48 34 



to this value; whereas. If 2Fi + * continually increases with the time, 

 d^Zc/dt^ must exceed the non-compressive value, while if 2F, + 4> decreases, 

 d^z^/dt^ must be somewhat smaller than the non-compressive value. Analogous 

 statements hold for dz^/dt. 



IMPULSIVE EFFECTS 



The following two special cases are of interest, partly because of 

 the light they throw upon the qualitative aspects of the action. 



Steady Pressure Suddenly Applied 



After a plate or diaphragm has been at rest and free from wave ac- 

 tion for a long time, let a wave of constant pressure suddenly begin to fall 

 upon it. During the quiescent period, = in Equation [l6] in order to 

 keep d^z/dt^ = 0, and for a short time thereafter 4> will be small. In the 

 neighborhood of any point of the plate, furthermore, the incident wave will 

 approximate to a plane wave incident at a certain angle. For a short time 

 after its arrival, therefore, the equation appropriate to plane waves. Equa- 

 tion [^1], can be used. Each element will begin moving according to this 

 equation independently of all others, and every element will execute the same 

 motion, but with a certain displacement in time if the Incidence is oblique. 



The plane-wave equation will hold until waves of relief pressure 

 arrive, coming from elements of the plate whose motion differs in other ways 

 than merely by a time difference due to oblique incidence. Thereafter the 

 action becomes more complicated and Equation [l6] must be used. In many 

 practical cases, however, the action of a shock wave is almost entirely com- 

 pleted before the simpler Equation [T7] begins to fail noticeably. 



If the plate is proportionally constrained, further light can be 

 thrown upon its later motion. In this case, for a plate mounted in a rigid 

 baffle, if = 0, Equation [29] becomes initially 



M^ = 2F,-pcA^ [51] 



where 



,2 



f[f(x,y)\ dS [52; 



and represents an effective area; see the Appendix, Equations [1*40] and [I'+l]. 

 This is the analog for the plate as a whole of Equation [ly] for the individ- 

 ual elements. If the mass per unit area m is uniform, A = M/m, where Mis 

 the effective mass as defined in Equation [30]. If the plate also moves pa- 

 rabololdally, as represented by Equations ['+0a, b], A = 7raV3, or a third of 

 the actual area. 



