Transertttcal Flow Past Slender Ships 
at any given cross section, We assume the following expansions for 
the disturbance potential and the free surface elevation 
ee eee oeakeitac: 
ay eee ae ate a raren er se (13) 
Following the same procedure as for the far field solution, 
we obtain a consistant approximation in the near field to 0 (e° é ) 
given by 
a (e)=e 7 ,~, (E ) SE sae B, eliGapie E 
Miia = 0 (eas =" a2) (14-a) 
pi, (KX ¥, -1)=0 ( all n's ) (14-b) 
ne Ot 0) 0 (Gn, = ee D (14-c) 
pene (ee Ale —haisias, 1-105 acy = A,/VIFA, (14-d) 
To the second approximation (n =2 ), the boundary value problems 
derived are identical to those of Tuck(2) as well as the order of 
magnitude estimate placed on the disturbance potential? However, 
we note there is a difference in the estimate placed on the elevation 
of the free surface and is 
aaa! Z 2 , 6/2 
S Tuck ae alla fie! ppl present } = GRE pee aaht) 
ti 2 ep 4 , 4/3 
¢ present Pe OE ae ) OD ooze g) Oe Tuck ) 
Thus we see that the surface disturbance is stronger here than in the 
linear case. 
Since the Neumann problems defined by equations (14) have 
already discussed in detailed by Tuck ') \2 » we shall make use of 
his results and using the restricted matching technique of Van Dyke 
4) to match one term far field to the two terms near field approxi- 
mation. The important result is the hull tangency condition for the 
far field equation on a equivalent body and gives 
a ! 
fy (x0) =S! (x) /2 (15) 
*We note here the difference in notation ¢ = = 3/2 
Tuck pres. pres. 
1535 
