Norrbin 



moment derivatives at F^l = 0«1 a-re increased by some 20 per cent 

 above their zero-speed values, an increase which is not fully 

 realized in model tests, A comparison of the results of this theory 

 with various experiments is presented by Newman, [41] . Newman 

 also points out that the free surface may give rise to a steady side 

 force as a thickness effect, and indicates a solution to that problem. 



From an inspection of the experimental results for the drift 

 moment, which are the more consistent, a first approximation to 

 the speed dependence is given by 



^K\K" = ^K.)o +iNi>" (6.5) 



where 2^|}^^y~ 1»3 (Ny'^)o . This suggests that the zero-speed values 

 will be some 20 per cent lower than those indicated by the mean line 

 of Fig. 11. 



Viscous Frequency Effects and Small-Value Non-Linearities 

 in Lateral Forces 



In dealing with the free-surface effects on added masses it 

 was concluded that so far the frequencies involved in manoeuvring 

 motions were to be regarded as low, but that frequency (or memory) 

 effects should be expected to appear in time histories were viscous 

 phenomena were of more concern. 



The extreme exemplification is furnished by the pitching 

 submarine, the stern planes of which are operating in the downwash 

 behind the bow planes, but in case of submarines as well as normal 

 surface ships also the very stern portion of the hull is exposed to 

 velocities induced by vortices trailing from upstream hull and 

 appendages. Moreover, local separation within the three-dimensional 

 boundary layer flow over the stern directly affects the cross -flow 

 momentum and the impulsive pressures. The forces and moments 

 experienced by the hull in transient motions can then only be calcu- 

 lated by use of convolution integrals over the entire time history, 

 such as derived by Brard in case of a special descriptive model, [ 42] . 



For application to the mathematical model defined by ordinary 

 differential equations it is again still possible to use frequency 

 dependent coefficients, but unfortunately this frequency dependence 

 is likely to be subjected to scale effects. It is therefore advisable 

 to design experiments for Strouhal numbers or reduced frequencies, 

 which are low enough to produce steady- state values. From a sum- 

 mary of published data in Ref . [ 41] the limiting frequency will be 

 expected to be sonaewhere in the region 1 < co' < 4, Fromi a more 

 recent analysis of sinusoidal free-sailing tanker model data Nomoto 

 suggests that this limiting frequency is approached already at 

 co' ~ 0.5, [43] . This indicates that the high-frequency part of a normal 



844 



