shallow Water Problems in Ship Hydrodynamics 



width curve, B (k) , xB (k) are Fourier transforras (cf. (4,1)) of 

 B(x) , xB(x) respectively, and a bar denotes a complex conjugate. 



The quantities F^ , F^ must be computed separately, e.g. 

 by computer programis such as those of Iless and Smith [ 1964] or 

 Tuck and von Kerczek [ 1968] , Alternatively, one may estimate them 

 experimentally. It is important to note that F?°, f2° are independent 

 of water depth and of Froude number; indeed, when divided by pU 

 these are constants which are a property of the hull geometry alone. 



An interesting special case is a ship with fore-and-aft sym- 

 metry, where (neglecting viscosity) F5 = 0. In addition, since 

 S(-x) = S(x) and -xB(-x) = -xB(x) , S"^ is real and even with 

 respect to k whereas xB is imaginary and odd. As a result, 

 only the imaginary part (4.12) of A'^(k) contributes to the integral 

 in (4, 1 5) , and we have 



tt2 r°° 



F, = -£^\ dk k^S*9(xB*)JA* (4.16) 



^ 2Tr^ Jo 



1 

 0, F = (/ch)"2 < 1 



(4,17) 



£S r°° d^^go^ S*(k)JxB*(k) , F > 1 



'0 Vqo - k 



Similar, but more complicated, results are obtained for F -F 

 and Fg-F^ when the ship does not possess fore-and-aft symnnetry. 

 Evaluation of the supercritical trina moment F^ from (4.17) re- 

 quires only a single numerical quadrature (apart from the prior 

 estimation of the Fourier transforms S*, xB ), However, in the 

 general case an additional numerical quadrature is needed to deter- 

 mine the real part of A'''(k) from (4.10), 



The shallow-water limit of the above finite-depth results 

 corresponds to letting kh -* , i.e. we let the depth tend to zero 

 relative to a typical effectiv^e wavelength Z-rr/k. In particular, from 

 (4. 11) we have qgh -* as kh — ^ and hence k — ^ ^fKh.qQ, or 

 qQ -► Fk. Thus for F > 1, (4,17) gives for ships with fore-and-aft 

 symmetry that 



Fg -. £g _£ \ dkkS*JxB^'' (4.18) 



= pU C S'(x)xB(x) dx, (4.19) 



ZttVf^- 1 -'-i 



643 



