(3), and the coefficients of the various powers of e equated to zero. From the co- 

 efficient of e one gets the linearized boundary condition on the ship : 



<pn(x,y } 0+, t) = - #1*0 - J'cdt,y). (21) 



From the coefficient of e 2 one gets 



<P2z(x,y,0 + , t) = $i x {x - J t cdt,y)i P i x (x,y,0+,t) + £ ly <p ly - fi<p u j (22) 



In the part of the plane z = outside of the centerplane section of the ship both y lz 

 and (p 2z must be zero, for we have assumed a symmetric ship and <p g must be zero there. 

 The boundary condition for y> is i > 2, will be of the form 



cp is (x,y,0+,t) = CY{fi,<pi, • • • , ^-_i}, (23) 



where C, represents some functional of the functions in braces. 



Several problems immediately occur to one, and, it seems to the author should 

 be considered if the procedure outlined above is to be regarded as more than formal. 

 The first concerns the class of functions to which one must restrict g t . Can, for 

 example, infinite slopes be allowed at the bow and stern? Although it does not interfere 

 with the formal procedure for linearization, it would seem to conflict with the require- 

 ment of small surface disturbance if the infinite slope is near the water-line; on the 

 other hand, exclusion of infinite slopes below the waterline would exclude treatment 

 of bulbous bows. Such questions are presumably connected with the convergence of 

 the perturbation series. As far as the author is aware, the only cases where convergence 

 has been proved do not involve solid boundaries penetrating the surface. A further 

 question concerns specification of the proper region of the (x, y) -plane in which condi- 

 tion (23) is to be satisfied. The simplest thing to do would be to take the part of the 

 centerplane section below the undisturbed water plane. However, this would neglect 

 the effect of the part of the hull above the waterplane on the wave-making, and this 

 must, of course, eventually be taken into account. If the region for ^ is to depend 

 upon the wave profile determined from <p i ^ 1 in the preceding step, as seems reasonable, 

 then (23) shows that one must first be prepared to extend one's knowledge of <p k , 

 k < i, beyond the regions in which they are originally defined. This problem does 

 not occur, of course, for a completely submerged "thin" body, for it arises from the 

 meeting of two boundaries with different boundary conditions. 



A point worth noting is that it results from the procedure of linearization that 

 the boundary conditions for y> 15 insofar as they concern the ship hull, are imposed on 

 the centerplane of the ship. Indeed, the same remark applies to ipi for i >1. Thus 

 recent attempts to improve the linearized theory by altering the boundary conditions 

 on the centerplane or by adding additional source distributions off the centerplane do 

 not seem to the author to be well founded as an attempt to find a better approxima- 

 tion to the exact solution. Such attempts usually have as goal to satisfy the exact 

 boundary conditions on the ship hull and the linearized boundary conditions on the 

 free surface. But this is contrary to the physical assumptions inherent in the use of 

 the linearized free-boundary conditions. The perturbation procedure described above 

 does give, subject to the noted difficulties, a systematic procedure for improving simul- 

 taneously and step by step the accuracy with which the boundary conditions on both 

 the hull and the free surface are satisfied. On the other hand, one should remain 

 aware that even if one finds ip 2 (or further cpi) in the preceding formulation, one is 

 still dealing with a "thin" ship and the method is not likely to be suitable for a form 

 which varies radically from a thin ship. For a form such as a barge a different method 

 of linearization would be used, i.e., a different choice of e would be made. Here one 

 should consult remarks of Stoker and Peters [1, pp. 14, 15]. 



112 



