Now, turn to the equation for pressure which is necessary to com- 

 pute the force on the body. 



Take the pressure to be zero at the free surface. Then Bernoulli's 

 equation may be written: 



P = - P If - 7 pvrv* - pgy. (E-19) 



Substituting the expansion for ((> : 



P = - p{e -^ . e -^— . 2 [^ C-^x-) 



^ ^ ^—^ ] + gy} + oce-"). 



ay 



Since (f) = 0, we can drop this term and proceed to separate the 

 equation by order: 



P^l) = -P-^F-- PSy (ii-20) 



^""^ r?1 aJl^ 2 (1) 2 



P - - I [C^-3 - C^) ]• (E-21) 



Substituting the velocity potential into the equation, one finds: 

 p'-^-' = - pglA^e'^l^ sin(k^x - w^t + 6^ 



+ A2e'^2^ sinCk^x - co2t + 62) + X) (E-22) 



for the first order, and 



P*-^-* = - I { [- oa^A^e^^l^ sin(k^x - co^t + 6^) 



k y ^ 



- w2A2e 2-^ sinfk^x - Wjt + 62)] 



k y - 

 + [a),A,e 1 cos(k X - a),t + 6,) 



+ u)„A„e 2^ cosCk^x - a^^t + ^2-'] -^ 

 for the second order. Note that this is identical to part of the 



154 



