Van Oortmevssen 



II. 1 Potential theory approach 



Consider a fluid, bounded by a partially or totally submerged 

 rigid body, a fixed bottom and a free surface. The undisturbed free 

 surface will be taken as XOY-plane of the co-ordinate system, with 

 the z-axis pointing vertically upwards. The fluid is assumed to be 

 inviscid, incompressible and irrotational. All motions will be infinitely 

 small. At infinity the fluid motion behaves as a single harmonic wave, 

 travelling in the positive direction of the x-axis. If the undisturbed wa- 

 ve has a frequency U, the velocity potential may be written as 



= Re 

 The function <p has to satisfy the Laplace equation : 



-icot 



Be 



(1) 



V 2 (p =0 (2) 



and the boundary conditions : 



- at the bottom -£-!- = for z = -d (3) 



Oz 



- in the free surface -r — = * r for z = (4) 



Oz 



- at the body contour N = for x = s (5) 



On — - 



in which 



d = water depth 



= CJ 2 /g 



g = the acceleration of gravity 



js_ = vector which describes the body contour 



n = vector normal to the contour 



The function <JP can be split into two components : 



p=p. + £ s (6) 



in which 



ips 



= the wave function of the undisturbed incident waves 

 = the wave function of the scattering waves 



Both components have to satisfy the Laplace equation. The 

 function for the incident wave, including the boundary conditions in the 

 free surface and at the bottom, is given by 

 f g coshk (d + z) 



^• =X ^ km e ^ 



l ex) cosh kd 



960 



