Observing a focused weak shock wave perform a Hilbert transform.

Abstract

The spatial profile of several types of focusing two-dimensional shock waves (weak shock regime) created in a thin liquid gap are imaged using a shadowgraph with strobe photography. Their profiles before and after focusing are shown to constitute approximately a Hilbert transform pair, regardless of the range of directional angles involved in the focusing and of whether the focusing is at a single point or an extended caustic. This Hilbert transform relation, observable here from a single image or a pair of images, can be explained in terms of the π/2 Gouy phase shift for focused waves in two dimensions, as was shown elsewhere with experiments involving terahertz pulses in the time domain.