Ogilvie 



Newman expands each of the dependent variables in multiple series expres- 

 sions. For example, the potential is made to depend on all three small param- 

 eters: 



(p(x,y,z,t) = J^ /3i yj 8"^ (p. .^(x,y,z,t) , 

 i , j ,k 



whereas the motions are represented by double series, e.g., for heave, 



' . j 



(The unsteady motions depend on 7, by the definition of this parameters, but we 

 also expect steady displacements, which will depend on /3. This is the reason 

 for including both parameters in the expansion here.) 



Since Newman develops his analysis on the assumption that it will be neces- 

 sary to include higher than first order effects, he carefully introduces other 

 needed expansions, which will not be written out here. For example, he trans- 

 forms from a body-fixed coordinate system to a steadily translating system, 

 both for calculation of the potential functions and for calculation of the pressure 

 and forces; this transformation involves the parameters ,3 and y. Finally he 

 obtains a sequence of problems, each homogeneous in each of the small param- 

 eters, and he solves explicitly for the following potential functions: ^p^q^, the 

 potential for the steady translation problem; Pqo j, the potential for incident 

 waves; Pho? the potential for small motions of the ship in an otherwise undis- 

 turbed ocean; cp 101, the diffraction potential. The first is just the Michell- 

 Havelock potential, and the second is the classical potential for sinusoidal 

 waves on an infinitely deep ocean. The third and fourth potential functions are 

 somewhat more interesting and deserve some further comment. 



If a ship model is forced to perform small oscillations of order of magni- 

 tude 7, perhaps by being driven by a mechanical oscillator, then the appropriate 

 potential function is (Puo • The one complication of interest here lies in the 

 specification of the body boundary condition. In the initial formulation of this 

 problem, it is necessary to state the boundary condition on the actual, instan- 

 taneous position of the body surface. Then, by a systematic procedure, this 

 condition can be translated into a (different) condition on the mean position of 

 the body. This problem has already been mentioned; see the discussion accom- 

 panying Eq. (10b). There is an interaction between the ship oscillations and the 

 steady flow past the ship which produces effects in the lowest order unsteady 

 solution. This interaction is lost if we assume immediately that the body bound- 

 ary condition can be satisfied on the mean position of the hull. Such an error 

 has occurred frequently in work in this field; the first correct treatment is ap- 

 parently due to Hanaoka (1957). Newman (1961) discusses the problem quite ex- 

 plicitly and shows that the difference in the potential fvinctions, corresponding 

 to the two methods of satisfying the hull condition, is equivalent to the potential 

 of a line distribution of oscillating sources located on the mean keel line. It 

 must be emphasized that this is not a higher order effect, and the problem is not 

 related, for example, to the arguments about how to satisfy the body bovindary 

 condition in the steady motion (resistance) problem. The elegant formulation of 



54 



