172 



have a radial velocity U outward, so that it moves a distance Udt in time 



dt. Hence, as is evident from Figure 3, the rate at which material is 



swept over by the wave is 



dL •= -R ♦ U (1) 



dt 



As the wave travels by, the material ring of width dL in effect is tilted 



impulsively into a truncated conical ring of vndth dl, and thiclcness H. In 



deforming, the inner edge of the annulus undergoes a displacembnt v dt normal 



to the original plane of the diaphragm while the outer edge rerains fixed. 



Actually, then, the bending vrave travels a distance dL along the generator 



of the conical element while it is traveling inward along the radius a 



distance -Rdt. Kence, the rate at which the wave travels along the profile 



of the deformed diaphragm is given by 



dL 

 dt 



-^[P* v2 (2) 



It is clear, from this and Figure 3, that the generator of the conical element 



makes an angle OC '/dth the outward pointing radius, defined by the relations 



dL cos cxC = R (3) 



dt 



dL sin oc: = V (4) 



dt 



which together are equivalent to (2). 



Let Srvr, be the total normal stress component parallel to a generator in the 



diaphragm at a point just behind the bending ivave, and let S! be the total 



shear stress component parallel to a plane element norr.ial to the generator 



at this point as in Figure 4. 3y total stress is meant here the sura of the 



plastic anci the constraint stresses. These compon-jnts exert forces of 



magnitude S^jj lUide and S^jj Hhd6 respectively on the outer edge of a segment 



of width dL and thicloiess 1! which subtends an angle d9 at the denter of the 



diaphragm, 



-9- 



