shallow Water Problems in Ship Hydro dynamics 



extending from the bottom z = - h to the top z = of the water. It 

 is apparent right from the outset that with this model of a ship we can 

 expect no prediction of squat, for the "ship" has vertical sides every- 

 where, and no fluid passes under it. Michell's only concern was with 

 wave resistance. 



The mathematical problem is specified by a disturbance veloc- 

 ity potential ^ such that the fluid velocity is V(Ux + <j)) , satisfying 

 Laplace's equation '^ 







- h < z < 0, (3.2) 



the bottom condition 



•1^= on z = - h, (3.3) 



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the linearized free surface condition 



g||^U^-§=0 on z = 0, (3.4) 



and the linearized hull boundary condition 



■|^=±iUb'(x) on y=0,. (3.5) 



Both equations (3.4), (3.5) are linearized on the basis that the ship 

 Is thin , i.e. that its slope b'(x) is everywhere small, so that ^ and 

 its derivatives are small, as Is the free surface elevation. 



We now apply the assumption that the depth h Is small. The 

 corresponding approximate equations may be obtained formally by 

 stretching the z-coordlnate with respect to h, then carrying out an 

 asymptotic expansion In terms of the small parameter h/i , see 

 Wehausen and Laltone [ I960] . However, the leading terms are 

 easily obtained by simply expanding (|) In a Taylor series with re- 

 spect to z, about the bottom value z = - h, I, e. 



<|>(x,y,z) = (()(x,y,-h) + (z+h)<|)j,(x,y , -h) 



+ |(z+h)%„(x,y,-h) + ... . (3.6) 



The second term in the expansion (3.6) vanishes by (3.3), 

 and we use (3.1) to express cf)^^ ^" terms of ^^^ and <|>yy , writing 



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