shallow Water Problems in Ship Hydrodynamics 



dimensional diffraction problems. 



We now suppose that, except for the scattering effect of the 

 ship, the flow field is described by an incident plane wave moving in 

 the y-direction, with wavelength Z-rr/k, frequency cr , and amplitude 

 A, where k and cr are related by the shallow- water dispersion 

 relation 



o-=Vghk. (5.1) 



The potential of this wave will be taken to be the real part of ^qS'^^^^ , 

 where 



(5.2) 



In fact, (5,1) and (5,2) are of course already approximations to the 

 exact formula (e.g. Wehausen and Laitone [ i960]) for small-ampli- 

 tude waves In finite depth h; for Instance the exact expression for 

 <j)o is that given by (5.2) multiplied by cosh k(z +h)/cosh kh, which 

 tends to unity as kz -♦ 0, 



This Incident wave is modified by the presence of the ship. 

 We suppose that the total field is then the real part of (cf)Q + <J))e"'*'"* , 

 where <^ = <|)(x,y,z) is the disturbance potential, which is to be 

 found. The exact equations satisfied by (|> are (3.2), (3,3), an 

 unsteady free surface condition analogous to (3.4), namely 



g|| - 0-^4) = on z = (5.3) 



and a boundary condition on the ship's hull of the form 



-|^(4>o+^)=0 (5.4) 



where 9/9n denotes differentiation normal to the hull. 



We construct first an outer shallow- water approximation to 

 <|) in the same way as in Section 3, i.e. by expanding in a Taylor 

 series* with respect to (z +h). Equation (3,7) still applies, but on 

 substitution of (3.7) in (5.3) we now find 



ax^" ^ 972/ <*>(^'y''^) " ^ <l'(x,y,-h) = 0, 



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