Some Aspects of Very Large Offshore Structures 



in which : 



o = incident wave amplitude 



a r , 



k = wave number = 2 Tl /y\ 



]\ = wave length 



The relation between wave frequency and wave length is given by the 

 dispersion equation : 



2 



CJ = kg tanh kd (8) 



The wave function <f & , corresponding to the motion of the scattered 

 waves must, besides the boundary condition in the free surface and 

 at the bottom, also satisfy the radiation condition. This condition 

 requires that, at infinity, Cp„ behaves as a radially outgoing pro- 

 gressive wave and imposes a uniqueness which would otherwise not 

 be present. 



In a system of local axes with cylindrical co-ordinates r, 6 

 and z, the radiation condition can be formulated as : 



il) <4> ) = (9) 



s 



9 = arctan (y/x) 



II. 2 Analytical solutions 



An analytical solution of the potential function can only be 

 given for certain bodies of which the geometry can be described by 

 means of a simple mathematical formula, such as the cylinder, the 

 sphere and the ellipsoid. Havelock [2 J for instance, has given the 

 solution for an infinitely long vertical cylinder of circular section. 

 This solution has been adapted for a cylinder fixed to the bottom in 

 shallow water by Mac Camy and Fuchs j^3]] and Flokstra \_4] . 

 According to Flokstra, the analytical solution of the potential in cy- 

 lindrical co-ordinates is - for this particular case - given by : 



,k / \ o a. , , / -,\ -iC*)t 



(D (r, 0, z, t) = — , = = cosh k (z+d) e 



cj cosh kd 



oo 



^ £ C (i) +n cos n 6 (10) 



n=0 n n 



961 



