82 



68 



>('-f) 



' c as ' ' T 



[10U] 



where z ,__5- Is the velocity at time t - s/c. 



c 



Hence, If the Integration Is restricted to a ring-shaped area be- 

 tween s = Sj and s = s^, 



f 2, , ^ = -2irc\ -^ i, .ds = ~2-nc'z, .1 



' = '1 "1 "1 



-^-W'-f)-('-f)] 



[105] 



whereas If the Integration covers the entire plane, and if the plate started 

 from rest so that z(- 00) = 0, 



Thus [T02] becomes 



\'z , a — — = 2nc z(t) 

 mi + pcz = 2p^ + <f> 



[106] 



[107] 



where all quantities refer to time (. This is an obvious generalization of 

 the one-dimensional equation; see Equations [10] and [11] on pages 24 and 26 

 of TMB Report 480 (10). 



Equation [107] has reference to plane waves at 

 normal incidence. It was pointed out by Taylor (U) that 

 the case of plane waves incident at any angle Q can eas- 

 ily be treated provided it is assumed that 0=0, so that 

 the elements of the diaphragm move independently. 



Let y denote distance measured along the dia- 

 phragm in the plane of incidence. In Figure 30 there is 

 shown an incident wave QQ'Q", at all points of which the 

 incident pressure has the same value. If Q strikes the 

 origin for y at time (, Q' will not strike the diaphragm 

 until a time ^ ^ " — later, where c is the speed of sound 

 in the liquid. Thus if p,(t) denotes the incident pres- 

 sure at y = 0, its value elsewhere on the diaphragm is p, (f - ^ — ^ — j. 



It is a natural surmise now, to be verified in tne sequel, that the 



displacement z will also be a function of the same argument or z U - -^ j 



Then all elements execute the same motion but in different phase; and 



Figure 30 



f .. di c ..I ^ s + y sin9\ 



dS 

 s 



