Lea and Feldman 
speed is a single wave of translation perpendicular to the free stream. 
In the absence of viscous dissipation this wave extend to infinity so 
that the disturbance in the lateral, y, direction is greater than in the 
axial, x, direction. Thus it seems logical that we should have x=0(1) 
and y=0 (ge ) at large distances from the ship where € is a small 
parameter related to the ''shallowness'" of the water. 
Shallowness implies that depth is small relative wave length 
(H/ Ly << 1) and slenderness implies that the wave maker, the 
ship, must be longer than it is either wide or deep. If we define B 
as the maximum beam, T as the maximum draft, then slenderness 
means 
Bi / ds ple de al 
where Lis the length of the ship. In order to proceed inan order- 
ly manner some estimates must be placed on the relative orders of 
magnitude between Ly and L. We note that the dispersion relation- 
ship for steady progressive free waves in two-dimensions is 
tanh (27H /e) 7/27 Piya) gH (2) 
which can be approximated by the following expression after making 
use of the long wave assumption (H/ L, << 1) 
Hy ey Sah (2a FF.) Fe =u. /gh (3) 
The behaviour of this expression in the transcritical region is esti- 
mated as 
SDB, Veale REERC TS (4) 
Furthermore, if we take the depth to ship length ratio H / Las gau- 
ged by the slenderness of the hull, i. e. H/ L =0(€), then 
2 
By fis 0 (ei Maser) (5) 
As the ship approaches the critical flow condition, the characteristic 
wave length of the surface wave decreases so that in order to retain 
the transcritical effects and at the same time impose slenderness 
assumption we take 
L,/L=0(€ fr ae Fo) = 0(1) 
We note that in Tuck's analysis(?) it was assumed that as the ship 
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