173 



In time dt, this segment loses all its momentum component, 

 pHRdOdLv, nor'nal to the original plane of the diaphragm. If the change 

 in momentum is equated to the impulse [-SnoHEdO sin ^ ••• S^uHRd© cos oC ] dt 

 delivered in this direction in time dt, there results the relation 

 p ^ V - S^ sin o^ - S^jj cos OC (5) 



SJ.niilarly, if Sj^ is the total principal radial stress component 

 in the diaphragm material just ahead of the bending wave, as in Figure l^, 

 the net radial force component on the same segment of the annuHur element 

 while it is bein,'^ tilted is -Sj^JlKde cos <X - Sjjfflde - Spj^HRdO sinoc . 

 In time dt this effects the re.aoval of the radial momentum pHRdO dLU from 

 the segnent. Again if the change in the radial riomentum component is 

 equated to the impulse, vte obtain the relation 



pdL U = Sjj * S^ cos o<: ♦ Sj'^jj sin ^ (6) 



dt 



As the bending wave sweeps inward over the element depicted in Figure U, 



the stress Sjj does work of amount -Sj^HRd© Udt on it, urtnile the stresses S^ 



I 



and Sj^u, being of the nature of constraint forces in the rigid stationary 



material behind the ivave, do no work. The work which is done in the time dt 

 equals the increase in ';inetic energy, so that 



^ p ^ (U2 . v2) i S^U- (7) 



Several consequences of interest can be deduced quickly from equations (l) 

 through (7) inclusive. Recognizing from (l) and (2) that 



U^ ♦ v^ = 2U(U - R) 

 and incorporating this in (7), and comhinlng the result with (6), we find 

 that p_^^ ft » S^ cos oC + S^jj sin oC (8) 



Q u 



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