10 



In general, however, we must look for a better mechanical model 

 than that presented by the rigid wall or the free surface neglecting gravity. 



2.4 It is natural to substitute systems of singularities (sources or 

 dipols) for the body (ship) and to calculate the hydrodynamic masses using 

 these images and the boundary condition, Equation [5]. Following a procedure 

 known in the theory of wave resistance, these images apparently could be de- 

 termined using a uniform flow in an unbounded medium. Since this approach is 

 successful when calculating the resistance at uniform speeds of translation, 

 it has been applied several times to investigations of accelerated transla- 

 tions and oscillations in calm water and in a seaway. 6 ' 7 The result is sur- 

 prising; the hydrodynamic inertia factors like k^ k found in such a way be- 



c o X 



come equal to k k , etc.; there is no dependence upon the Froude number or 

 other characteristic parameters. 



This obvious contradiction to simple physical reasoning has not 

 been detected for some time. It induced. the author to make some similitude 

 investigations; from these the importance of such parameters as F, F w , and 

 a/g and the history of the flow field has been realized when calculating the 

 added masses. In the meantime, three fundamental papers have been published 

 which clarified, at least in principle, the intricate problem, although an 

 exhaustive solution is still lacking. 



2.5 For the special case of a horizontal accelerated motion of a circu- 

 lar cylinder, Sir T.H. Havelock proved that the hydrodynamic masses can only 

 be found by using a second approximation for the image distribution. 8 



He further obtained explicit values when the acceleration of the 



cylinder a is constant. Beside the depth of immersion ratio r/f discussed 



before, the ratio a/g becomes a decisive parameter. An interesting diagram 8 



shows the coefficient k as function of the Froude number F when the cylinder 



has been accelerated from rest with a given a/g = constant. It is surprising 



that k = k holds not only for F > °o, which is well known, but also for 



y y ' 



F -> 0. Otherwise expressed, in the present case of a uniform acceleration, 



the hydrodynamic masses in the starting condition are the same as for an im- 

 pulsive motion; the influence of gravity can be neglected. Starting condi- 

 tions are especially important in the theory of directional stability. 



Unfortunately, at present we are not yet able to generalize the 

 results obtained under the assumption a/g = constant for other acceleration 

 laws which may be more realistic. But following Havelock, another generaliza- 



o 



tion seems to be plausible: the result k = k remains valid for the start 



y y 



of the constantly accelerated motion when the acceleration begins not at U = 0, 

 but even at U # . 



