66 52 



should probably be more nearly proportional to ^/R in this case than in 

 either of the other two cases, but should increase with W move rapidly than 

 in Case 2. 



The applicability of Equation [84] in actual cases is doubtful, 

 however, because of the artificial assumption that has been made as to the 

 motion of the water. If closure of the cavitation in reality progresses 

 from the edge inward, it is possible that support of the water by the outer 

 part of the diaphragm may greatly decrease the development of kinetic energy 

 in the water. Furthermore, a fixed baffle has been assumed. If there is no 

 baffle, or if it yields, the kinetic energy acquired by the water and the re- 

 sulting increase in the plastic work will be less. 



CASE 3: Negligible Diffraction Time T^ but Wave Not Short; 7^ « T^ and 

 T^« T^. Under these circumstances non-compressive theory can be used. If 

 also T^« r,, or the time constant of the wave is much less than the swing 

 time, the situation is that of Case 1 . Otherwise the action of the wave 

 overlaps on that of the stress forces, and the motion of the diaphragm is 

 more complicated. 



For a proportionally moving diaphragm mounted in a large plane 

 fixed baffle, quantitative results are easily obtained. According to the 

 simple assumptions that were made in the beginning, the net stress-force re- 

 sisting its motion will be proportional to its deflection; hence it is pos- 

 sible to write <t> ■= -kz^ where A; is a constant. Then Equation [35] becomes 



(M + M,)^ + kz, = 2Fi [85] 



which is of the same form as for a forced harmonic oscillator. For the ex- 

 ponential wave or p^ = Pm^"°'» ^i ^^^ ^® written F, = F^e'"' where Fg is a 

 constant. The appropriate solution of Equation [85], when z^ = z^ = at 

 t = 0, is then 



2F 



'" {M+R 



where 



or 



"'^wf^ 187) 



M + M, 



The final deflection z^,„ will be the first maximum value o*' ^ ; 

 find it requires the solution of a transcendental equation. It may conver- 

 lently be expressed in terms of the deflection under a static load of ragnl.' 

 tude F^, that is, under a static pressure equal to the maximum incident 



