The conditions necessary for discontinuous motion in gases

Abstract

The possibility of the propagation of a surface of discontinuity in a gas was first considered by Stokes in his paper “On a Difficulty in the Theory of Sound.” This paper begins with a physical interpretation of Poisson’s integral of the equation of motion of a gas in one dimension. The integral in question is w = f { z —( a + w ) t }; and it represents a disturbance of finite amplitude moving in a gas for which the velocity of propagation of an infinitesimal disturbance is a ; w is the velocity of the gas in the direction of the axis z . It is shown that the parts of the waves in which the velocity of the gas is w travel forward with a velocity a + w , and that there is in consequence a tendency for the crests to catch up the troughs. After a certain time, and at a certain point in space, the value of ∂ w /∂ z will become negatively infinite; a discontinuity will then occur, and Poisson’s integral will cease to apply. Stokes then leaves the subject of oscillatory waves and proceeds to consider whether it is possible to maintain a sharp discontinuity in a gas which obeys Boyle’s law ( p = a 2 ρ ). His argument, slightly modified by Lord Rayleigh, is as follows:— Suppose that a travelling discontinuity can exist. Give the whole gas such a motion that the discontinuity is brought to rest. Consider then a gas which is moving with uniform velocity u 1 up to a discontinuity. At this point the velocity suddenly changes to u 2 ; and the gas moves on uniformly at this speed. Let ρ 1 and ρ 2 be the corresponding densities, p 1 and p 2 the corresponding pressures.