Propagation of weak shock waves in an elliptical cavity

Abstract: The propagation of weak shock waves in a planar elliptical cavity is studied theoretically and experimentally. The results are of interest in the design of an elliptical high-speed water jet generator operating in a transient fashion with electrically generated shocks in the acoustic regime. Perturbation theory and geometrical acoustics are used to find the resulting shock wave from the discharge. From this the reflected wave from the cavity walls is found through a condition incorporating the wall admittance. Padé approximants are employed to extend the validity of the solutions. The reflected wave is shown to be strongly dependent on the elliptical eccentricity, and the energy in its front is found to have a minimum for a particular, high value of this parameter. With the use of high-speed photography and schlieren technique experiments with a water filled cavity reveal a complex wave system in the liquid. A prominent feature is the appearance of a lateral wave. This is caused by wave motion in the wall, and conditions for when it is induced are derived. It will be absent if the product of eccentricity and propagation speed of the mode that causes it is less than the liquid sound speed. Experiments with a simple jet generator show that liquid jet speeds of the order of 100 m/s are easily attained. The long-time development of perturbations on the shock wave structure is derived. Different rates of approach to the asymptotic shock shape is found. The exact behavior appears to depend on certain combinations of parameters.