46 



32 



It may be noted that for the circular piston and for the parabo- 

 loldal form the Integrodlfferentlal equation can be replaced without great 

 error by a more easily handled difference-differential equation; for example. 

 Equation [29] Is replaced by 





or Equation [33] by 



2 2 



., d z^ , , dZc , „ dzi, , -. d Zf, 



dt^ dt ^ dt ' dt^ 



-'['.-'...-rh^^^ (^9) 



Here Zj^,-j. denotes the value of z^ at time t - T, where T is a retardation 

 time of the order of the diffraction time T^, while all other quantities re- 

 fer to time t. If thinning of the diaphragm is neglected, k and 6 are con- 

 stants; see Equation [125] in the Appendix. 



An equation rather similar to Equation [48] but containing an in- 

 tegral was used by Klrkwood in developing a theory of damage in the absence 

 of cavitation (6) (7) (8). His equation was obtained for the central ele- 

 ment of the diaphragm on the assumption of a paraboloidal form, without the 

 provision of any mechanism for the maintenance of this form. In the theory 

 as developed in the present report, the form is assumed to be maintained by 

 suitable constraints and an equation of motion for the entire diaphragm is 

 obtained. The results in practical cases differ little, however, and it is 

 doubtful whether either type of theory represents the motion of an actual 

 diaphragm very closely. 



THE REDUCTION PRINCIPLE 



It has already been noted that under suitable circumstances suf- 

 ficiently accurate results can be obtained from non-compressive theory, in 

 which the compressibility of the liquid is Ignored. This is In reality a 

 special case of a more general principle. The action of a wave tends con- 

 tinually to change into or reduce to the type of action that is characteris- 

 tic of incompressible liquid. For convenience, this principle is called in 

 this report the reduction principle. 



Consider, for example, a flat-topped wave form in which the pres- 

 sure rises dlscontinuously to a value Pj and then remains at this value for 

 a considerable time. The discontinuous wave front is propagated past an ob- 

 stacle in strictly rectilinear fashion, leaving a perfect shadow behind the 



