Ship Waves and Wave Resistance 



«c = «e + «lP 



(v X n) X c 



dsy = R^ + p 



N 



(v • c)n - (c • n)v 



dS 



(20) 



Therefore, with (c • n) = (v • n) assumed even for the first-order flow, 



/' 



P(ei • n) 



(c • v) - (v • v)/2 



dS 



(21) 



But this is just fig, the resistance defined from pressure integration over the 

 hull, because the nonstatic pressure is /o[(v • c) - (v • v)/2] and we thus have 

 from the contribution of S,.? 



R^ + P J (Px [(c • n) - (v . n)] dS . 



(22) 



To R^ as defined above we now add an appropriate correction for the influence 

 of the wave profile along the water line, as only the wetted part of the hull can 

 experience pressure from the fluid, and then define the quantity obtained as 

 "third-order wave resistance." Up to second order, pressure is atmospheric 

 pressure plus hydrostatic pressure due to the wave elevation ^. Integrating the 

 last quantity over dZdY, the projection of the surface element on the plane ver- 

 tical to X axis, we find a correction as 



AR = pg ^ (Z- O dZd77 = pg j. ^ ^^^^^^ dv . 



(23) 



Now the perturbation procedure gives the first-order wave elevation (2) as 



^'''- ^t^X. i.e.,^=^0i^\x,Y,O). 



(24) 



and adding this in nondimensional form to Eq. (19), we see that this correction 

 just cancels the line integral around Lp, which we therefore can happily discard 

 (8). 



A further simplification will be made by extending the integration of s over 

 the whole plane (,= 0, -T < rj < t , and ^ < X^, which means an error of order 

 e"*, as the water plane area is of order e . Inserting now ^ = ei/'^ *^ + e V^ ^\ 



§ = e 2s ( 2 ) ^ and R = e 2r( 2 ) + e 3r( 3 ) we have 



{(2) = r 





./,(!)' + 0O)^ _ ,,,(1) 



2 ,(In2 



27 



dL 



drj 



(25) 



andR^^^ = R^i^^ + rI,^^ (corresponding to the partition ^^^= i/;^!^^ +i//^^^), with 



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