Variational Approaches to Steady Ship Wave Problems 



APPENDIX A 

 The Linearized Velocity Potential [2,23] 



Let us consider the flow of water around a ship S, taking the 

 coordinate system as In Fig. 1 and the velocity of the stream at up- 

 stream Infinity to be unity. 



(FREE SURFACE) 



(UNDISTURBED) 



UNIT 

 VELOCITY 



Fig. 1. Coordinate System 



The pressure p(x,y) on the water surface Is given by 



-p(x,y) = - <|>(x,y,0) - g^(x,y). 



(A.l) 



In the linearized theory, where p Is the water density; g, the 

 gravity constant; t,, the surface elevation, and <|), the perturbation 

 potential (d(|) =-udx-vdy-w dz). The suffix stands for differ- 

 entiation. 



The kinematic condition on the water surface Is 

 6(x,y,0) = ^ (x,y). 



(A. 2) 



Since the pressure on the free surface Is constant, the potential 

 must satisfy the condition 



<»)Jx,y,0) + g<j)_(x,y,0) = 0. 



(A. 3) 



A solution which has a source singularity at a point Q and 



565 



