Ogilvie 



for example, if we drop a pebble into a pond, waves continue to move about for a 

 very long time. If the fluid were not viscous, the waves would appear forever. 

 This is in considerable contrast to the common situation in which a body moves 

 through an ideal fluid filling all space. In such cases, all motion stops instantly 

 if the body stops. Thus it can be seen that the impulse response method exhibits 

 very clearly the basic contribution of the free surface to the problem. 



Following Cummins, we consider first the case of a ship with no forward 

 speed. Let x denote the position vector of a point on the hull surface, s, meas- 

 ured in a fixed reference system, and let x' be the position vector of the same 

 point on the hull surface, measured in a reference system moving with the hull. 

 The two systems of axes are assumed to coincide when the ship is in its equi- 

 librium position. When the hull is displaced from equilibrium, the deflection of 

 any point of the hull can then be expressed: 



6 

 x-x' = ]2 a. (x,t) , 



k = 1 



where 



r a^(t) i^ , k = 1, 2, 3, 



^kCx.t) = \ (5) 



I ^i(^t) [ik-axx] , k = 4, 5, 6. 



a^(t) is a deflection in surge, sway, or heave, respectively, for k = 1, 2, or 3, 

 or a rotation in roll, pitch, or yaw, respectively, for k = 4, 5, or 6. It is as- 

 sumed that all a^(t) are small enough that only second order errors are in- 

 curred in the vector addition of rotations. Also, we can use x' as the first 

 argument of a^ , causing thereby only second order differences in the results. 



The velocity potential, $(x, t), must satisfy the following conditions:* 



6 



a) -^ = n • y a, (x, t) on the hull, 



on - '—' '^ ~ 



k = l 



b) ^+g^ = onx3 = o, (6) 



c) a radiation condition for x^^ + ^^ -► co^ 



d) I vol -• 00 , as X3 ^ -03. 



It is easily seen that only second order errors arise if the body boundary condi- 

 tion is applied at the mean position of the hull, rather than on its actual moving 

 surface; t all of the boundary conditions are then applied on fixed domains. 



*I am defining $ such that its gradient equals the velocity vector. For a thorough 

 derivation of the free surface conditions, see Stoker (1957) or Wehausen and 

 Laitone (I960). 



tThis will not be the case when forward speed is included. 



26 



