180 



EXPLOSIONS AS SOURCES OF SOUND 



AD oi the cylinder to be very small but still very 

 much greater than the thickness of the shock front, 

 that is, of the region within which the rapid rise of 

 pressure takes place. It will then be legitimate to 

 treat the pressure, density, and velocity as having 

 constant values pi, pi, ui, over that part ABFS of the 

 cylinder which lies behind the shock front. Ahead of 

 the shock front, of course, the fluid is at rest with 

 undisturbed values po, po, of the pressure and density. 

 Remembering that the cylinder has unit cross sec- 

 tion, the mass of fluid in it may be written as 



P,IS + PoSD. 



If we let the boundary of the cyhnder move with the 

 water, this cannot change in the course of time. Now, 

 after the interval dt. shown in the figure, the mass is 



PiA^' + poWD 

 and since SD - WD = Vdt 



and A'S' - AS ^ (V - ui)dt 



where V is the speed with which the shock front ad- 

 vances, the two expressions given for the mass can 

 be equal only if 



Pi(V - Ml) = poF. (12) 



Similar equations can be derived by applying, in- 

 stead of the law of conservation of mass, the laws of 

 conservation of momentum or energy. Thus, the 

 change in the momentum of the cylinder of Figure 4 

 in the time dt is 



PiMi(F — ui)dt 



and this must be equated to dt times the force 

 (pi — Po) acting on the cylinder which gives 



PlMi(F — Ml) = Pi — Po. (13) 



For the energy equation, if we denote the internal 

 energy per unit mass of the fluid in front of and be- 

 hind the shock front respectively by eo and €i and 

 remember that the moving part of the fluid has 

 kinetic energy 3^it? per unit mass, we can write for 

 the change in the total energy of the cylinder during 

 the time dt 



4-1) 



(V — ui)dt — potoVdt. 



This must be equal to the product of the pressure pi 

 by the distance uidt through which the rear boundary 

 of the cylinder has been pushed. Thus, we get the 

 final equation 



4-1) 



(V - Ml) — pocoF = piMi. (14) 



The three equations (12), (13), and (14), when 

 augmented by known relations between the thermo- 

 dynamic parameters of the fluid, can be shown to 

 determine all the quantities pi,pi,Mi,F,ti in terms of 

 any one of them, when po and po are given. The equa- 

 tions may be put in a more explicit form as follows. 

 From equation (12), 



Ml = 



V. 



Pi 



Inserting this and equation (12) into equation (13), 



Pi 



Po 



Po- 



Pl 



F^ = Pi 



Po 



' PoVpi — Po / 



Po\Pi — Po 

 whence, from equation (15), 



Ml 



]' 



(Pi — Po)(pi - Po) 



PoPi 



(17) 



Finally, if we insert the expression (12) into the first 

 term of (14), and the expression (15) into the right- 

 hand side, 



Pol'l «i i- — J — Pol^€o = poK — 

 whence, using equation (17), 



PoPi 



«i — «o = Pi- 



Pi — Po Ml 

 PoPi 2 



(18) 



Pi — PC (pi — Po)/pi — Po\ 



= Vi ^ — ( )' 



PoPi ^ \ PoPi / 



or ei - «o = ^(pi -I- po){ )• 



\Po Pi/ 



In the discussion leading to these equations, we 

 have spoken of the region behind the advancing shock 

 front as the "high-pressure region," although all the 

 equations which have been written would still be 

 valid if pi were less than po instead of greater. How- 

 ever, it is easy to show from the energy equation (18) 

 that in ordinary fluids a "rarefactional shock wave," 

 that is, one for which pi < po, cannot exist. To prove 

 this, consider the pressure-volume diagrams shown 

 in Figure 5. The state of the undisturbed fluid of the 

 shock front is represented by the point So. If the fluid 

 were gradually and adiabatically compressed to den- 

 sity pi, it would reach the state S'l. Now, according to 

 the second law of thermodynamics, any sudden com- 

 pression to this density involving irreversible proc- 

 esses must leave the fluid in a state which is hotter 

 than S'ly in other words, since pressure normally in- 



