NONLINEAR PRESSURE WAVES AND SHOCK FRONTS 



181 



creases with increasing temperature, the point Si 

 corresponding to the state of the fluid behind the 

 shock front must He above S'l, as shown. That this 

 is entirely consistent with equation (18) for a com- 

 pressional shock wave can be seen from the upper 

 half of Figure 5. The right side of equation (18) 

 represents the area of the trapezoid under the straight 

 line SijSo, while the energy difference between S[ and 

 So is represented by the area under the adiabatic 

 curve between these two points. If the adiabatic 

 curve is concave upward, as it is for all normal fluids, 

 the area under the trapezoid will exceed the area 

 under the curve, and this is consistent with the 

 known fact that <Si has a higher temperature, hence 

 a higher energy, than SJ. For a rarefactional shock 

 wave, on the other hand, the energy of S'l is lower 

 than that of So by an amount represented by the area 

 under the adiabatic curve between these two points 

 in the lower half of Figure 5, and since the area of the 

 trapezoid is again greater than this, equation (18) 

 could not be satisfied unless Si had less energy than 

 S'l. Thus, a rarefactional shock is impossible for a 

 normal fluid,'' as is indeed to be expected from the 

 fact, proved in Section 8.4.1, that regions of high 

 pressure advance faster than those of low pressure. 

 It is instructive to compare the equations (16) and 

 (17) with the corresponding relations of acoustical 

 theory to which they reduce in the limit. To verify 

 the latter statement we may note that for a disturb- 

 ance of infinitesimal amplitude, equation (18) re- 

 duces to the law of adiabatic compression, while 

 equations (16) and (13) become respectively" 



\dp/ 



and 



Ml 



^dp. 

 Pi - Po 



Pi - Po 



pi(F — Ml) pc 



For disturbances of large amplitude, however, it is 



I' A British report'" questions this conclusion on the basis of 

 certain theoretical calculations and of some photographs of 

 rarefactional waves. However, the computed example cited 

 there of a "negative shock front" is merely a normal com- 

 pressional shock when viewed in a system of reference in 

 which the fluid ahead of it is at rest. Moreover, the rarefac- 

 tional waves which have been photographed must be regarded 

 as mere acoustic disturbances; the resolution of the photo- 

 graphs is insufficient to distinguish a discontinuous shock 

 front from a gradual pressure front which has a fairly appre- 

 ciable thickness. 



" Throughout this chapter the symbol ~' will be used to 

 denote asymptotic equality; in other words, it implies that the 

 quantity on the left equals the quantity on the right plus other 

 terms whose ratio to the quantity written approaches zero in 

 the limiting process being considered. 



easily seen from the top half of Figure 5 that for all 

 ordinary fluids the value of V given by equation (16) 

 is greater than the value Co of (dp /dp)'' in the undis- 

 turbed fluid. For the quantity under the radical in 

 equation (16) is just 1/pl times the slope of the 

 straight line SiSo while Co is 1/pl times the slope of 

 the tangent to the adiabatic curve at So- By a similar 

 argument it can be shown that for a fluid such as 



PRESSURE 



COMPRESSIONAL SHOCK WAVE 



J± »■ SPECIFIC VOLUME Vp 



PRESSURE 



RAREFACTIONAL SHOCK WAVE 

 (IMPOSSIBLE FOR THE FLUID SHOWN) 



SPECIFIC VOLUME \/p 



Figure 5. Pressure-volume diagram of the changes 

 occurring in a shock front. 



water, for which S'l and *Si are very close together 

 (F — Ml) is less than ci, the value of (dp/ dp)- im- 

 mediately behind the shock front. These results mean 

 that small disturbances created behind the shock 

 front may overtake it, but that no small disturbance 

 can be propagated from the shock front into the un- 

 disturbed fluid ahead. 



In fluids such as water and air, the dissipative 

 phenomena taking place in the shock front gradually 

 convert the mechanical energy of a shock wave into 

 heat, causing the amplitude of the wave to decrease 

 as it advances by an amount additional to the 

 familiar decrease due to spherical divergence. A 

 sufficiently intense shock wave traversing a high ex- 

 plosive, however, can maintain itself indefinitely be- 

 cause of the energy supplied by the chemical con- 

 version of the explosive into gaseous products; such 

 a shock wave would constitute a detonation wave. 



