40 26 



where p^ is the Incident excess of pressure above hydrostatic pressure, or, 

 if there Is no baffle, approximately 



I = jp^dt [26] 



Here JPidt represents the Incident Impulse per unit area. 



To obtain this result. It Is only necessary to multiply the equa- 

 tion of motion of the plate by dt and Integrate. From Equation [tU] 



d'z 



I=fp^dt=f(m^-<t,)dt 



When the value of d^z/dt^ is substituted here from the equation of motion, 

 the double integral in dt and dS vanishes, as is seen at once upon inverting 

 the order of integration. For example 



= 



since every point on the plate begins and eventually ends in a state of rest. 

 Thus, from Equation [l6] or [l8], / = zjp-dt, as stated. Or, if Equation 

 [21] is used, since \[dpjdt)dt = Ap^ = 0, / = jp^dt, at least approximately, 

 in the absence of a baffle. 



Similar treatment of Equation [24] gives for the sun ace of the 

 liquid, whether free or not, 



jip-p^)dt = 2fp^dt [27] 



for the total impulse in excess of hydrostatic pressure due to external 

 forces, on unit area of the surface, provided the surfs, e is at rest except 

 during a certain finite interval of time. 



The effect of the relief pressure, and hence the effect of diffrac- 

 tion, thus vanishes in the end if the motion of the surface is limited in 

 time. 



It must be assumed also, however, that the motion is such as to 

 make the integrals containing dS converge. 



THE PROPORTIONALLY CONSTRAINED PLATE OR DIAPHRAGM 



Equations [l6], [l8], and [21] to [2U] are of the integrodlfferen- 

 tial type, and they are difficult to solve because z Is a function both of 

 the time and of position on the plate. For this reason Interest attaches to 

 the solutions of the following artificially simplified problem, which can be 

 handled more readily. 



Let it be assumed that all parts of the plate execute proportional 

 motions. Then z can be written in the form 



