Ogilvie 



3,1. The Mode rate- Speed, Steady- Motion Problem 



The theory presented here is due to Tuck [ i963a] . The 

 analysis --as far as I carry it here --is not very much more diffi- 

 cult than the analysis of the infinite -fluid problem, and so it will 

 only be sketched here. 



The theory is attractive for its simplicity and its elegance, 

 but unfortunately it has not been successful in predicting wave 

 resistance. The reasons are not entirely clear, although they have 

 been discussed for many years. See, for exannple , Kotik and Thonnsen 

 [ 1963] . The difficulty could very well be that real ships are just 

 not slender enough for a one-term expansion (or perhaps any number 

 of terms) to give an accurate prediction of wave resistance. This is 

 the old question, "How small must the 'small' parameter be?" 

 Another possibility is that the error arises because the lowest-order 

 slender-body theory places the source of the disturbance precisely 

 on the level of the undisturbed free surface, and so there are no 

 attenuation effects due to finite submergence of parts of the hull. 

 (These two possible causes of error are not entirely separate.) 

 Still another possible cause Is considered In Section 3.2. 



The hull surface will be specified by the equation: 



r = ro(x,e). (3-1) 



Now it will be convenient to measure 9 from the negative z axis, 

 since most ships are symmetrical about the mldplane. We assume 

 that rQ = 0(c) and that 8"rQ/8x" = 0(c) , as needed. 



There Is a velocity potentlcil satisfying the Laplace equation 

 and the seune kinematic body boundary condition, (2-58), as in the 

 infinite -fluid problem. The Incident stream Is again taken In the 

 positive X direction, that Is , with velocity potential Ux. The two 

 free- surface conditions are: 



g4 +-^[<i>x + *y + <l>5 = 7U^ on z = C(x,y); (3-2) 



;A"^ ^<t>y- 4>j = 0, on z = ;(x,y). (3-3) 



Finally, there is a radiation condition to be satisfied. 



This reference Is not readily available, but the material which Is of 

 Interest here can also be found In Tuck [ 1963b] , Tuck [ 1964a] , and 

 Tuck [ 1964b] , all of which are gathered Into Tuck [ 1965a] . 



732 



