Variational Approaahes to Steady Ship Wave Problems 



?Jx,y,0) + g\{y.,Y,0) = 0, (A. 22) 



so that the fundamental singularity is the same as that for the direct 

 flow, except that the wave follows on the downstream side. This may 

 be ejqpressed as 



S(P,Q) =S(Q,P), (A. 23) 



we also have 



Sx.(P,Q) = S,(Q,P). (A. 24) 



The boundary conditions for this case are 



<|>y = - <})v = Xy on S, (A. 25) 



and 



(^j = - w = - <j)j = w = - t,jj on S. (A. 26) 



APPENDIX B 

 The Progressing Wave 



Let us obtain the solution for aperiodic progressing wave, 

 moving at constant unit speed, by the variational method of §3. 



We take the form of the complex potential to be 



<1> + iijj = - ia exp (kz - ikx) , (B. 1) 



where the origin is on the undisturbed water level. 

 The integrcils to be evaluated are 



.ir/k 

 M = - \ 



P = M - T - V, iM=-\ dx \ <j)xdz, 



7^ = lJJ (V*)'dxdz, iv= ij^'dx. 



(B.2) 



569 



