Transerttteal Flow Past Slender Shtps 
The total velocity in the flow field is given by : 
Sh Ewer Clee ey + grad — 
where is the disturbance potential due to the presence of the ship. 
The dimensional governing equations are the Laplace equation, free 
surface kinematic and pressure equations, bottom condition and hull 
tangency condition, These are as follows : 
igh bw tp ghee ia 
er a Pak aoe 
2t¢/U~, = -(2 0+. +9. +p°) z=$(x,y) (l-c) 
line qvtzplat pds Tid janie 
BR PARIS Ap ena LS WEA? ie) 
where § (x,y) is the unknown free surface and A (x, z) is the given 
surface of the ship hull. If we are to proceed in a systematic fashion 
the relative orders of magnitude of the various terms must be establi- 
shed. One way of accomplishing this is by selecting proper scales for 
all the dependent as well as independent variables and thereby intro- 
duce non-dimensional variables of order unity. This does not mean 
that all quantities will have its maximum of one, but rather that if we 
choose the correct scale the maximum value could be large as ten 
units but not one thousand units. We take Uq_ as the velocity scale 
and the undisturbed depth, H, as the vertical length scale. The selec- 
tion of a horizontal length scale is a bit more involved as it must re- 
flect the shallow water approximation include the transcritical nonli- 
nearities produced by a slender hull, 
Now, shallow water theory assumes as a first approximation 
that the vertical pressure variation is purely hydrostatic or that ver- 
tical accelerations are negligible compare to horizontal accelerations. 
This can be derived in a systematic manner assuming that the depth 
to characteristic wave length (H / Ly << 1) is small and utilize 
Ly, as the length scale in x and y directions and expand — as a power 
series in H / Ly . We note that shallow water theory is not necessa- 
rily linearization and the latter results from restrictions that we pla- 
ce on the type of ''wave maker"! present in the problem. Furthermore, 
we note that the surface wave system generated by a ship at critical 
1529 
