Ogilvie 



increases linearly with distance from the x axis. The latter is a 

 rather strange kind of potential function; it represents a wave which 

 becomes larger and larger, without limit, at large distance. How- 

 ever, one must remember that this is the inner expansion of the 

 outer expansion of i|j(x,y,z ,t); it means that there are waves near 

 the X axis which seem to increase in size when viewed in the near 

 field . At very great distances, one must revert to the previous 

 integral expressions for i|j(x,y,z,t). 



We must next find an inner expansion which satisfies con- 

 ditions appropriate to the near field and which matches the above 

 far -field expansion. One finds readily that: 



i|jyy + 4^2z - ^ ^^ the fluid region, 



to the order of magnitude that we consistently retain. Thus, the 

 partial differential equation is again reduced to one in two dimensions, 

 and so we seek to restate all boundary conditions in a form appropri- 

 ate to a 2-D problem. 



The body boundary condition must be carefully expressed in 

 terms of a relationship to be satisfied on the instantaneous position 

 of the body. This condition can then be restated as a different con- 

 dition to be applied on the mean position of the hull. It can be shown 

 that: 



^~ ^a - ^^S + - U^5 + U(e?-xe§)(hOyXvz -X„) on z=d(x,y). 

 ^ (l+dy^)'/2 (1 +d2)'/2 



[c'/^S] [e6] ^^-^^^ 



The derivative on the left has the same meaning as in the previous 

 slender-body analyses: It is the rate of change in a cross section 

 plane, in the direction normal to the hull contour in that plane. The 

 first term on the right-hand side is the same as in the zero-speed 

 problem; see (3-26). The quantity - Uig has a simple physical 

 interpretation: it is a cross-flow velocity caused by the instantaneous 

 angle of attack. The remaining terms all arise as a correction on the 

 steady-motion potential function, Ux ; the latter satisfies a boundary 

 condition on the mean position of the ship, which is not generally the 

 actual position of the ship, and so it must be modified. 



Intuitive derivations of strip theory usually omit the terms 

 involving x« However, in a consistent slender-body derivation, 

 they are the same order of magnitude as the angle -of- attack term. 

 (This says nothing about which is the more nearly valid approach!) 



The free- surface condition reduces ultimately to: 



762 



