87 101 



*„ = - Znahz^ [l66] 



If the hydrostatic pressures on the two sides of the diaphragm are equal, 

 * = *„ in [l64]. 



If the diaphragm, flat initially, remains within the elastic range, 

 it is readily shown that 



approximately, where E is Young's modulus and ^. is Poisson's ratio; see TMB 

 Report 490, Equations [^^], [^'[] . In this case, after evaluation of the in- 

 tegral with = 0, as given in [l66]. Equation [l64] gives 



. 2 ^ 1 nEh 



^' M + Af, 2(1 -/i)a2^^"' 



(C.-^/) C^S] 



where z^^ is the value of z^ at which z^ = 0. The swing time then Involves 

 the integral 



dx 1.311 



' ■'cm *<: "^'" 



The values of M and M; may also be inserted from [120] and [128a], in which 

 m = p^h and p = p^ in terms of the density p^ of the diaphragm and the den- 

 sity p^ of the adjacent liquid, or 



M = ypj/ia^, M, = 0.813p,a^ [170a, b] 



With these values, [165] and [l68] give for the elas.tic swing time 



^.-^\/^('.-«"»f'.) i^^^i 



Thus in the elastic range the swing time varies with the amplitude z^^. If 

 the Initial velocity z^„ is known, the amplitude z^„ can be found by setting 

 z^ = z^„ and z^ = in [168] and solving for z^„. 



As an alternative, if the diaphragm stretches plastically under a 

 constant yield stress a and if the initial elastic range of the motion can be 

 neglected, from [l6U] and [l66] 



. 2 _ 2nah / 2 2v moi 



and the integral that is needed is 



'cm 



r dz, { 



1 



dx 



'*cm *c 



