366 



fti^, ^M,o^x^^- -^f^ 



X = — ^-7 r-v H-Ct) . o fe % i. » - 



(30) 



where 





^o 



The first of Eqs. (30) is the equation of motion of a free plate. Thus we 

 see that the central area of the diaphragm moves initially as a free plate 

 and is consequently flat; however, the flatness is destroyed by a transverse 

 wave represented by the second of Eqs. (30) traveling in fTom the periphery 

 with a velocity R^Wj, (* + ">)" ( Roic><,is the velocity of transverse plas- 

 tic waves in the unloaded diaphragm, the factor C\-*7l)" takes account of 

 the water loading). 



If PoC*"^ is bounded, it follows that \y.(,t) is continuous and has 

 a continuous derivative; consequently^ in such a case the deflection z. and its 

 derivatives with respect to the time t and the reduced radius X 

 respectively are continuous. However, suppose that p^d't'^ is not bounded and . 

 to be more specific, suppose it has a singularity at t-o such that 



Jo ^o<--^')^t' = I >o ^ -fc >o J 



= , "6 ^ o . 



Then the time derivative of the deflection is discontinuous at the initial 

 instant and the radial derivative and time derivative are both discontinuous 

 at 



53 



