Beck and Tuck 
This question is examined further in Appendix III, where it is 
argued that, at least in so far as the surge exciting force T,, is 
concerned, the theory remains valid for ''blunt'' ships, provided we 
discard the terms arising from integration by parts. It seems likely 
that a similar consideration applies to all expressions involving S'(x). 
Of course in the absence of transoms etc., i.e. when S(t) = 0, this 
difficulty of interpretation does not arise, and this is true of the com- 
putations to be presented here for the Series 60, block 0.80 hull. 
Figures 3.1 - 3.3 show vertical motions computations in all 
3 modes for head seas (f= 180°). The horizontal scale chosen is 
ship length divided by wavelength, while the vertical scales represent 
linear displacement amplitudes divided by wave amplitudes. In pitch 
this is equivalent to vertical bow motion due to pitching alone. The 
results are given for depths of 1.0 and 2.5 times the draft of the 
ship (0.062 and 0.15 times the ship length). A depth equal to the 
draft is of course not safely achievable, but no difficulty arises theo- 
retically in this case for vertical modes (not so for horizontal modes) 
and this case may be viewed as a limiting one in practice. 
The motions shown are those resulting from use of all available 
information about terms in the equations of motion. In spite of the 
imbalance in orders of magnitude as indicated in the previous section, 
no terms have been neglected, and all couplings between all three 
modes have been included. 
For comparison purposes however, the first-order results are 
also shown, these being balances between hydrostatic and Froude- 
Krylov forces only in heave and pitch, and between natural inertia and 
Froude-Krylov forces only in surge. In heave and pitch the first-order 
result is independent of depth at fixed wave-length, whereas the first- 
order surge varies inversely as the depth. 
The effects of the second (and third) order terms are quite 
varied, but some general comments can be made. The main difference 
between the first order and full heave results in Figure 3.1 is due to 
the diffraction exciting force. This is particularly true near the mi- 
nimum of the first order heave (about L/\ = 1.2) , where the heave is 
substantially increased by diffraction effects. 
The general trend of the heave results is remarkably similar 
to those of Newman and Tuck (1964) for infinite depth. The first order 
heave minimum at about L/A= 1.2 appears in both cases to be shift- 
ed by second order effects, especially diffraction, to about L/) = 1.4. 
This is not too surprising numerically in view of the similarity bet- 
ween (3.8) and the equation of motion in infinite depth. 
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