102 



88 



Then 



^» = ^ Vi(^^ + ^-^^^ ! ^') 



[173] 



Inclusion of both the elastic and the plastic, ranges leads to very 

 complicated formulas. The error is not large, however, if the plastic formu- 

 la [175] Is used for all motions that extend into the plastic range. The er- 

 ror is greatest when the maximum displacement z^^ just attains the elastic 

 limit z^, , which is found by substituting z^^ for z^ in [167] and interpret- 

 ing a as the yield stress: 



c.= «/ 



2(1-1^) 



U)a 



E 

 When Ze„ = z^, , the correct elastic formula, [171], gives 



T. = 0.76 I 



a|/;^(p, + 0.776 fp,) 



[174] 



[175] 



whereas the plastic formula [173] would change the coefficient from O.76 to 

 0.64. 



Swing times for a similar diaphragm not loaded by liquid on either 

 side and with equal pressures on the two faces can be obtained by setting 

 p, = in [171] and [173]. Or, if there is liquid on both sides of the dia- 

 phragm, with densities Pj and p^ on the two sides, respectively, p^ is to be 

 replaced by pj + p^ ^o'" ^^e reason explained on page 79- 



SECOND-ORDER EFFECTS IN REFLECTION 



In linear or first-order acoustic theory, when either plane or 

 spherical waves fall upon a rigid wall, the boundary condition can be satis- 

 fied by assuming reflected waves which are the mirror image in the surface of 

 the incident waves. Thus even the afterflow part* of the particle velocity 

 in a spherical wave has equal and opposite components perpendicular to the 

 surface in the two waves, so that the resultant component in this direction 

 vanishes. The pressure on the surface due to the waves is exactly doubled 

 by reflection. 



The case of large amplitudes can easily be Investigated, for plane 

 waves at normal incidence, by the method of Riemann, which is explained In 

 Section 28? of Lamb's "Hydrodynamics" (23). It can be imagined that, in the 

 medium carrying the waves, values of the quantity Q = fi + p^c^v are propa- 

 gated forward without change, while values of S = // - p^^c^v are at the same 



For the terminology, see TMB Report 480, page 39 (10). 



