On the Decay of Plane Shock Waves

by Subrahmanyan Chandrasekhar

BRL REPORT NO. 423 NOVEMBER 1943. ON THE DECAY OF PLANE SHOCK WAVES. S. Chandrasekhar. BALLISTIC RESEARCH LABORATORIES. ABERDEEN PROVING GROUND, MARYLAND

Ballistic Research
Laboratory Report No. 423

 

Chandrasekhar/emh
Aberdeen Proving Ground, Md.
8 November 1943

 

ON THE DECAY OF PLANE SHOCK WAVES

 

Abstract

 

In this report the problem of the decay of plane shock waves is considered and it is shown that there exists a special class of shock pulses for which the problem of decay admits of an explicit solution. The case in question arises when shocks of moderate intensities (Mach numbers less than 1.5 if an accuracy of one per cent is demanded) are considered. In these cases the changes in entropy as well as in the quantity 5 c — u (where c denotes the local velocity of sound and u the mass velocity) as the shock front is crossed can be neglected. The case of linear shock pulses in which both u and c are linear functions of x behind the shock front is considered particularly in some detail.

1. Introduction. The only case in the theory of the decay of plane shock waves which appears to have been studied in any detail is the one which has recently been investigated by W. G. Penney and H. K. Dasgupta (R.C. 301) by numerical methods. And it appears not to have been recognized that there exists a special class of shock pulses for which the problem of decay admits of an explicit solution. The case in question arises when shock waves of moderate intensities (i.e., shocks with Mach numbers less than 1.5) are considered. For, under these circumstances (when the ratio of pressures on either side of the shock front is less than 2.5) the increase in entropy of an element of gas as it crosses the shock front can be ignored. And moreover the change in the quantity

(1)$Q={\frac {2}{\gamma -1}}c-u$

(where c denotes the local velocity of sound and u the mass velocity) as we cross the shock can also be neglected. That this is so is apparent from Table 1 where the velocity of sound immediately behind the shock front (in units of the velocity of sound, a, in front of the shock) as determined by the Rankine-Hugoniot equation

(2)

${\frac {c}{a}}$	$=\left[y{\frac {\gamma +1+(\gamma -1)y}{\gamma -1+(\gamma +1)y}}\right]^{1/2}$
	$=\left[{\frac {y(6+y)}{1+6y}}\right]^{1/2}$	for γ = 1.4,

where y denotes the ratio of pressures on either side of the shock front, is compared with that given by

(3)

${\frac {c}{a}}$	$=y^{(\gamma -1)/2\gamma }$
	$=y^{1/7}$	for γ = 1.4,

which will be valid if entropy changes be ignored. It is seen that in agreement with what we have stated, the values of c for γ = 1.4 determined by equations (2) and (3) differ by less than one per cent for y < 2.5. Similarly we also notice that the change in Q as we cross the shock front is also less than one per cent under the same circumstances.

TABLE 1

In this table y denotes the ratio of the pressures on the two sides of the shock front, U the shock velocity, u the mass velocity behind the front (both in a frame of reference in which the mass velocity in the undisturbed region is zero), c the velocity of sound immediately behind the shock front, P = 5c+u and Q = 5c-u.

 

y	U/a	u/a	c/a	y1/7	P/a	Q/a
1.0	1.000	0.000	1.000	1.000	5.000	5.000
1.1	1.042	0.069	1.014	1.014	5.137	5.000
1.2	1.082	0.132	1.026	1.026	5.265	5.001
1.3	1.121	0.191	1.038	1.038	5.384	5.001
1.4	1.159	0.247	1.050	1.049	5.496	5.002
1.5	1.195	0.299	1.061	1.059	5.602	5.005
1.6	1.231	0.348	1.071	1.069	5.704	5.007
1.7	1.265	0.395	1.081	1.078	5.801	5.010
1.8	1.298	0.440	1.091	1.087	5.894	5.014
1.9	1.331	0.483	1.100	1.095	5.984	5.018
2.0	1.363	0.524	1.109	1.104	6.071	5.023
2.1	1.394	0.564	1.118	1.112	6.155	5.027
2.2	1.424	0.602	1.127	1.119	6.237	5.034
2.3	1.454	0.639	1.136	1.127	6.318	5.040
2.4	1.483	0.674	1.144	1.133	6.395	5.047
2.5	1.512	0.709	1.152	1.140	6.470	5.053

 

It would accordingly appear that a significant case of shock pulses is provided by neglecting entropy changes and considering Q as a constant throughout. As we shall see this leads to an interesting class of shock solutions which does not appear to have been isolated so far.

2. The Solutions for a Special Class of Shock Pulses for which Q = Constant. Measuring the velocities in units of the velocity of sound in the undisturbed air in front of the shock, the equations of motion in Riemann's form are (cf. Penney, R. C., 260; also, the Appendix to this paper)

$dP={\frac {\partial P}{\partial x}}\left[dx-(u+c)dt\right]+{\frac {1}{\gamma -1}}c^{2}{\frac {\partial \log \theta }{\partial x}}dt$,(4)

and

$dQ={\frac {\partial Q}{\partial x}}\left[dx-(u-c)dt\right]+{\frac {1}{\gamma -1}}c^{2}{\frac {\partial \log \theta }{\partial x}}dt$,(5)

where

(6)$P{=}{\frac {2}{\gamma -1}}c+u$;$Q{=}{\frac {2}{\gamma -1}}c-u$

and $\theta$ denotes the potential temperature (i.e., the temperature which the element of gas under consideration would have if reduced adiabatically to a certain standard pressure). For shocks of moderate Intensities the term in $\theta$ which, incorporates the changes in entropy can be ignored and we have

(7)$dP={\frac {\partial P}{\partial x}}\left[dx-(u+c)dt\right]$,

and

(8)$dQ={\frac {\partial Q}{\partial x}}\left[dx-(u-c)dt\right]$,

The foregoing equations are equivalent to

(9)${\frac {\partial P}{\partial t}}=-(u+c){\frac {\partial P}{\partial x}}$,

and

(10)${\frac {\partial Q}{\partial t}}=-(u-c){\frac {\partial Q}{\partial x}}$,

The particular significance of the case

(11)$Q{=}{\frac {2}{\gamma -1}}c-u{=}$ constant ${=}{\frac {2}{\gamma -1}}$

is now apparent: Q has the value 5 outside the shock pulse, it retains its value (to within 1% for y ≤ 2.5) as we cross the shock. Moreover, according to equation (8) since

(12)$dQ=0$ for $dx=(u-c)dt$

it appears that after a sufficient length of time the shock pulse must be characterized by Q = 5 throughout. In any event it is clear that in considering the case Q = constant = 5 we are not limiting ourselves to too 'trivial' a class of shock pulses. Assuming then the validity of equation (11) we have

(13)$u={\frac {2}{\gamma -1}}\left(c-1\right)$;$P={\frac {2}{\gamma -1}}\left(2c-1\right)$;$u+c={\frac {\gamma +1}{\gamma -1}}c={\frac {2}{\gamma -1}}$,

and equation (9) reduces to

(14)${\frac {\partial c}{\partial t}}=-\left({\frac {\gamma +1}{\gamma -1}}c-{\frac {2}{\gamma -1}}\right){\frac {\partial c}{\partial x}}$.

Letting

(15)$\phi ={\frac {\gamma +1}{\gamma -1}}c-{\frac {2}{\gamma -1}}$

we have

(16)${\frac {\partial \phi }{\partial t}}+\phi {\frac {\partial \phi }{\partial x}}=0$.

A complete integral of the foregoing equation can be written down at once. We have

(17)$\phi ={\frac {1+qx}{b+qt}}$

where b and q are two arbitrary constants. In terms of this complete integral we can readily write down the general solution of equation (16). But postponing this discussion to §3 and limiting ourselves for the present to the solution (17), we have (cf. eqs. (13) and (15))

(18)$c={\frac {\gamma -1}{\gamma +1}}\left({\frac {1+qx}{b+qt}}+{\frac {2}{\gamma -1}}\right)$

and

(19)$u={\frac {2}{\gamma +1}}\left({\frac {1+qx}{b+qt}}-1\right)$

According to equations (18) and (19), at any given instant both c and u are linear functions of x behind the shock front. By a translation of the time axis we can rewrite equations (18) and (19) more conveniently in the forms

(20)$c={\frac {\gamma -1}{\gamma +1}}\left({\frac {1+qx}{1+qt}}+{\frac {2}{\gamma -1}}\right)$,

and

(21)$u={\frac {2}{\gamma +1}}\left({\frac {1+qx}{1+qt}}-1\right)$.

From the foregoing equations it follows that

(22)u = 0, c = 1 for x = t

In other words the point at which u = 0 in the pulse moves forward with a uniform velocity equal to the velocity of sound in the undisturbed regions. Starting at this point, x = t, u and c increase linearly with x till they attain their maximum values immediately behind the shock front. And moreover the shock velocity U is related to the mass velocity u_max behind the shock front by the Rankine-Hugoniot equation

(23)$u_{\max }={\frac {2}{\gamma +1}}\left(U-{\frac {1}{U}}\right)$.

Thus at any given instant the positive phase of the pulse extends from^[1]

(24)x = t to $qx_{\max }=(1+qt)\left({\frac {\gamma +1}{2}}u_{\max }+1\right)-1$.

It now remains to determine u_max (or equivalently U) as a function of time. We proceed now to establish this relation.

Fig. 1

Fig. 1

Let ABC denote the pulse at time t. At time t + Δt, A moves to A′ and C to C′ where

(25)$AA'=\Delta t$ and $CC'=U\Delta t$,

where U denotes the shock velocity at time t. During the same time the slope of AB also changes. For at t + Δt the velocity field will be governed by

(26)$u={\frac {2}{\gamma +1}}\left({\frac {1+qx}{1+q\left(t+\Delta t\right)}}-1\right)$

Accordingly the value of u at the new position B′C′ of the shock front is given by

(27)$u_{\max }\left(t+\Delta t\right)={\frac {2}{\gamma +1}}\left[{\frac {1+q\left(x+U\Delta t\right)}{1+q\left(t+\Delta t\right)}}-1\right]$.

The increment in u_max during the time Δt is therefore given by

(28)

$\Delta u_{\max }$	$={\frac {2}{\gamma +1}}\left[{\frac {1+q\left(x+U\Delta t\right)}{1+q\left(t+\Delta t\right)}}-1\right]-{\frac {2}{\gamma +1}}\left[{\frac {1+qx}{1+qt}}-1\right],$
	$={\frac {2}{\gamma +1}}{\frac {q}{1+qt}}\left[U-{\frac {\gamma +1}{2}}u_{\max }-1\right]\Delta t$.

Using equation (23) we can rewrite the foregoing equation in the form

(29)${\frac {d}{dt}}\left(U-{\frac {1}{U}}\right)={\frac {q}{1+qt}}\left({\frac {1}{U}}-1\right)$.

Thus the differential equation governing the dependence of U on time is

(30)$\left(1+{\frac {1}{U^{2}}}\right){\frac {dU}{dt}}=-{\frac {q}{1+qt}}\left({\frac {U-1}{U}}\right)$.

This equation can be integrated to give

(31)$1+qt={\frac {\left(U_{0}-1\right)^{2}}{U_{0}}}e^{\left(U_{0}-U\right)}$

where U₀ denotes the shock velocity at time t = 0. The dependence of the ratio of pressures on the two sides of the shock front on time can be readily written down from equation (31). We have (for γ = 1.4)

(32)$1+qt={\frac {\left(U_{0}-1\right)^{2}}{U_{0}}}{\frac {\left[\left(6y+1\right)/7\right]^{1/2}}{\left[\left\{\left(6y+1\right)/7\right\}^{1/2}-1\right]^{2}}}e^{U_{0}-\left[\left(6y+1\right)/7\right]^{1/2}}$.

Equations (20), (21), (23), (24), (31) and (32) together describe completely the behavior of a linear shock pulse. The only limitation on this solution is that U₀ ≤ 1.5 for γ = 1.4 if an accuracy of the order of 1% is demanded.

In Fig. 2 we have illustrated the dependence of U on t for the case U₀ = 1.24 and γ = 1.4. Similarly in Fig. 3 the velocity field in the positive phase of the shock pulse at various instants is illustrated for the same case. And finally in Table II we have tabulated x_max, u_max, and U as functions of time also for the case U₀ = 1.24 and γ = 1.4.

Fig. 2

FIG. 2

Fig. 3

FIG. 3

It is of interest to compare these results with those of Penney and Dasgupta. Though these authors considered an initial shock pulse with a Mach number in the neighborhood of 3, it is seen that their curves illustrating the velocity in the pulse at various instants are very similar to those illustrated in Fig. 3.

We may further note that, according to equations (31) and (32), for γ = 1.4

$\left.{\begin{array}{lll}qt\sim {\dfrac {\left(U_{0}-1\right)^{2}}{U_{0}}}\ \mathrm {e} ^{U_{0}}\ {\dfrac {1}{\left(U-1\right)^{2}}}&\ &\left(t\rightarrow \infty ,U\rightarrow 1\right)\\\\qt\sim {\dfrac {49\left(U_{0}-1\right)^{2}}{9U_{0}}}\ \mathrm {e} ^{U_{0}}\ {\dfrac {1}{\left(y-1\right)^{2}}}&\ &\left(t\rightarrow \infty ,y\rightarrow 1\right)\end{array}}\right.$

(33)

It is possible that the laws

$\left(U-1\right)\propto {\frac {1}{\sqrt {t}}}\ \mathrm {and} \left(y-1\right)\propto {\frac {1}{\sqrt {t}}}\left(t\rightarrow \infty \right)$

(34)

are more general than their derivations from equation (31) and (32) would suggest.

Table II

qt	qxmax	U	umax	qt	qxmax	U	umax
0	0.434	1.24	0.361	004.878	006.000	1.10	0.159
0.195	0.673	1.22	0.334	008.199	009.616	1.08	0.128
0.450	0.982	1.20	0.306	015.375	017.28	1.06	0.097
0.796	1.394	1.18	0.277	035.88	038.77	1.04	0.065
1.280	1.960	1.16	0.248	146.6	152.5	1.02	0.033
1.986	2.771	1.14	0.219	∞	∞	1.00	0.000
3.074	3.999	1.12	0.189

 

3. On the General Solution of Shock-Pulses with Q = Constant. In §2 we have discussed the special case of linear shock pulses which are characterized by Q = Constant $\left(=2/\left(\gamma -1\right)\right)$ throughout. In this section we shall briefly indicate how the most general shock pulses under the circumstances Q = constant can be constructed. For this purpose we require the general integral of equation (16). As is well known (cf. A.R. Forsyth, Differential Equations, pp. 375-380) the general integral can be readily written down in terms of a complete integral, i.e., an integral which contains as many constants as independent variables. Writing the complete integral of equation (16) in the form (cf. eq. (17))

(38)$\phi ={\frac {a_{1}+x}{a_{2}+t}}$

where a₁ and a₂ are two arbitrary constants, the general integral of equation (16) can be expressed as the eliminant between the equations

(36)

$\phi \chi (a)+t=a+x$	,
$\phi {\frac {d\chi }{da}}=1$	,

where χ is any arbitrary function of a. It is now evident that with the solution in the form (36) we can make φ satisfy any arbitrary distribution at time t = 0. Alternatively we may say that the distribution of c (or equivalently u) at time t = 0 will determine χ(a) thus making the solution determinate. In this fashion the most general form of shock pulses under the assumptions made in §1 can be constructed. In a later report we propose to give examples of shock pulses belonging to this more general class.

S. Chandrasekhar

APPENDIX

THE EQUATIONS OF MOTION IN RIEMANN'S FORM ALLOWING FOR CHANGES IN ENTROPY

In the text equations (4) and (5) were quoted from Penney (R.C., 260). Since Penney's report may not be generally accessible, it has been thought worth while to include here a brief outline of his derivation of these equations.

The equations of motion of a linear pulse are

${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}{=}-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}$,

(1)

${\frac {\partial \rho }{\partial t}}+u{\frac {\partial \rho }{\partial x}}{=}-\rho {\frac {\partial u}{\partial x}}$,

where p denotes the pressure, ρ the density, and u the mass velocity at any point. Introduce the two functions

$P{=}\Delta _{1}{=}f(\rho ,\theta )+u$,

(2)

$Q{=}\Delta _{2}{=}f(\rho ,\theta )-u$,

where

(3)$f{=}\int _{0}^{\rho }{\sqrt {\left({\frac {\partial p}{\partial \rho }}\right)_{\theta }}}d\log \rho$;$c^{2}{=}\left({\frac {\partial p}{\partial \rho }}\right)_{\theta }$,

and c the local sound velocity. In equations (3), θ denotes the temperature which the element of gas under consideration would have when it is reduced to a standard pressure adiabatically. It is evident that θ remains constant during the motion of any element of gas (except when it crosses a discontinuity).

Consider the total differential

(4)$d\Delta _{i}{=}{\frac {\partial \Delta _{i}}{\partial t}}dt+{\frac {\partial \Delta _{i}}{\partial x}}dx$(i=1,2).

Rewriting this in the form

(4′)$d\Delta _{i}={\frac {\partial \Delta _{i}}{\partial x}}\left[dx-\left(u\pm c\right)dt\right]+\left[{\frac {\partial \Delta _{i}}{\partial x}}\left(u\pm c\right)+{\frac {\partial \Delta _{i}}{\partial t}}\right]dt$

(the + or - sign going respectively with i = 1 or 2) we shall consider the second term in square brackets occurring as the coefficient of dt. We have (cf. eq. (3))

${\frac {\partial \Delta _{i}}{\partial x}}\left(u\pm c\right)+{\frac {\partial \Delta _{i}}{\partial t}}$

(5)$=\pm c{\frac {\partial \Delta _{i}}{\partial x}}+\left({\frac {\partial f}{\partial \rho }}{\frac {\partial \rho }{\partial x}}+{\frac {\partial f}{\partial \theta }}{\frac {\partial \theta }{\partial x}}\right)u+\left({\frac {\partial f}{\partial \rho }}{\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial \theta }}{\frac {\partial \theta }{\partial t}}\right)$

$\pm \left(u{\frac {\partial u}{\partial x}}+{\frac {\partial u}{\partial t}}\right)$.

Remembering that θ remains constant during the motion apd using the equations (1) we find that

(6)${\frac {\partial \Delta _{i}}{\partial x}}\left(u\pm c\right)+{\frac {\partial \Delta _{i}}{\partial t}}=\left(\pm c{\frac {\partial f}{\partial \theta }}\mp {\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}\right){\frac {\partial \theta }{\partial x}}$.

Thus,

(7)$dP={\frac {\partial P}{\partial x}}\left[dx-\left(u+c\right)dt\right]+\left(c{\frac {\partial f}{\partial \theta }}-{\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}\right){\frac {\partial \theta }{\partial x}}dt$.

and

(7)$dQ={\frac {\partial Q}{\partial x}}\left[dx-\left(u-c\right)dt\right]-\left(c{\frac {\partial f}{\partial \theta }}-{\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}\right){\frac {\partial \theta }{\partial x}}dt$.

Now for a perfect gas (with a ratio of specific heats γ) we have

(9)$p=A^{2}\theta ^{\gamma }\rho ^{\gamma }$.

where A is a constant. From this equation we readily derive the formulae

(10)$c=A\gamma ^{1/2}\theta ^{\gamma /2}\rho ^{(\gamma -1)/2}$,

and

(11)$f={\frac {2}{\gamma -1}}A\gamma ^{1/2}\theta ^{\gamma /2}$$\rho ^{(\gamma -1)/2}={\frac {2}{\gamma -1}}c$.

Thus for the case under consideration

(12)$P={\frac {2}{\gamma -1}}c+u$ and $Q={\frac {2}{\gamma -1}}c-u$.

Moreover we readily verify from the foregoing that

(13)$c{\frac {\partial f}{\partial \theta }}-{\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}={\frac {1}{\gamma -1}}{\frac {c^{2}}{\theta }}$.

Combining equations (7), (12) and (13) we obtain the equations quoted in the text.

---

1. In this paper we do not explicitly discuss the negative (or the suction) phase of the pulse. It is however clear that the discussion of the suction phase will proceed on lines exactly similar to that given for the positive phase.