Computatton of Shallow Water Shtp Mottons 
modes, The main force balance in heave and pitch is between hydro- 
statics and Froude-Krylov exciting forces, exactly as in infinite depth 
(Newman & Tuck 1964) . Remarkably however, in shallow water the 
added inertia dominates the natural inertia. This conclusion is only 
valid solongas o2= 0(€), i.e. for reasonably low frequencies, 
certainly well below resonance. This question is discussed further 
in Section 3. 
In sway and yaw most forces are comparable in magnitude so 
long as the clearance is not small, the only negligible effect being 
roll coupling. The latter is only significant if the roll magnitude is 
0( 208) , two orders of magnitude higher than that predicted. If the 
clearance is small, the added inertia dominates the natural inertia 
by an extra factor of depth/clearance. 
Finally, the amplitude of roll is profoundly affected by the 
order of magnitude of the metacentric height. The orders given cor- 
respond to Z); - ZG = O(e) , but broadly similar conclusions apply 
for (say) zy - ZG = O(e2). Only if z), - ZG = 0(e3) does the roll 
amplitude become large enough for roll to affect sway and yaw, or 
for roll inertia to be important. 
Another way of looking at this effect is to observe that the 
roll resonance frequency is roughly given by 
ae g(z,, = Za) 
ae 2 
x; 
Since k, = 0(e) , the frequencies of interest such that o“ = 0 (e) are 
necessarily far below resonance in roll, unless the metacentric height 
is as little as 0(&). 
A final rather more intuitive argument for neglect of roll is 
that the shallow water assumption requires that the incident wave 
pressure be uniform with depth. Thus the resultant force on a verti- 
cal wall (modelling a ship with a very small clearance in shallow 
water, beam seas) acts through the mid-point of the wall. One should 
anticipate a pure swaying motion of the ship section due to sucha 
uniform distribution of pressure. In terms of a roll angle measured 
about an axis in the waterplane, this amounts to the conclusion that 
the sway coupling term from M,, in (2.3) cancels out the net in- 
cident pressure moment about the waterline. 
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