b. The ship oscillates harmonically in position. If one sets 

 *(x,y,z,t) = Q e iut 



one obtains 



* o -^^=0 [7] 



Furthermore, if we introduce an appropriate length I and if we 

 choose I equal to the ship length L in Case a so that 



<f> = d> L, x = x L 

 1 i 



this immediately gives 



®!*i + Sk ?*i = o [6a] 



dx 2 u 2 az ! 



^ + lji=0 [6b] 



where F = — — is the usual Froude number. 



In the oscillation problem, Case b, it is obvious that I should be 

 set equal to the beam of the ship B, i.e., I = B. With * = * B, z = z B, 



Oil 



Equation [7] becomes 



" B^ dz x 



__g_ a* L= = [7a] 



* _ _L ^*l= o z = o [7b] 



1 F„ 2 dz x 



Here the important dimensionless oscillation parameter F u = wVB/g 

 is introduced. 



We see that in both of the special cases considered the potential 

 and therefore the hydrodynamic masses depend on the Froude number F and the 

 oscillation parameter F u , respectively. These cases can be extended to the 

 case of a uniform or accelerated translational motion with harmonic oscilla- 

 tion; for our purpose, however, Equations [6] and [7], which will now be dis- 

 cussed together, are sufficient. In this joint discussion of the two equa- 

 tions, we shall use the same symbol <f> for both. 



