2.3 Let us consider two limiting cases: 



a. The velocity or the frequency is extremely small, i.e., one 

 can set F 2 ■*■ or F 2 ■*• 0. Equations [6] and [7] transform into the simple 

 boundary condition 



M= z = [8] 



dz 



i.e., the vertical velocities vanish or the surface acts as a solid boundary. 



b. In the case of extremely high velocities or frequencies, i.e., 

 F 2 -*» oo or F 2 ■»■ oo, the boundary conditions simplify to 



3J = and *=0 z = [9] 



ax 2 



It can be shown, although with some difficulties, that the condition ?_r- = 



dx 2 

 likewise leads to the condition (j> = 0. 



Physically, this boundary condition indicates, as is well known, 

 that we can neglect gravity in comparison to the inertia forces. Therefore, 

 this is true in particular for impact phenomena on the surface. 



The solution for the hydrodynamic masses obtained on the basis of 



the simplified boundary conditions ^ - and <i> = are self -evidently no 



oz 



longer dependent on the parameters F and F u , and consequently fail as general 

 solutions of the problem. An indiscriminate application of the results, as 

 frequently occurs at the present time, Is therefore misleading. On the other 

 hand, they represent important limiting cases and reveal some interesting 

 facts so that it is worthwhile to discuss them in some detail. 



First of all, we shall complete the system of our symbols. To 



u 



differentiate from the previously introduced concepts k , ..., m , ... (briefly 

 called the "deeply immersed" values) and k ,..., m„ ,... (the "wave" values), 



Rett 



we shall designate the hydrodynamic mass values for the solid boundary jp = 



by k .... , m , . . . and for the free surface neglecting gravity <j> = by 

 o x o x 

 k x ,..., m x ,... . 



We obtain a physical understanding of the relations between the 

 sketches: 



values k , k , etc., and the deeply immersed values k from the following 



We mirror first the immersed part of the body S, which in principle 

 may have an arbitrary shape, at the line 0Y. Thus above the axis a body S' , 

 symmetric to S, is generated. 



In Figures 1 and 2 we consider a vertical translation of the body 

 and indicate the resulting motion of the body by introducing sources and 

 sinks. Clearly, the disturbances are larger for the rigid wall than for the 



