The reflection of a shock pulse at a liquid-gas interface

We present an analytical method for computing the reflection and transmission of a non-linear pulse of finite width, i.e. an impulsive shock wave, at a liquid-gas interface. The problem is treated analytically by considering idealised pulses and solving a series of consecutive Riemann problems. In the acoustic limit, the method produces identical results to linear acoustic theory, where reflection and transmission coefficients depend only on the impedance difference. However, the characteristics of the reflected and transmitted waves depart from linear theory as the pulse strength increases. It is shown that the reflection problem cannot only consider the interface, and wave interactions between the incident pulse and reflected waves must also be considered. For a water-air interface, we explain how a reflecting pulse can put the water into tension without any incident negative pressure. It is further shown that the magnitude of the reflection coefficient decreases with increasing incident shock pressure, and the reflected pulse widens. Reflections of pulses with positive and negative pressures temporarily create negative pressure regions with greater magnitude than the incident pulse. Finally, the method is used to predict the reflection of non-idealised waves. Comparisons with numerical simulations show that the reflection characteristics can be qualitatively explained using the analytical method, and the reflection coefficients are accurately predicted.