A theory of the stability of plane shock waves

Abstract

The stability of form of a plane shock, obtained when a 'corrugated' piston is moved impulsively from rest with constant velocity, is investigated mathematically. Linearization of the problem is accomplished by assuming the corrugations to be small. The solution is built up by methods of Fourier analysis from 'conefield' solutions of the analogous 'wedge'-shaped piston problem, solved by methods due to Lighthill. The plane shock is shown to be stable, perturbations from plane decaying with time in an oscillatory manner like t$^{-\frac{3}{2}}$ for large ta$_{1}$/$\lambda $ (where a$_{1}$ is the velocity of sound behind the shock and $\lambda $ the wave-length of the corrugations). The stability, measured by the amplitude of this oscillation after the shock has traversed a given distance, decreases both as the shock Mach number increases above and decreases below the value 1$\cdot $14. Shocks of this strength exhibit strongest stability. Asymptotic forms for large time are given for both the shock shape and pressure distribution for shocks of moderate strength in section 4. A more complicated asymptotic form for the shock shape holds at large Mach numbers (section 5) which in the limiting case of infinite Mach number gives the result that the perturbations of shape decay like t$^{-\frac{1}{2}}$ only. Complete solutions are obtained for weak shocks in terms of Bessel functions (section 6).