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v/ith the foregoing picture in nind, it is found possible to 

 develop a consistent nathematical formulation of the theory of the main 



notion so as to satisfy the fundamental laws of mechanics.* This is done In 



the following sections. 



alathematical Formulation of the Theory 



The basic suppositions and ass'jr.ptions introduced in the preceding 

 section lead to certain quantitative relations between the parameters which 

 describe the configurations and state of the diaphragm material. The 

 development of these relations can be divided into four main parts. Part 1 

 has to do vjith conservation of momentum, energy, and mass (volume) in the 

 neighborhood of the bending wave. In Part 2, the equations of motion of 

 the material in the central flat plastic oortion are derived, and in Part 3> 

 plasticity theory is introduced and applied. Part 4 is concerned with the 

 specification of the initial state of the motion. 



Part 1. Conditions in the Neighborhood of the Bending Wave . 



Suppose that, at any time t, after the start of the motion, the 

 distance of the bending wave from the center of the diaphragm is R, as 

 shown in Figure 3. Let dL be the width of an elementary ring of thickness H 

 just ahead of the bending wave. During the time interval dt, this ring is 

 swept over and, in effect, tilted by the bending wave as it is propagated 

 inward. The radial distance traveled by the wave in this tiae is -H dt, 

 the negative sign being affixed because R, the velocity of the bending wave, 

 is negative. Let the material at a distance dL ahead of the bending wave 



* The treatment of the initial elastic action is perhaps not quite as 

 satisfactory, except when the hypothesis of a zero elastic strain range 

 is rigidly maintained. This appears to be a subject for further investigation, 



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