62 



48 



should vary more rapidly than as ^ /R. Furthermore, the duration of the pres- 

 sure wave at a given point should increase som'ewhat with an increase in R; 

 for an exponential wave this is represented by a slow decrease in a. These 

 two changes have opposite effects upon 7, but the first should predominate. 

 Thus Ix{R') should vary somewhat more rapidly than as l/i?'. 



A further complication arises in practical cases from the spherical 

 form of the wave. This further decreases the deflection somewhat; and the 

 decrease should be greater at small distances. Work hardening and increased 

 strain-rate effects in the diaphragm will also have the effect of decreasing 

 the larger deflections as compared with the smaller. 



The final result seems to be that Equation [771 implies a varia- 

 tion of 2 ^'s.W^F[W'yR) where FiW^R) equals I ^{R/VJ^ multiplied by a factor 



to correct for sphericity and other minor factors; and FW^/R) might vary 



1. 

 either more rapidly or less rapidly than as WyR. The variation of z might 



n + 1 



happen to be nearly as IV ^ /R^ where n is a constant either a little greater 

 or a little less than unity. 



Other cases in which cavitation does not occur may be treated by 

 integrating one of the other equations of motion. Kirkwood solved his equa- 

 tion for the paraboloidal diaphragm, which was mentioned in connection with 

 Equation [48], with the help of Fourier Analysis; the results may be found in 

 his reports. References (6) (7) and (8). 



CASE 2: Prompt and Lasting Cavitation at the Diaphragm Only; T^<^Ti,T^^T, 

 or the compliance time is much less than either the diffraction time or th^ 



swing time, as illustrated in Figure 2t 



Time 



Figure 26 - Illustration of Case 2, 



Prompt and Lasting Cavitation 



at the Diaphragm 



At the compliance time r^ the diaphragm has 

 reached maximum velocity; cavitation is then 

 assumed to occur and to last at least until 

 the diaphragm has completed its outward swing. 



It is assumed here that cavitation 

 sets in so quickly that the diaphragm 

 acquires maximum velocity before the 

 pressure field has been appreciably 

 modified by diffraction, and also be- 

 fore the diaphragm has moved far 

 enough to call appreciable stress 

 forces into play. It is also assumed 

 that no further deflection Is pro- 

 duced when the cavitation disappears. 

 These conditions as to times are com- 

 monly satisfied in test assemblies; 

 if cavitation occurs at all. It 

 should usually occur relatively early 

 in the damaging process. 



Under the conditions stated, 

 all parts of the diaphragm will be 



