Shallow Water Problems in Ship Hydrodynamics 



of p times the slope b'(x) that the net force (wave resistance) is 



f 0, F < 1, 



F, = \ (3.10) 



2^ 



r C [b'(x)f dx, F> 1 

 1 ^ -i 



No doubt Michell was disappointed in his conclusion of zero 

 wave resistance in the more important sub-critical regime, and 

 indeed this conclusion may have contributed to the neglect of his 

 shallow-water results. However, we can expect no other result 

 from the present theory, which lacks a dissipation mechanism in the 

 sub-critical reginne to leading order. This feature it has in common 

 with linearized aerodynamics. However, in aerodynamics the drag 

 vanishes even according to nonlinear theory for Mach numbers every- 

 where less than unity, whereas in the present water-wave problem 

 it is only to leading order that the wave-nnaking dissipation mecha- 

 nism disappears. No second-order calculations seem to have been 

 carried out to find the non-zero subcritlcal wave-resistance, and 

 this is a problem which merits attention. 



Michell's analysis for a wall-sided "ship" was extended to 

 ships of arbitrary cross-section by Tuck [ 1966] . In this case we 

 can expect to predict a squat effect, and, although the analysis in the 

 1966 paper is rather complicated, the main conclusion is quite 

 simple. By the method of matched asymptotic expansions (Van Dyke, 

 [ 1964]), Tuck showed essentially that Michell's result (3.9) for the 

 pressure still holds , providing we interpret the function b(x) as the 

 mean thickness of the ship at station x, averaged over the full depth 

 of the water, i.e. set 



•b(x) =^ S(x) (3.11) 



where S(x) is, as in Section 2, the cross-sectional area of the ship 

 at station x. Thus, for example, we obtain again Michell's wave 

 resistance formula (3.10) but with (3. 1 1) used to rewrite it in terms 

 of S(x). 



On the other hand, the modified geometry of the ship does now 

 allow non-zero vertical-plane forces and moments, and we find an 

 upward heaving force 



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