SURFACE AND OTHER EFFECTS S95 



be estimated from the bubble migration or, conversely, used to infer the 

 migration. 



The results so far discussed are evidently rather incomplete from 

 several points of view. A more thorough analysis should include con- 

 sideration of the lateral extent of dome formation, the internal structure 

 of the dome, and the effect of charge depth. The phenomenon in its 

 entirety is a fairly elaborate one, as yet imperfectly understood. Some 

 of the factors involved are considered in section 10.2, but much of the 

 existing evidence has had to be omitted from the present discussion for 

 reasons of space and unavailability of the reports. 



The venting of the gas bubble and other disturbances also lead to 

 the development of surface waves, as distinct from waves of compres- 

 sion. These have been the subject of some study and are found to be 

 insignificant for all but extremely large charges. For 1,800 pounds of 

 TNT fired at optimum depth, the crest to trough height is of the order 

 of 5 inches 500 feet from the charge and falls off rapidly with increasing 

 distance. 



10.2. Dome Formation 



A. Initial formation of the spray dome. The particle velocity at a 

 free surface can be thought of as the resultant velocity due to the inci- 

 dent shock wave and the reflected tension wave. The particle veloci- 

 ties in these waves have equal and opposite horizontal components, 

 which therefore cancel, and equal upward components of magnitude 

 P cos b/poCo, where 5 is the angle with the vertical of a line from the 

 charge to the point on the surface. The resultant velocity Uz of a par- 

 ticle at the surface is therefore 



f.f..., 2Pcos5 

 (10.1) Uz = 



PoCo 



a result which of course is also obtained from the acoustic formula of 

 section 2.2. 



If the incident wave is exponential, the pressure P falls rapidly from 

 its peak value Pm and it would appear from Eq. (10.1) that u^ would 

 decrease correspondingly and give rise to very small displacements. 

 However, the head of the rarefaction wave reflected into the water falls 

 progressively behind the direct wave at increasing depths. The re- 

 sultant pressure just behind the rarefaction front thus decreases below 

 the hydrostatic pressure Po and soon becomes a negative pressure, or 

 tension. Water can, however, withstand only a limited tension, and at 

 some value {Pb) of resultant pressure, cavitation bubbles form which 

 prevent the pressure from decreasing further. The depth at which 

 cavitation first forms corresponds to the time t' at which the incident 



