The Piston Problem:sonic and supersonic velocities

Kevorkian and Cole, (Perturbation Methods in Applied Mathematics; Springer Verlag,1985), formulate a mathematical model for the acoustic response of a gas at rest in a semi-infinite space to an imposed subsonic piston velocity. The model combines classical linear acoustic analysis with asymptotic methods to develop solutions for the induced fluid motion and the accompanying thermodynamics.

The objective of current model is to employ the Euler equations written in terms of the logarithms (ln) of the thermodynamic variables to reveal the physics of the flow induced by a dimensional (') time-dependent piston velocity, u'_p(t')= αa₀f(t') whereα is a non-dimensional parameter and a'₀ is the speed of sound in the undisturbed gas. The change in fluid particle density, pressure and temperature is caused directly by the spatial derivative of the induced fluid speed, u'_x. The analysis is developed for subsonic, α<>O(1), pistons. The first reproduces the linear Kevorkian and Cole results. The sonic case is described by weakly nonlinear acoustic equations for the induced speed and the thermodynamic variables. The supersonic case utilizes an unusual asymptotic formalism facilitated, for example, by the nondimensional mass conservation equation, D(lnρ)/Dt=-αu_x where the parameter can be removed from the equation with the non-linear variable transformation for the density, ρ=exp(αR), where R is the rescaled density. Ditto for T and p.