Computatton of Shallow Water Shtp Mottons 
square root singularity at \= 1, removed by the changes of varia- 
bles A=1 } u2, respectively. The integrals are then evaluated by 
the trapezoidal rule, with a fixed given interval in u . In practice an 
interval 20-30 points per unit of \ have been found sufficient. The 
infinite range of the integral is accounted for by testing for conver- 
gence after integrating through about one unit of \} ata time, stopp- 
ing when the answer changes by less than 0.1%. 
The program is accurate but inefficient and expensive to run, 
taking about one minute (CDC 6400) fora run ata single depth and 
a single heading angle, each run including 8 frequencies, This time 
is at least half due to diffraction force computations, which doubles 
the number of Fourier transforms to be evaluated because of the fac- 
tor e!KX COSB in (3.6). Also, were it not for the diffraction force, 
the integrals (3.2) would be independent of depth and heading angle, 
enabling more information to be obtained cheaply for each run, Clear- 
ly much can be done to improve the efficiency of this integration pro- 
cedure; 
APPENDEX iil 
END-EFFECTS IN SURGE FOR TRANSOM STERNS 
In evaluating the surge exciting force T,;g we appear to re- 
quire the derivative S'( x) of the section area curve. To avoid nu- 
merical differentiation, we can integrate (3.1) by parts in this case, 
obtaining 
M Q 
T,) = Ps [S(x) art ape] gr PBiK cos | dxS(x) e 
: 9 (AIII. 1) 
ikx cosB 
tt “S{ £ Qf) 7 0, the question arises as to whether retention of the in- 
tegrated part in (AIII.1) is correct. This question is not easy to an- 
swer, in view of the fact that such extreme bluntmess of the ends of 
the ship ought to be precluded on slender body grounds. 
However, itis clear by considering the following special case 
that a decision on this matter can be made. Suppose the ship is a rect- 
angular box, with S(x) = Sp = constant , |x| <X. Then clearly the 
Froude-Krylov exciting force arises solely from pressure differences 
between the two flat ends of the ship. That is, the force amplitude per 
unit wave amplitude is 
Bey amicr Sp. LE ee i (AIII. 2) 
L671 
