33 47 



obstacle. After the front has passed, however, lateral equalization of 

 pressure sets in and produces the phenomena knovm collectively as diffrac- 

 tion. Pressure builds up in the shadow, and all modifications of the pres- 

 sure field that may have been caused by reflection in front of the obstacle 

 fade out. The final result is a uniform pressure of magnitude p^ all around 

 the obstacle, such as would be inferred from the ordinary hydrostatic, non- 

 compressive theory. The time required for approximate equalization of the 

 pressure is roughly equal to the diffraction time for the obstacle, or to 

 its radius divided by the speed of sound in the liquid. 



Any sudden increment of pressure, positive or negative, behaves in 

 a similar manner. At first, its effects exhibit the characteristics of wave 

 action; then the effect changes in continuous fashion until it reduces to the 

 effect that would have been produced in incompressible liquid by the same in- 

 crement of pressure. 



Furthermore, any pressure wave can be regarded as a succession of 

 small increments. Thus the usual conclusion is reached that waves much 

 shorter than the diameter of an obstacle will behave in a manner strongly 

 resembling rectilinear propagation, whereas waves that are much longer will 

 act more nearly like a static pressure. The non-compressive case previously 

 noted is one in which changes of pressure occur so slowly that reduction is 

 practically complete all of the time. 



The reduction principle is difficult to formulate mathematically in 

 the general case, but an exact expression of it is easily obtained for a 

 proportionally constrained plate. In this case the chief content of the 

 principle, as deduced In the Appendix, is the following. Suppose that the 

 plate has been at rest for a time exceeding D/c where D is its greatest di- 

 ameter. Suppose also that thinning of the plate may be neglected, so that M 

 and Af, may be treated as constants. Then, during any subsequent interval of 

 time equal to D/c, both acceleration and velocity take on at least once the 

 non-compressive values as calculated for the time t at the end of that in- 

 terval, namely, from Equation [35]. 



d'z, 2F. + * dz, i(2F,+ <P)dt 



dt^ M +Mt ' dt M + Ml 



[50a, b] 



Here M, is the mass due to loading by the liquid as given by Equation [36], 

 F; and the derivatives of z^ stand for values at time t, and J F^dt extends 

 from the beginning of the action up to that time. 



From this statement it is fairly clear, after a little reflection, 

 that, if 2F, + * is constant, d^Zc/dt^ must oscillate about the non- 

 compressive value as given by Equation [50a] and gradually settle down 



