Page:Chandrasekhar - On the decay of plane shock waves.djvu/5

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.

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