Computation of Shallow Water Shtp Mottons 
d if.cosB 
tz S(é) Serhan oe —e 
inno, oa (k Ce : ) 
The terms on the left of (3.8) constitute natural inertia, 
hydrostatics, and Froude-Krylov exciting forces, all hydrodynamic 
effects being on the right. The expression 
eikécosp 
o (3. 9) 
is the relative vertical displacement between ship and wave at station 
£, whereas the term 
if cos ; 
Oo P iktcosé 
jen Shae ant) 
is the relative horizontal displacement between the (surging) ship and 
the water particles in the wave. 
This display of the equation of motion is similar to that given 
by Newman and Tuck (1964) for infinite depth, except that in infinite 
depth the horizontal motion terms do not appear. It should be noted 
that the surge motion ¢, and the horizontal fluid particle motions 
are large in shallow water, of 0(€-1), which is the reason why the 
relative horizontal motion is now potentially as important as the re- 
lative vertical motion in determining hydrodynamic effects. 
The first step to actual solution for the motions is numerical 
evaluation of the coefficients Tj; . This is a moderately difficult task, 
especially as regards the double integrals in (3.2) . This task is 
carried out indirectly, by Fourier transform techniques as described 
in Appendix II. 
An apparently trivial but actually significant point about the 
numerical computations is the fact that we may wish to avoid nume- 
rical differentiation of the section area curve S(x) to give S'(x) in 
(3.3) and (3,6) . In fact a simple integration by parts avoids this 
difficulty, but raises another question. If the section area S(x) does 
not vanish at the ends x =+f (e.g. with transom sterns), what do we 
do about the "integrated part'' after integration by parts ? Thisisa 
classical end-effect problem in slender body theory, since at least 
in principle slender body theory is inapplicable to such blunt ships. 
P55 
