416 SURFACE AND OTHER EFFECTS 



The maximum deflection zdtm) thus is proportional to the impulse 

 PmO of the incident wave, but unless the plastic time is large only a 

 fraction of the impulse is utilized. If dp is much smaller than the time 

 constant 6, the deflection approaches a limiting value Zc{tm) = ^P^Op/poCo 

 as tm becomes large compared to dp, and the deformation is proportional 

 to the peak pressure Pm- The time Op for a diaphragm 10 inches in 

 diameter and 0.1 inch thick with a yield stress of 60,000 Ib./in.^ is 600 

 /xsec. For small charges with 6 less than 100 )usec. (less than 5 pounds 

 of explosive), the deformation would thus be determined by impulse. 

 For charges of the order of 300 pounds or more, with 6 of the same order 

 of magnitude as dp, only the earlier portions of the pressure-time curve 

 would be effective. These considerations give an indication of the 

 changes in effect of the incident wave. They are, however, often in- 

 applicable, as they neglect diffraction effects, which are important for 

 large charges, and the possible development of cavitation, which makes 

 the solution invalid in many practical cases. 



The effect of diffracted waves is simply illustrated in the non- 

 compressive approximation, for which the equation of motion is 





-t/d 



The solution of this equation is 



Zc{t) = ~ " ^ [cOo^C"'/^ + sin Cx)ot — OOod COS OJot] 



PoCo 1 + {(^od)" 



where coo^ = 4:aoh/ma"(l + 2poa/3m) and dp is the plastic time previously 

 defined. The quantity coo has the characteristics of an angular fre- 

 quency of motion of the plastic diaphragm loaded by the moving water 

 near its surface. The time tm for maximum deformation (for which 

 dzc/dt = 0) is given by the transcendental equation 



Q-t,n/e = cos ix)ot„i + o)od sin o)ot,n 

 and the deflection Zc{f,n) is then 



2P,nd 



Zc(tm) = — OOodp sin O^otm 



PoCo 



This maximum deformation is for the case of an infinite rigid boundary 

 external to the plate. If there is no baffle, the pressure doubling from 

 reflection is reduced by diffraction, and in the noncompressive approxi- 

 mation the deformation is very nearly half as great. 



If the natural period 2Tr/coo of the structure is large compared with d, 

 tm is nearly 34 of this period or 7r/2wo, and the deformation is propor- 



