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Figure 'I - Vertical Motion, 

 Rigid Wall 





Figure 2 - Vertical Motion, 

 Free Surface 



free surface without gravity; we expect therefore that 



One derives further from Figure 2 that 



k z =k z [11] 







since S + S' together behave like a double body. Hence, the coefficient k 

 is known when k„ is given. Similarly, as for Equation [11], we conclude 



yy 



yy 



ilia] 



k xx ~ k xx 



nib 



Figures 3 and '4 explain conditions when the body moves horizontally . 

 The rigid wall (Figure 3) has the function of a plane of symmetry; S + S' 

 move again together as a double model. We conclude therefore that in this 

 case 



k y =k y [12] 



and by analogy 



k 



k x = k x 



J2a] 

 [lib] 



Figure 4 pictures the conditions at the free surface; obviously, one follows 

 that k <k and therefore k < k . The same applies to the other components, 



y y y y v * 



The difference in the results obtained for motions in a vertical 

 and a horizontal plane is essential. Using one of the approximate boundary 

 conditions, we can substitute a double model for the ship only for a single 

 direction of motion. The difference in the effects involved is appreciable: 

 in the case of an elliptic cylinder which floats at its plane of symmetry, 

 the ratio £/k is only h/if . 



k 



y 



0.4 



[12c] 



