Beck and Tuck 
One interesting feature of the heave equation of motion in 
beam seas (f= 90°) for a fore-and-aft symmetric ship is that all 
second order terms disappear, so that were it not for the third-order 
mass terms the heave would exactly equal the wave amplitude. Thus 
at B= 90° ,@(3.6) gives A, (x) = -B(x) = -A,(x) , and we have 
T,, = -T,3 - Butalso T,, = C3, hence assuming fore-and-aft sym- 
30 
metry the heave equation of motion is 
2 
-o MS, + (Ge. rE), wd bbe ib =" 05, (3, Wp 
Hence 
(4 
Corre tof [2 - au | 
3 0 C33 or 
3 (3. 12) 
= abel doc Of ems) 
Similarly, if we do neglect all second and third order effects, the 
first order theory predicts zero surge and pitch, and heave fy = 5 
even in the absence of fore-and-aft symmetry. 
Figures 3.4 - 3.5 show computed heave and pitch motions in 
beam seas. There is a substantial (60%) increase in heave over the 
first-order value {| = ‘= as the depth increases, especially at about 
L/\ = 1.5. The pitch (in bow motion) remains below 25% of the 
wave amplitude, however, and surge is quite negligible, never more 
than 2% of the wave amplitude at any frequency. 
IV. MOORING FORCES 
As an example of the type of analysis required in order to ac- 
count for the effect of mooring lines on motions (and perhaps more 
importantly, vice versa !), we give below a simple discussion of the 
effect of a single linear bow mooring line on vertical plane motions. 
More realistic and complicated types of mooring systems can be 
studied with similar procedures and conclusions, The general con- 
clusion is that of Wilson and Yarbaccio (1969), who find that ''the 
spring is quite weak compared to the mass, and the ship can be con- 
sidered to be floating unrestrained except for restraint against conti- 
nuous drifting''. 
If we consider only linear effects of mooring lines, the appro- 
priate modifications to the equation of motion simply require contri- 
butions to the restoring force coefficients Cjj in equation (2.2) 
1556 
