HYDRODYNAMICAL RELATIONS 25 



found in practice have times of rise less than the resolving time of experi- 

 mental measurements, less than 0.5 microseconds in some cases. It is 

 important to note that while nothing has been said about the velocity 

 of such shock fronts, it is always greater than Co, the velocity of small 

 amplitude sound waves. 



It is easy to see, by reversing the argument given for waves of com- 

 pression, that a wave of rarefaction, in which later portions of the wave 

 are regions of lower pressure and the particle velocity is away from the 

 direction of advance, will broaden out as it advances and shock waves 

 of rarefaction cannot develop. 



Although the arguments just given apply to plane waves, we should 

 expect the same sort of effect in spherical waves except that the ampli- 

 tude will be weakened by the spreading out of the wave, and the effect 

 will become less important as the distance from the source increases. 

 It would be erroneous, however, to conclude that effects of finite ampli- 

 tude at a shock front are important only within a few feet of an explo- 

 sion. 



A. Plane waves of finite amjplitude. The qualitative argument just 

 given for plane waves can be made more explicit by means of a relatively 

 simple argument due to Riemann.^ The equation of continuity (2.2) 

 and equation of motion (2.4) for motion in one dimension are 



(2.14) |^ + ,Jf + ,^=0 



dt dx dx 



du , du , dP ^ 



p — + p^— + — =0 



dt dx dx 



Riemann's treatment is based on introducing a new variable a defined 

 by the integral 



(2.15) <r = j" c{p) ^ 



where po is the initial density in the absence of a disturbance and the 

 sound velocity c = vdP/dp is a function of density p. If the variables 

 a and c are used to replace P and p in Eqs. (2.14), we obtain the rela- 

 tions^ 



1 This development follows essentially the treatment by Lamb (65). A complete 

 discussion for plane waves in air has been given by Rayleigh (90) . 



2 The relations follow from Eqs. (2.14) by noting that 



