NSWC/WOL TR 76-15 5 



where p is the density of the water and I = Jp dt, the applied 

 impulse." 5 By the methods of Appendix A we can now calculate the 

 radius ratio AMAX/AMIN for the oscillating bubble. See Figure 3.5.1. 



The solution -- by means of a tabulated function -- is of the 



form 



AMAX 

 AMIN 



= FUNCTION 



A.^fpp- 



(3.5.2) 



where p. is the initial value of the ambient water pressure. 



Since ^1 p . does not vary greatly for shallow explosion 

 geometries and since in most bladder fish the swim bladder comprises 

 a roughly constant fraction of the total volume (about 6%) , Equation 

 3.5.2 can be written 



AMAX 

 AMIN 



= FUNCTION 



1/3 



M 



( approximately) 



(3.5.3) 



where we have substituted M, the mass of the fish, for the volume, 

 since all fish are approximately neutrally buoyant. Eauation 3.5.3 

 shows that the Bladder Oscillation Parameter and the Impulse Damage 

 Parameter are for practical purposes equivalent, for the special 

 condition of shallow fish depth and impulsive pressure loading. 



Taken together, the results of the Lovelace Foundation for 

 fish-kill as a function of the impulse and the present results 

 described in terms of the bladder oscillation parameter (and, in 

 part by the impulse damage parameter) give us confidence that we 

 have achieved a correct approximate solution to the problem of 



5. Kennard, E. H., "Radial Motion of Water Surrounding a Sphere of Gas in 

 Relation to Pressure Waves," 1943, published in Vol. II of "Underwater Explosion 

 Research," Office of Naval Research, 1950- 



50 



