172 S. Schuster and H. Schwanecke 
(3) 
The pressure distribution at the profile, i.e., the function f(€) with f(x) being the meanline 
of the profile, can be found from the relation 
u(x,-h*) f'(x) = w(x,-h") (4) 
with w and w taken from Eq. (1). 
The boundary condition (2) and hence the velocity field (1) is valid for smooth water 
above the foil as well as ahead and aft, i.e., for a condition which permits the existence of 
deep water waves. 
Within the technically interesting range of the Froude depth number F, = u, gh there 
is as shown by experimental investigations rapid flow prevailing above the foil with the ex- 
ception of the direct neighborhood of the leading edge. For this case the boundary condition 
(2) and therewith the terms for the velocity field (1) are no longer valid. Behind the foil, 
with the exception of the direct neighborhood of the trailing edge, the flow is tranquil again, 
for here the physical suppositions for rapid flow are missing due to the water presumed as 
being deep. For this range Eqs. (1) and (2) are valid again. 
The consequence of the rapid flow is that the water surface above the foil remains 
smooth, while for small distances between foil and water surface the water flows parallel to 
the upper side of the profile, whereby the well-known apparent camber reduction results. That 
means above the foil there is, dependent on the Froude depth number, at relative low speeds 
already a state of flow which will occur behind the foil only at considerably higher speeds. 
Therefore for the range of the foil (-1 > x > +1) in Eqs. (1) the terms containing g/u,?2 may 
equal 0. The equations simplify to 
+1 5 
Sale e | £(E) (aa aes 
e0 ie (x- €)2 + (zt+h*)y? 
+1 
1 (z-h") dé 
a f soe ee a 
= | Oper ReeeSTT (5a) 
+1 
w(X,Z) 1 Gesejeds 
= = —— f Be ee NO 
Uy ar . () (x-€)2 + (z+h*)? 
ie (x= €) dé 
tay ; OP So Se Slo BES b 
arr fe 4 (x= Gy + (a- hy? a 
At the surface (z = 0) the additional velocity u, in Eqs. (5) disappears, so that in the case 
of rapid flow at the upper side of the profile the boundary condition 
