Lea and Feldman 
where m is a constant which will be determined by the matching of 
far field solution to near field solution. It must be remembered that 
this restriction placed on € and 6& does not imply the existence of 
a functional relationship between slenderness of hull and depth of free 
stream. We have chosen the shallowness parameter, € , as a conve- 
nient gauge function*and use it as a standard for order of magnitude 
comparisons, 
The following non-dimensional and scaled variables are in- 
troduced : 
Dale =o ee, -* Ge Re 
x=xL,y=yLe 
and the non-dimensional variables 
q=qauU yy it) ate We Lig gycrks = 1 ¢£K.; K =0 (ogg 
The full inviscid equations become : 
Potential Equation 
2 Zatti Zip 
+ € + =~ (0) - 
Pe ots! : yy (7 =) 
Bottom Tangency 
p(x, Yoni) 3 (7 -b) 
Free Surface Kinematic 
rige 2+2p bs 
PREN(Lb poe tO py fo ones bx, y) (7-c) 
Free Surface Pressure 
2 
2 €*¢/ 1 4+eK = ay — [2¢,. +(p_)° +e°P (y)*] 
en aoe 6 Cx oan) (7-d) 
We assume the following far field expansions for the disturbance po- 
tential and the free surface elevation 
e~ De" eo (x,y,z), t~ 2, emt (x, x) (8) 
n=] n n 
Substituting these expansions into the full inviscid equations and equa- 
ting like powers of € results gives the following : 
* See Van Dyke (4), pages 23-28 
L532 
