Instability of a plane centered rarefaction wave

An analytic small‐amplitude theory of the instability of a plane centered rarefaction wave (which has recently been discovered numerically by Yang et al.) is presented. A finite‐difference (FCT) calculation is performed and compares well with the theory. The instability manifests itself as perturbation growth on the wave’s trailing edge. The asymptotic value approached by the perturbed velocity of the trailing edge is expressed as kδx₀a₀u_∞(M,γ), where k is the perturbation wave number, δx₀ is the constant perturbation amplitude of the leading edge, a₀ is the sound speed in the unperturbed gas, and u_∞(M,γ) is a dimensionless function that depends on the adiabatic exponent, γ, and the strength of the rarefaction wave, M, taken as the ratio of sound speeds behind and ahead of it. This function is essentially determined by the way the perturbed rarefaction wave is formed, e.g., by moving a corrugated piston from a gas‐filled space or by interaction of a plane shock wave with a rippled contact interface between two different gases. The function u_∞(M,γ) is shown to vanish in the limiting cases of strong and weak rarefaction waves, its asymptotic form in the latter case (M→1) being u_∞(M,γ)=[(γ+1)(1−M)/(γ−1)]^1/2. This instability, which, for example, might develop in high‐gain nuclear fusion pellets accelerated by laser beams, appears to be relatively harmless for these types of applications, since neither the perturbations of density and pressure at any given value of x/t, nor the integral ∫(δρ)dx taken over the width of the rarefaction wave increase in time.