Lea and Feldman 
note that the Laplacian operator in the horizontal plane does not occur 
to 0 ( ¢4) thus in this respect the present expansion for the distur- 
bance potential is simpler than the linear theory. On the other hand, 
the free surface kinematic and pressure conditions for —, are deri- 
ved from higher order approximation which lead directly to the non - 
linearity in the problem. It would appear that in the transcritical 
region the nonlinear free surface conditions are dominate and the po- 
tential nature of the flow is only secondary. In passing we note that 
equation (10) is mathematically identical to the equation governing 
transonic flow past two dimensional airfoils. 
We can now define the relative orders of magnitude between 
the shallow water parameter ( © ) and the slender body parameter 
( 6 ) by examination of the behavior of the far field solution on the 
body surface. While this can actually be done by the formal matching 
process, we choose to do it here to simplify the algebra. Substituting 
the far field variables and expansion into the hull tangency condition, 
we obtain for the leading terms 
3/2 
CECE Ghia pee 
fe (x,0A)= 6A + €Off, (x; A) Ab ol. 
2y x 
- (z+1) f yey 104) a (12) 
where the ''slenderness"' of the ship hull is exhibited explicitly 
through 6 A with A = 0(1). Guided by the two-dimensional aerodyna- 
mic slender air foil theory, we take € 72 -%§ which satisfies our 
earlier requirement that lim_.o (6/€) =O. Thus it seems to imply 
that the shallow water problem is analogus to high aspect ratio air- 
foil problem while the deepwater problem is analogus to the low aspect 
ratio problem. 
III NEAR FIELD APPROXIMATION 
The nonlinear effects are not expected to be important in the 
near field region where the basic flow pattern is strongly influenced 
by the hull form. Asa result, one would expect that the near field 
expansion would yield a series of Neumann problems in the ( y-z ) 
cross flow plane similar to those derived by Tuck (ey 2 the following 
non-dimensional and scaled variables are introduced : 
x=XL, y=ELY, z =€LZ,f =E LS N =E€LN 
All the remaining variables are as in the case of far field approxima- 
tion. An additional variable N is introduced such that the unit vector 
en in the N direction is normal to on and the hull contour (6 A(Y, Z)) 
1534 
