 



J' 



■Icz 



cos {gk(t-T) 2 Wk 



cos h{ (x — vt)—cl} 



da, . (27) 



i ft ,-. -Jcz x i 



&{x,z,t) = {2g*a 2 h/'7r)\ dr\ € A* sin {gk{t-T) 2 }*dJc 



j: 



+ O0 



Wa> — ~ da - 



(28) 



The integrations with respect to a can readily be performed,, 

 and we obtain 



* 



Jo JO 



e cos Jc(x—vt) cos{gk{t — T) (*dfc» 



dr\ e " ^oos*(*-w)sin{^(e-T) , } t ^. 



*/0 Jo 



(29) 



Hlc. 

 (30) 



The motion here represented may be regarded as that 

 reflected from the free surface when a cylinder moves 

 steadily with velocity v at a fixed depth h. Equations (29) 

 and (30) may be interpreted to mean that this reflected 

 motion is the same as that produced by a line pressure 

 acting on the surface which contains the axis^ of the image 

 cylinder and coinciding at each instant with that axis. 

 This brings our solution into line with the usual interpre- 

 tation of the motion given by (23) as equivalent to a 

 line pressure acting at each instant at the axis of the 

 cylinder. 



The integrations required in (29) and (30) can in each 

 case be carried out by means of the stationary phase principle.. 



