Tuak and Taylor 



ratio of 1 In 20 is quite adequately shallow for the use of shallow- 

 water theory. But the results at twice this depth are also in reason- 

 ably good agreement with shallow-water theory, the latter theory 

 underestimating sinkage by about 20%. The corresponding under- 

 estimate at 100 ft d§pth is about 40%, so that one should consider a 

 water-depth/ship length ratio of 1 in 6 as too great to use shallow- 

 water theory for sinkage. 



However, it must be pointed out that the difference between 

 the finite depth and shallow water predictions of sinkage is pre- 

 dominantly due to the influence of the term Ff* in (4. 14). If this 

 (positive) term is left out of (4.14) the finite depth computations at 

 cdl depths merely oscillate about a mean which is quite close to the 

 shallow-water curve. These oscillations are clearly visible in 

 Fig. 2; they are quite similar to the humps and hollows in theoretical 

 wave resistance curves, and have the same explanation, as an inter- 

 ference effect. One may thus speculate that, by ignoring these oscil- 

 lations, we may obtain a useful empirical scheme for computing 

 finite depth sinkage by adding the shallow water estimate (e.g. (3,21)) 

 to the infinite fluid zero Froude number estimate computed by (4.23). 

 Further work needs to be done to test this suggestion, which is of 

 some significance since computations based on the theory of the pres- 

 ent section are too complicated and expensive of computer time for 

 general use. 



No direct comparisons of the finite depth computations with 

 experiment have yet been made, but the differences between the finite- 

 depth and shallow-water results appear to be in the right direction 

 to explain most of the discrepancies already noted by Tuck [ 1966] 

 between shallow-water theory and the experiments of Graff et al. 

 [ 1964] . In particular, more detailed computations for a depth of 

 100 ft, (h/L = 0.167) show that the peaks in the sinkage and trim 

 curves occur at about the right Froude numbers, 0.94 and 0.98 

 respectively, and that the trim starts to become significant at a 

 Froude number as low as 0.8. 



V. THE ACOUSTIC ANALOGY FOR UNSTEADY LATERAL FLOW 



In the remainder of this paper we shall be concerned with a 

 very special aspect of the problem of ship motions in shallow water, 

 namely computation of the exciting force on a stationary ship under 

 the influence of regular beam seas. A more general formulation 

 and partial solution of the problem of ship motions in shallow water 

 is given by Tuck [ 1970] . Most other work on ship motions in shallow 

 water, e.g. Freakes and Keay [ 1966] , Kim [ 1968] , concerns finite 

 depth rather than shallow water. An exception is a number of papers 

 by Wilson (e.g. [ 1959]) on responses of moored ships in harbors, 

 but no account is taken there of ship geometry. Mention must also 

 be made of the thesis by Ogilvie [ I960] , in which the shallow- water 

 asym.ptotic expansion was developed rigorously for a class of two- 



646 



