Eggers 



resistance expressions was not performed, credit is generally given to Sisov (4) 

 for first dealing with these. We should, nevertheless, be aware that expressions 

 given by Sisov so far essentially contain divergent integrals due to the selection 

 of an improper radiation condition for Green's function of the pressure point. In 

 our present investigation, we will rederive some of Sisov 's results from a 

 Green's theorem approach essentially following Wehausen. We will, in particu- 

 lar, show some simplifications which make calculations straightforward once a 

 Fourier representation of first-order flow components is given. It will become 

 evident that integration over the undisturbed free surface has to be performed 

 only in a small domain where local flow is significant; third-order wave resist- 

 ance is, therefore, much more tractable to numerical evaluation than is appar- 

 ent from what was formulated by Sisov, provided we decide on an appropriate 

 definition of wave resistance. 



We decided to deviate from Wehausen 's approach by some simplifications 

 regarding the actual flow boundaries. However, the resulting expressions found 

 for third-order resistance depend in a simple manner only on the ship's offsets 

 and on first-order velocity components. We, therefore, feel that these devia- 

 tions at least have not introduced artificial complications against results still to 

 be found from more refined analysis. 



Regarding the third problem, i.e., ships of minimum resistance within low- 

 est order theory, our investigation should throw light on the question as to what 

 degree third-order contributions might counteract the tendencies predicted. At 

 the present stage, however, our calculations are limited to a two-parameter 

 class of hull forms having parabolic waterlines. This is mainly due to the fact 

 that we preferred analytical evaluation of integrals over the geometry of the 

 ship. An extension of our program for local flow, to include contributions from 

 empirical surface elements is feasible, but loss of closed integration would 

 probably increase time for computation and weaken control of accuracy. More- 

 over, the necessary degree of hull subdivision will in general depend on the 

 Froude number and is not known beforehand. Even for analytical ship forms, 

 the development of formal expressions for closed integration cannot be done by 

 the computer and provides many opportunities for errors in evaluation of singu- 

 lar regions of integrands for local flow. 



DESCRIPTION OF ANALYSIS TO DERIVE POTENTIALS 

 AND WAVE RESISTANCE 



We shall essentially follow the approach of Wehausen (3), but modify it for 

 flow in a tank of rectangular cross section. This will simplify the formulation 

 of radiation conditions for the flow and allows the use of a Green's function in a 

 Fourier series representation regarding the coordinate y chosen in the direc- 

 tion perpendicular to the vertical tank walls. The ship's motion is in the +x 

 direction with speed c ; the z coordinate is taken vertically upward to conform 

 with earlier work (6). As far as possible we otherwise use notation consistent 

 with Ref. 3. However, the direction of normal vectors is reversed resulting 

 from our definition of Green's function with an opposite sign. Extension of the 

 results to unrestricted water is straightforward. 



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