NONLINEAR PRESSURE WAVES AND SHOCK FRONTS 



179 



is different from zero will advance toward increasing 

 X while the region in which (xp — u^) differs from zero 

 will recede in the opposite direction. Eventually 

 these two regions will separate and leave between 

 them an interval within which both (^ + Uj,) and 

 (^ — Mi) are zero so that the Huid is at rest at its 

 normal density po- The original disturbance has thus 

 been split up into two progressive waves traveling in 

 opposite directions. In the wave which travels in the 

 positive X direction (^ + Ux) is finite while (^ — u^) 

 is zero ; in this wave, therefore, ^ = Ux and both the 

 density and the particle velocity are propagated for- 

 ward with the velocity (mj + c) = (^ + c). Since for 

 all ordinary fluids both xf/ and c increase with in- 

 creasing pressure, this velocity of propagation will be 

 greater the greater the pressure, and any disturbance 

 \\'\\\ ultimately develop as shown in Figure 3. After 

 the wave front becomes very steep, of course, the 

 basic equation (2) or (6) is no longer valid and must 

 be modified to take account of the effects of viscosity 

 and heat conduction, which are negligible for waves 

 of more gradual profile. 



Most practical applications, such as pressure waves 

 produced by explosions, involve spherical waves di- 

 verging from a source rather than plane waves of the 

 type we have been discussing. It can be shown, how- 

 ever, that spherical waves have properties very simi- 

 lar to those just established for plane waves in that 

 the high-pressure regions travel faster than the low- 

 pressure regions and tend to overtake them. This 

 overtaking effect becomes slower and slower as the 

 wave advances farther from its source because of 

 the decreasing amplitude of the disturbance due to 

 spherical spreading. For this reason, a pressure pulse 

 radiating from a small source has to be extremely 

 intense if it is to develop a shock front by means of 

 the overtaking effect; rough calculations^ have shown 

 that pressure pulses of the amplitudes ordinarily ob- 

 tained from echo-ranging transducers will be only 

 very slightly distorted by the overtaking effect and 

 will not develop shock fronts.^ However, in the case 

 of transmissions at supersonic frequencies this slight 

 distortion of the wave profile might be detectable by 

 a receiver timed to twice the original frequency. 



8.4.2 The Rankine-Hugoniot 

 Theory of Shock Fronts 



We have seen in the preceding sections and Figure 3 

 how any pressure wave of sufficiently large amplitude 

 ultimately develops an extremely steep shock front 



within which the motion of the fluid will be strongly 

 influenced by factors such as visco.sity and heat con- 

 duction which do not appear in the equations of 

 motion of perfect fluids. Certain characteristics of 

 such a shock front can be predicted only by a theory 

 which takes account of these additional factors ex- 

 plicitly; one such characteristic, which will be dis- 

 cussed in Section 8.4.4, is the thickness of the region 

 within which the abrupt rise in pressure takes place. 

 Fortunately, however, it was discovered by Rankine 

 and Hugoniot in the last century that many valuable 

 conclusions could be drawn merely by applying the 

 laws of conservation of mass, momentum, and energy 

 to the motion of the fluid, without bothering at all 

 about the details of phenomena in the shock front. 

 To show how this can be done, let us consider the 

 mass of fluid contained in a flat cylinder having unit 

 cross-sectional area and having its end planes parallel 

 to, and respectively ahead of and behind, the shock 

 front. A side view of such a cylinder is shown at a 

 particular time t by the full line ABCD in Figure 4, 



HIGH PRESSURE REGION 

 P,,^,. u, 



UNDISTURBED FLUID 



Figure 4. 

 front. 



Cylinder of fluid travereed by a shock 



AB and CD being projections of the end faces. At a 

 time dt later, the fluid which was originally in ABCD 

 will occupy the cylinder A'B'CD, shown with a 

 dotted boundary. Now if the pressure variation in 

 the shock wave is similar to that shown in Figure 3D, 

 the pressure, density, and other factors will change 

 very rapidly in a very thin region near the plane SF, 

 but will change much more gradually everywhere 

 else. If this is the case, we can assume the thickness 



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