Instability of converging shock waves and sonoluminescence

We study the problem of the stability of a nearly spherical converging shock wave in a van der Waals gas and consider the implications for sonoluminescence. An approximate geometrical theory of shock propogation, due to Whitham [Linear and Non-linear Waves (Wiley, New York, 1974); J. Fluid. Mech. 2, 146 (1957); 5, 369 (1959)], is used. A first-order treatment of deviations from spherical symmetry, similar to one performed by Gardner, Brook, and Bernstein [J. Fluid. Mech. 114, 41 (1982)] for an ideal gas, shows that these deviations are unstable, coming to dominate the shape of a shock wave as it converges. The instability is weak, although not as weak as in an ideal gas. Perturbations grow as a small inverse power of the radius. The mechanism for concentration of energy in sonoluminescence involves a spherical converging shock. The validity of the theory given here is checked by comparing the results for spherically symmetric shocks with a simulation by Kondic, Gersten, and Yuan [Phys. Rev. E 52, 4976 (1995)]. We then estimate the degree of bubble symmetry necessary for sonoluminescence and relate this result to the experimental robustness of sonoluminescence.