Transertttcal Flow Past Slender Shtps 
approaches a line (€—+*0) that} 1 - Fe | remains fixed and of order 
unity which implies that L , |L = 0( € ) <<1 and is equivalent to 
the condition Sg! a /V gl = 0 (VE). Thus outside the transcritical 
region the proper scale length in any horizontal plane is the length of 
the hull L. However, the situation within the transcritical region is 
L=0(L, ) so that we can choose either L or Ly as a horizontal 
characteristic length with the restriction € /VTl - el = Ot 1 jaar 
= "T+ €& (6) 
ae 
where K is some similarity constant which is of order unity. The 
particular form chosen here is guided by the transonic aerodynamics 
analysis of the slender airfoil theory since we anticipate a close ana- 
logy between it and the present shallow water problem. It should be 
noted that in aerodynamic theory K is not uniquely determined by any 
analytical approach but depends on the correlation of experimental 
data. 
II. FAR FIELD APPROXIMATION 
Singular perturbation solution is a systematic procedure by 
which successive estimates to the solutions can be made in the various 
regions of the domain of solution. If properly applied, the dominate 
features of each of these regions will be magnified and secondary fea- 
tures suppressed by scaling of variables. We expect that in the far 
field details of the ship hull will be lost and that the dominate feature 
of the problem is that of the surface wave system. As noted in the 
previous section Lave / Loin = 0( 1) in transcritical region so that 
for scaling purposes either one would be appropriate and we shall 
refer to it as simply the characteristic length L. The shallowness 
parameter € and the slenderness parameter 6 are given by 
Eso /. 1, 
= B/ jae yi 
We shall restrict our attention to that class of problems in which the 
hull must be more slender than the water is shallow, i. e. the maxi- 
mum draft be less than the depth, thus 
lim (O7* a0 
A simple relation which satisfies this condition is 
Sd a nOn- keane el 
los 
