57 71 



where C Is a nearly constant coefficient, so long as elastic effects can be 

 neglected. The factor W^z in this expression represents a variation interme- 

 diate between the H^^ for the pressure and the H^ 3 for the impulse, or a varia- 

 tion as the square root of the product of maximum pressure and impulse. 



Observations generally show a variation of the deflection more or 

 less as In Equation [96]. The plastic work commonly differs, in fact, by 

 less than a factor of 2 from the energy brought up by the wave. A discussion 

 of the data is contained in TMB Report U92 (l8). 



From the analytical standpoint, however, correlation of damage with 

 the energy in the wave appears to be somewhat of an accident, contingent upon 

 the range of magnitude of various factors as they occur in practice, rather 

 than a direct consequence of the conservation of energy. There exists no 

 general necessity for the plastic work done on a structure to equal the ener- 

 gy that is directly incident upon it according to the laws of the rectilinear 

 propagation of waves. Part of the incident energy may be reflected; or, on 

 the other hand, if the motion approximates to the non-compressive type, it is 

 possible for the energy absorbed by the structure greatly to exceed that 

 which is brought up by the wave. 



In TMB Report 489 ('''') It was inferred, nevertheless, from the ex- 

 ample of the free plate, that damage to a diaphragm should probably correlate 

 better with the Incident energy than with the Incident momentum. The argu- 

 ment is substantially that by which it was concluded In Case 2 that the set 

 deflection z might vary about as W^. Or, it might be that the more rapid 

 variation introduced by reloading, as in Case 2a, would assist in bringing 

 about rough proportionality of z to W^. 



The analytical formulas indicate, furthermore, that in most tests 

 on diaphragms the plastic work should not differ greatly from actual equal- 

 ity with the energy that is brought up to the diaphragm by the incident wave. 

 For an exponential wave, Equation [2], and a circular diaphragm of radius a, 

 this energy will be 



= na'f 



Pldt=^^ [91] 



pc 2apc 



Thus the ratio of the energy absorbed by the diaphragm, estimated as nahz^, 

 to that brought up by the incident wave will be, for three cases treated In 

 the last section, from Equation [771. [8l], or [84], respectively. 



(1 ) non-compressive: tt- = r — — 



^" 1 + 0.776-^T- 



(2) lasting cavitation: ^= Ux'"' 



