## Appendix A Empirical fitting of the droplet diameter data with an analytical function. Since the experimental droplet diameter data was recorded from independent single-shot measurements, it included experimental noise in the range of 20 to 50 nm peak-to-peak for undisturbed drops. Also, since the time delays were sampled at ~10 ps or less, but the delay data was recorded with a timer-counter with ~20 ps standard deviation, the delay measurements also had significant shot-to-shot noise. This degree of experimental noise precluded a direct differentiation of diameter data to extract surface velocities. Although shock kinematics data can in principle be calculated from the raw data using Savitzki-Golay fitting [17; 52], here we chose to fit the raw data using an analytical function because it provided a more compact result and the fit quality was good across all data. The fit function, which is given by Equation (1), is the integral of the Friedlander function [53; 54]: \((1-z)\mathrm{exp}(-z)\), with \(z>0\) and \(z=(t-\mathrm{t_{shock}})/\mathrm{t_{decay}}\). The Friedlander function represents a waveform with a discontinuous pressure jump at \(z=0\), followed by an immediate decay that lowers the pressure to a minimum that is below the initial pressure. The Friedlander function was found to describe accurately the pressure-time history from hemispherical blast waves in air [54], and here we implicitly assumed that this function can also describe with sufficient accuracy the particle velocity-time history for the cylindrical shocks in water. To fit the data with Equation (1), we first subtracted from the raw data the drop diameter prior to the arrival of the shock, \(\mathrm{D_{0}}(t)\), which we expressed as linear function of time because it drifted during the experimental runs. The fit procedure is illustrated in Figure 2, and Figure A1 shows the same procedure for a different experimental run. The run shown in Figure A1 had the largest drift in \(\mathrm{D_{0}}(t)\) among all measurements presented here. Nevertheless, it was still possible to correct the data for the drifts because the relative change in the droplet diameter over the fitted time delays was less than 0.002, which is smaller than the 0.02 relative spread of undisturbed drop diameters among the runs using the smaller drops (29.9-30.5 um diameters). Variations in the drop diameter between different runs can affect the merging of different data sets because in a larger drop the shock will arrive later, and will have a lower pressure, than in a smaller drop. Nevertheless, for our measurements the drifts and the run-to-run variations in drop diameters had only a small effect on the accuracy of the derived data. For example, Figure 3 shows that the peak pressure scales by less than the inverse of the propagation distance, therefore the relative systematic error in the peak pressure, caused by a slightly different droplet diameter, will be smaller than the relative differences (~0.02) between droplet diameters in different runs. Using surface velocity measurements in droplets, the properties of the cylindrical shock generated by XFEL pulses can be determined directly only up to a limited time delay, because two additional types of pressure waves can arrive at the surface and overlap with the cylindrical shock: (i) Oblique shock reflections from surface regions away from the equatorial zone, and (ii) if spallation occurs, reflections from the surface of the void generated by spallation. To minimize the chance of measuring the contribution of additional waves, the data was fitted only up to either (i) the 'acoustic' time delay \(\mathrm{t_{acoustic}}\), which is equal to the time in which a low-amplitude wave propagates at the speed of soundfrom the center to the equator of an undisturbed drop, or (ii) until the arrival of the spallation reflection, if earlier than the acoustic time. Also, the shock waveforms were computed only up to the acoustic time.