Submerged Two-Dimensional Bodies 



where only terms to the order /S^ have been included. Hence, the pressure is 

 constant (p = 0) on the free surface if 



V - — 



(A8) 



Let us now investigate if it is possible to have a Stokes wave behind a sub- 

 merged two-dimensional body in a uniform stream. The two following condi- 

 tions must be satisfied. 



1. The pressure at the horizontal plane y = -h ash-»co must be the same far 

 upstream and far downstream. 



2. Conservation of mass: The inflow through a section upstream must be 

 equal to the outflow through any section downstream. 



The pressure at a depth y = -h far downstream is by the Bernoulli equation, 

 Eq. (A5), 



- + ^ U^ (-2z./3e-^'' cos vk + v^ fi^ e' ^''^) - gh = , (A9) 



and therefore when h is very large 



P = Pgh . (AlO) 



Hence, to satisfy the pressure condition 1, we have that the undisturbed free 

 surface far upstream is given by y = 0. 



The conservation of mass condition requires that 



T7 



[ udy r I'xdy 



lim ; = 1 im ; 



(All) 



h -» CO 



where the left side is 



j Udy 



(A12) 



u 



h -♦ 00 



and the right side is 



625 



