3^ 





A0 = ' >X - fn v ZoWe 



Figure 20. The whipping of a ship. 



forward acoustic problem. The boundary conditions for the wave equation must be 

 satisfied at the surface of the shell taking into account the translation, dilatation and 

 the higher order flexural vibrations of the shell. Although the hydrodynamic concept 

 seems simple, the analysis as well as the questions regarding the behavior of the mate- 

 rial and the failure are not simple by any means and offer many challenging problems. 



The Whipping of Ships 



When an underwater explosion takes place under the keel of a ship, violent 

 and dangerous transverse vibrations of the ship as a whole are excited, as is indicated 

 by the bent line in Fig. 20. The girder strength of many ships is not great enough to 

 withstand such a loading and often ships have been broken in two by such an explosive 

 attack. 



Chertock [22] has published an analysis of this phenomena. He assumed 

 incompressible fluid motion, namely that the velocity potential has to satisfy the 

 Laplace-Equation and the boundary conditions on the surface of the gas bubble and 

 on the elastic beam which represents the ship. 



The assumption of incompressible fluid motion seems at first glance very crude, 

 since the shockwave and the bubble pulses which are compression waves are certainly 

 important factors in producing these transverse vibrations. But such a variable velocity 

 potential produces pressures in an incompressible fluid similar to those of the shockwave 

 and bubble pulses. Of course the pressure-distance relation is somewhat different; one 

 difference being that there is no energy dissipation in an incompressible fluid such as 

 occurs at the shockfront of high amplitude waves. However, all that must be done to 

 make this analysis quite realistic, is to adjust the velocity potential in such a way that 

 it produces the same pressures near the target which the shockwave and the bubble 

 pulses would have produced at this place. 



Then the only discrepancy is the time lag which a compression wave has due 

 to its finite propagation velocity, whereas in an incompressible medium the "wave" 

 arrives at all points at the same time. Therefore, it becomes plausible that this seem- 

 ingly crude approximation is not bad, as long as the period of the oscillations of the 

 beam is large in comparison with the transit time of the wave. In most practical cases 

 this condition is satisfied. 



345 



