Tuak and Taylor 



in that plane. Furthermore, the free surface condition reduces to a 

 rigid-wall boundary condition 



1^=0 or z = 0. (6.2) 



oz 



Thus the inner problem is identical to that treated by Newman [ 1969] , 

 who assumed that (6.2) was valid everywhere. 



The boundary condition at "infinity" for this inner solution Is 

 that the inner solution should match the behavior of the outer solution 

 in a common domain of validity, say many beams away from y = 0, 

 but not so far away that y is as large as H or 2iT/k. In effect, 

 this simply means that the inner boundary condition (5.7) for the outer 

 solution becomes the outer boundary condition for the Inner solution. 

 Thus the inner approximation to the disturbance potential cj) must 

 satisfy 



|^+a/|-=fP(J> as y-±oo (6.3) 



which Is satisfied If ^ Is asymptotically Independent of y, I.e. 



^^^±[IK as y-±oo (6.4) 



Implying 



4^-0 as y-±co. (6.5) 



The boundary condition on the hull Is (5,4) but where now 

 8/9n denotes differentiation normal to the hull cross section T at 

 station X, and where, since ky Is small In the Inner region, we 

 may replace the incident wave ^^ by 



*o - A if (w ^ y) • f^-^) 



i.e. by an incident stream of speed AVg/h. Then (5.4) becomes 



|i= _ A /l"!^ on r. (6.7) 



8n "y/h an 



Thus the Inner approximation to (j) Is the potential for flow due to 



650 



