Doctors 



t oo oo 



*U,y,z,t) = --^- //dS' / dr/ dw / dup S r (r,y; T ) 



S' -oo -°° 



coshj k(z + d)} sin)7(t-T)} 

 cosh(kd) " 7 ' ' ' 



... exp{i(wU -$') + u(y - y'))f (22) 



The pressure distribution, p 5 , as measured in the stationary- 

 reference frame is a function of time. It is related to the pressure 

 in the moving frame, p , by the following equation : 



g 



p U .y,t) = p(x,y) 



± p(* - s(t),y) (23) 



III. 2. The Wave Resistance 



The resistance of the pressure distribution is defined as the 

 longitudinal component of the force acting on the free surface, and is 

 therefore given by 



R(t) = / P S U .y.t) ^ d£ dy 



[24) 



'S 

 Substituting Eq. (11), we obtain 



R = 



5 



The second term in this expression contributes nothing to 

 the integral providing the pressure drops to zero at £ =ioo. The 

 result for <f> , Eq. (22), is now used. If one expresses the pressure 

 in terms of the moving frame by Eq. (23), then the wave resistance 

 becomes : 



t OO oo 



R= l - //p dS // p'dS' / c(r)dr|dw |duw?cos|T(t-T)| 



S S' -oo -oo 



exp|i(w(x - x' + s(t) - s(t)) + u(y - y'))( 



The real part of the integrand is now expanded. Then it is simplified 

 by invoking properties of even and odd functions. The final result is : 



44 



