NONLINEAR PRESSURE WAVES AND SHOCK FRONTS 



183 



influencing the thickne.ss of a shock front can be 

 valuable in that it may help us to evaluate and 

 interpret experiments which purport to measure the 

 time of rise of the pressure at the front of a shock 

 wave. 



As has been explained previously, the Rieniann 

 overtaking effect tends to make any pressure pulse 

 develop an infinitely steep front in the course of time, 

 and this tendency can be counteracted only by factors 

 neglected in the equations of motion of a perfect 

 fluid, on which Riemann's analysis is based. These 

 factors, of which viscosity and heat conduction are 

 the most obvious, must include the dissipative phe- 

 nomena responsible for the fact that the internal 

 energy of the fluid behind the shock front, as given 

 by equation (18), is greater than that which the fluid 

 would have if it were compressed reversibly and 

 adiabatically to the density pi. This fact provides a 

 clue which we can use to get a rough estimate of the 

 thickness of the shock front. For, the amount of 

 energy dissipated into heat per unit time by any 

 given dissipative mechanism will be dependent on 

 the thickness of the shock front, in other words, de- 

 pendent on the rapidity with which the pressure 

 changes from po to pi. This dissipated energy must 

 be equal to the product of the mass of water crossing 

 the shock front per unit time by the amount of 

 energy dissipated per unit mass; this quantity being 

 for all practical purposes simply the difference in 

 energy between the states Si and S'l in Figure 5. As 

 will be shown later, the latter quantity can be calcu- 

 lated from equation (18) in terms of the pressure 

 jump (pi — po) and the known properties of water; for 

 small amplitudes it is proportional to the cube of the 

 pressure jump. Since all reasonable dissipative phe- 

 nomena create heat more rapidly the more suddenly 

 they are made to take place, the greater the pressure 

 jump the thinner the shock front must be in order 

 to dissipate the required amount of energy. Thus if 

 we can show that the shock front should be quite thin 

 for a fairly weak shock wave, it must be even thinner 

 for a strong one. 



We shall therefore begin by calculating the ap- 

 proximate value of the dissipated energy for a weak 

 shock wave. Referring to the upper diagram in 

 Figure 5, we wish to calculate the energy difference 

 («! — e'l) between the states Si and S'l. 



Since by equation (18) the difference (ei — to) 

 equals the area of the trapezoid under the line SoSi 

 while the difference {t[ — eo) equals the area under 

 the adiabatic curve from S'l to iSo, we must have 



id — t'i)= area of region between iSi<So and adia- 

 batic curve 

 = area of segment between S[So and adia- 

 batic curve -|- area of triangle S'lSoSi. 

 Now the area of the triangle S'lSoSi is equal to half 

 the product of its altitude (1/po — 1/pi) by its base 

 (Vi ~ Pi)- Since the latter quantity is in the limit of 

 .small amplitudes proportional to (ei — ej), we can 

 make the area of the triangle as small as we like com- 

 pared to (fi — ei) by taking the amplitude of the 

 shock wave sufficiently small. Thus, for sufficiently 

 weak shock waves 



(ci — ei) -^ area of segment between S'lSo and adia- 

 batic curve 



\po Pi/ 



M - - - ) (19) 



<Po Pi/ 



where the constant k is proportional to the curvature 

 of the adiabatic curve and has the numerical value 

 1.5 X 10'" in cgs units for pure water. 



Let us now consider the mechanism by which dis- 

 sipation of energy occurs in the shock front. For any 

 assumed mechanism the rate of this dissipation can be 

 calculated, at least roughly, as a function of the thick- 

 ness of the shock front and the magnitude of the 

 pressure jump. Of the two most obvious mechanisms, 

 viscosity and heat conduction, the former gives much 

 the greater dissipation, and accordingly we shall carry 

 through the calculation only for the case of dissipa- 

 tion by viscosity. According to the theory of viscous 

 fluids, the mechanical energy converted into heat per 

 unit time in a fluid having a coefficient of shear viscos- 

 ity fj. is given, for one-dimensional motion such as that 

 in a plane wave traveling in the x direction, by 

 Dissipation per unit volume per unit time 

 'duX 

 dx / \p 



by the equation of continuity. If 8 is the thickness of 

 the shock front — the thickness of the region within 

 which most of the change in density from the value 

 Po to the value pi takes place — we have, roughly, for 

 a weak shock wave, 



Idp _^ c(pi — po) 



p dt poS 



Multiplying equation (20) by the thickness 5 gives a 

 rough value for the dissipated energy per unit time 

 per unit area of the shock front : 

 Dissipation per unit area per unit time 



2mc^(pi — po)^ 



=-(5y=-ef 



(20) 



5po 



(21) 



