Nonlinear acoustical transmission through a weak shock wave

Recent experiments (Ducousso et al., Phys. Rev. Appl., L051002, 2021) demonstrated the possibility to image weak shock propagation in solids by an ultrasonic probe wave. Wave interaction with a steady, ideal step shock in air has been previously described (Burgers, Selected Papers, Springer, 478–486, 1995—McKenzie and Westphal, Phys. Fluids, 11, 2350, 1968), without consideration for the particular case of a weak shock nor for the influence of the medium. The present paper considers a weak shock interacting in any inviscid fluid with an incident probe wave. No reflected wave arises. The transmitted wave, vortex and entropy modes behind the shock, and the shock front disturbance, are determined by the linearisation of the Rankine-Hugoniot relations. For a weak shock, entropy mode and energy jump relation can be omitted. The shock motion induces a Doppler effect dependant on the medium, air and water giving opposite trends. The transmitted wave amplitude is either increased or reduced through energy exchanges with the shock. For an incidence beyond the critical angle, instead of a total reflexion, an inversion of the direction of the transmitted wave occurs, propagating in the same direction as the shock. This phenomenon seems specific to weak shocks.