Beek and Tuck 
of the orders of magnitude of various terms in the equations of motion, 
We find that not all terms are of equal importance, and some may be 
neglected to an acceptable order of accuracy. This is a conclusion 
which may be arrived at formally by asymptotic expansion with res- 
pect toa small parameter ¢€ <<1 such that the beam and draft of the 
ship and the depth of the water are all small O(€) quantities relative 
to both the length of the ship and the wavelength of the incident waves. 
In some cases in addition we provide in later sections direct confir- 
mation of the smallness of the numerical effect on motions of terms 
which are asymptotically small. 
The orders of magnitude of Tij with respect to € are 
a 0 ea 0 Se 0 = 
0 Pe 0 a 0 Be “ 
3 2 2 2 
€ 0 € 0 € 0 € 
0(T,.) = Z eB golig (2. 9) 
J Or Aéigf (Ginnies wo bavSaw < 
2 0 ef 0 ef 0 me 
3 4 3 2 
Recall that the first column gives Froude-Krylov exciting forces, the 
last column diffraction exciting forces, and the remainder of the 
matrix added inertia and damping forces. The above orders of magni- 
tude are quite difficult to estimate, and the following observations are 
by way of explanation. 
Lateral symmetry of the ship provides the zero entries, de- 
coupling horizontal and vertical modes, and also affects some ex- 
citing force orders of magnitude. For example, if the ship did not 
possess lateral symmetry there would be an 0(€) contribution to the 
Froude-Krylov force T, 0 in sway. 
The water depth h, assumed 0O0(¢€), has a significant effect 
on these orders of magnitude, especially in the vertical modes. 
Whereas in infinite depth of water a unit magnitude vertical motion of 
a slender ship produces a small 0(€) motion of the water in cross- 
flow planes, such a ship motion produces a significant 0(1) lateral 
motion of water of O(e€) depth. Thus the surge, heave and pitch self 
and diffraction forces are all one order of magnitude larger than the 
corresponding estimates (Newman & Tuck 1964) for infinite depth. 
In horizontal modes (sway, roll and yaw) the assumption has 
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