Lea and Feldman 
depth) can be interpreted as the ratio of a characteristic speed to the 
propagation speed of small disturbances on the water surface in shal- 
low water theory. On the other hand in aerodynamic theory, the Mach 
number (M,, = Uy /ao » where U, is the free stream speed and 
ag is the isentropic speed of sound) is the ratio of the characteristic 
speed to the propagation speed of acoustic signals in the gas. Thus we 
see that Froude and Mach numbers play similar roles and this is re- 
flected in the mathematical formulation of the two problems. For ex- 
ample, Tuck 2 gave for the first approximation a hyperbolic equa- 
tion for supercritical flow F, > 1 and an elliptic equation for subcri- 
tical flow F, < 1. We find the same situation in inviscid compressi- 
ble flow past slender bodies for supersonic M,,> 1 and subsonic M <= 1 
flows. His results for vertical force, trim moment as well as drag 
contains integrals which relates the source sink distribution to the 
local hull area and multiplied by the following factor : 
2 2 
(ie ie ey ie F, for subcritical flow and 
2 2— 
aes pany FE -1 for supercritical flow 
This factor seem to indicate catastrophic failure at critical flow 
F, = 1. However, it should be pointed out that aside from the trans- 
critical region, where | Rie —1 | is small, Tuck's results appears 
to be more than adequate for most engineering purposes. 
We shall seek a singular perturbation solution to the problem 
of shallow water flow past a slender ship with the requirement that 
the solution must be valid within the transcritical region, This ap- 
proach is that followed by Tuck'2) and is well documented in books 
by Cole(?) and Van Dyke @) . The important difference between what 
follows and the works of Tuck is that two small parameters appear 
in the formulation, slenderness ratio and Froude number parameter 
me —1 |, instead of only the slenderness ratio, Appearence of an 
additional parameter drastically alters the mathematical representa- 
tion of the problem and the nonlinear effects suggested by Tuck’) are 
indeed present, 
I, EXACT STATEMENT OF THE INVISCID PROBLEM 
Consider a ship immersed ina steady stream of inviscid, 
incompressible stream with free stream velocity of Un Fy . Atcars 
tesian coordinate system is afixed to the ship with its origin at the 
bow and at the undisturbed waterline. The positive x-axis is directed 
toward the stern of the ship and z-axis is directed vertically upward. 
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