In this theory the longitudinal interaction term is computed by matching the inner 

 and outer solutions. This term and the results of the strip theory encompass the 

 solution of the unified slender-body theory. 



The numerical results for added mass are larger than those from strip theory 

 alone and the damping coefficients are generally smaller than the results from strip 

 theory. The heave and pitch motions generally compare well with experimental results 

 when the encounter frequency is small. For high encounter frequencies, the results 

 become extremely large, especially for pitch motion. 



The large discrepancies at high frequencies might be explained by the fact that 



in solving the two-dimensional potential (strip theory) , we have applied the Frank 



3 

 close-fit method, while Newman applied the method of multipole expansion. In multi- 

 pole expansion, there is a clear separation of the source strength at the origin 

 from that of the wave-free potential while in the Frank close-fit method there is no 

 separation. These different approaches to the solution of the two-dimension poten- 

 tial function is believed to cause these large discrepancies. 



BOUNDARY-VALUE PROBLEM 



We define two coordinate systems: the first, X Y Z is fixed in space and 



■' o o o o ^ 



the second, oxyz, is fixed with respect to the ship which moves with forward speed, 

 U, in the positive X -axis. The oz-axis is directed vertically upward and the 

 ox-axis is positive in the direction of the ship's forward velocity; see Figure 

 1. The oxy-plane is the plane of the undisturbed free surface. These two co- 

 ordinate systems coincide when the ship is at rest. 



Figure 1 - Coordinate System 



