Computatton of Shallow Water Shtp Mottons 
Pitch is almost unaffected by diffraction effects, the substan- 
tial increases shown in Figure 3.2 over the first-order theory at 
about the pitch maximum (L/) = 0.8) being instead largely due to 
inertia, especially added inertia. A rough explanation of the nume- 
rical smallness of the pitch diffraction force is that the first term of 
(3.6) involving B(x) is a nearly even function for a nearly fore-and- 
aft symmetric ship ; for reasonably low values of k (thus low L/\) 
the corresponding value of T is small because A, = -xB is near- 
ly an odd function of x. The terms of (3.6) involving S(x) corres- 
pond broadly to surge motion, and lead to small effects when the 
surging is small, as it is at these wavelengths. 
All second order effects on surge appear to be small, the 
first-order balance between natural inertia and Froude-Krylov ex- 
citing force being (Figure 3.3) remarkably close to the full result. 
The magnitude of the surge motion is, as expected, quite large for 
the longer waves, Tyo. < 0.8, say, for which the horizontal particle 
motion in the wave is large. At fixed L/\ , the first-order surge 
varies exactly inversely with depth, and the full equations give a 
similar trend except at high frequencies where surging is in any case 
quite small. 
Computations have also been carried out in oblique seas, i.e. 
for values of B other than 180°. In general, effects of reasonable 
heading angle on vertical plane motions are mostly accounted for by 
use of head seas results, but with the effective wavelength 2d sec 6 
instead of \ in the horizontal scale. 
This is exactly true for the first-order results in heave and 
pitch, and nearly so when second and higher order terms are includ- 
ed. In surge this effect is combined with a "'cos?8" factor, tending 
to reduce surging. However, since the effective wavelength is longer 
than the true wavelength and surge is greatest in longest waves, we 
should anticipate increased surging, were it not for the cos fac- 
tor. The net effect at fixed (true) wavelength isa ''cosfB'' reduction 
factor on surge. 
Since the computed results agree well with the above qualita- 
tive discussion, we omit presenting computations for bow seas 
(B= 135°) . Note however, that the "stretching out'' of the head seas 
curves due to the sec 8 factor means that heave and pitch are both 
increased at values of L/A (true) of about 1.2 - 1.6, where the 
head seas responses were small, In this important range bow seas 
produce significantly greater net vertical bow motions than do head 
seas (see Beck and Tuck, 1971). 
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