35 49 



As the elapsed time approaches the diffraction time. Equation [51] 

 falls and the complete Equation [29] must be used. As soon as the time con- 

 siderably exceeds the diffraction time, however, a simple description of the 

 motion again becomes possible. The motion then approximates rapidly to the 

 motion that would have occurred if the water had been incompressible. This 

 conclusion may be Inferred with sufficient cogency from the reduction princi- 

 ple just described. 



From this principle, and, in particular, from Equation [50a], it 

 is sufficiently clear that the acceleration of the plate will take on the 

 non-compressive value as stated in Equation [35] within a time less than D/c, 

 and will oscillate thereafter about this value with a rapidly diminishing am- 

 plitude of oscillation. The Initial acceleration, which is 2FyAf from Equa- 

 tion [51]. Is relatively high because the effective mass is at first that of 

 the diaphragm alone, but as the loading by the liquid takes effect the accel- 

 eration decreases toward the non-compresslve value. Because of the high ini- 

 tial acceleration, however, the velocity remains permanently somewhat in 

 excess of the non-compressive velocity. 



The transition from one type of motion to the other is easily fol- 

 lowed in detail if the accurate integrodifferential equation is replaced by 

 the approximately equivalent difference-differential equation. Equation [48]. 

 This equation is readily solved in simple cases, provided thinning of the dia- 

 phragm is neglected, so that fc and b are constants. 



In the case under discussion, z^ = and z^ = up to a certain in- 

 stant, which may be taken as ( = 0, and thereafter *= and ZF^/M Is equal 

 to a constant. An example of the results obtained from Equation [U8] for 

 this case is shown in Figure 20. The curves represent the central accelera- 

 tion 'z\ and velocity z^ of the diaphragm as functions of the time t ; the non- 

 compresslve values as given by Equations [50a] and [50b] are shown by straight 

 lines. The unit of time is taken to be the diffraction time, or T^ = a/c, 

 where c is the speed of sound in the adjacent liquid and a is the radius of 

 the diaphragm, assumed circular; and the incident pressure is assumed to have 

 such a value that the initial acceleration, ZF^/M, is unity. With a constant 

 incident pressure of different magnitude, all ordlnates would be changed in 

 proportion to the pressure. The figure refers to the special case in which 

 po/m= 12.5 and hence .M; = 9.7 M; then fc= 13.4 and 6 = 9.34. 



The figure would be applicable, for example, to a 10-inch steel 

 diaphragm of thickness 0.05 inch, acted on by a steep-fronted wave in which 

 the pressure behind the front is uniformly 1700 pounds per square inch. Then 

 z, is in inches, and the unit of time is T^ = 5/59 = O.O85 millisecond. 



The figure confirms the statements Just made as to the approach to 

 non-compresslve values, which is very rapid in the case represented. The 



