Transerttical Flow Past Slender Shtps 
Disturbance Potential : 
prt, (aye +£, Gayle” 4+ le, Gey) - (et /2) 4, Je%... 
4 2 
+t [£, Gey) - (et / 2) tf) |+0( 7) (ea) 
Free Surface Kinematic : 
+ rk SM 2 ee! 
ap tie a! ng rs 5 a 1 ayy out 
POE pe 0 (9-b) 
( bret fon Nokia ck CF 
Free Surface Pressure : 
(9-c) 
The bottom tangency condition is satisfied toO ( € if and the) ''stretch" 
in the y-coordinate is taken as y = yL /€?= yL /VE or p= 1/2. This 
is determined by the observation that if p<1/2, then the expansions 
should proceed as fractional powers of € which cannot be matched to 
the near field solution, On the other hand, if p > 1/2, then the term 
fy would not appear to 0 ( 4) anda degenerate case results. Thus 
the choice of p = 1/2 results ina "distinguished limit process'' as 
€ —» 0? The governing equation for the first approximation to the 
disturbance potential (», =f, (x,y) is obtained by elimination 
of second order variables (f, and{,) between the free surface kine- 
matic and pressure conditions to 0 (£4) and is 
(K+ 3f),) ave ; Tene 3" i) 
The mathematical structure of this equation could change locally in 
the domain of solution depending on the algebraic sign of the term 
(K + Sfay ). This equation can describe locally subcritical flow 
(elliptic equation) when ( K + 3f)x ) < 9, supercritical flow (hyperbo- 
lic equation) when ( K + 3h, ) > Oand the local characteristics have 
the slope 
(dy/dx )=+[K + ate) “1/2 (11) 
The expansion for the disturbance potential (  ) given by 
equation (9-a) is similar to Tuck's outer expansion; however, it must 
be noted that our small parameter is based on depth (& =H/L) 
while Tuck's parameter is the slenderness ratio ( 6 =B / i.) We 
* See Cole (3), page 46 
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