DIFFRACTION POTENTIAL 



With a similar method to solve the potentials V. for 1=3 and 5, the diffrac- 



J 

 tion potential can be solved. In the outer region we solve the three-dimensional 



Laplace equation and in the inner region we apply strip theory. Through matching, 

 the unknown source strength and coefficient are obtained. In the application of 

 strip theory, however, there are two different approaches. One is to solve the two- 

 dimensional Laplace equation and the other is to solve the two-dimensional wave 



equation or the Helmholtz equation. The first approach has been applied by Salvesen 



12 3 13 



et al. and others. Newman, Troesch, and others have solved the Helmholtz 



equation for computing the diffraction forces. In this method there is a singu- 

 larity in the solution when the angle of the incoming wave is or 180 degrees. To 



3 

 avoid this singularity, Newman has introduced the long-wavelength solution. How- 

 ever, there is difficulty in solving the equation for short waves. 



In this study we shall solve the two-dimensional Laplace equation to compute 

 the diffraction force as the solution of the strip theory. The inner solution for 

 the diffraction potential can be given in the form (see Reference 3) 



V?^ = <P^^^ + C^(x) ((j) +^ ) (70) 



II I s s 



where C (x) = a function to be determined by matching with the outer solution 



(s) 

 ^ -. = the diffraction potential solved by the strip theory 



(s) 

 d) = the symmetric function of ^ -. 

 s I 



(s) 

 The diffraction potential, '^ -. , satisfies the two-dimensional Laplace equation with 



the boundary condition given in Equation (18) . If we express the potential of the 



incoming wave as 



•^ = - ^^ exp (K z+i K X cos B-i K y sin 3) (71) 



(s) 

 the boundary condition of ^^ is given by 



(^ (^^ = - K (n„-i n„ sin 3) V (72) 



/n o 3 2 o 



16 



