## Appendix A Empirical fitting of the droplet diameter data with an analytical function.
The diameter data is fitted for time delays that exclude data within a few hundred picoseconds of the shock arrival, see Figures 3b and A1b. In a perfectly aligned drop the slope of the diameter data should change instantaneously upon the shock arrival, but the recorded data shows a gradual increase in slope due to the jitter and the systematic misalignment of the drops. The droplet centers are aligned with the beam to better than 1 \(\upmu\)m, but even misalignments of 0.1 \(\upmu\)m will make the shocks arrive at measurably different times at the two equatorial surfaces, leading to smearing of the data recorded at delays within a few hundred picoseconds of the shock arrival time in a perfectly centered drop. To choose the first data point for fitting, we first identified the earliest-delay point from where all measurements at longer delays were larger than the peak (diameter-subtracted) diameter of undisturbed drops. These peak values, caused by drop diameter noise, were typically around 20 nm for the \(\sim\)30-\(\upmu\)m drops. For runs collected at the smallest pulse energies, this first point was located after the gradually rising slope and was thus chosen as the first point for fitting. At larger pulse energies, we empirically increased the threshold value up to 100 nm to avoid fitting data near the shock arrival time. The true rise times of the shock waves were too short to be measured. Since the run in which the drops were best aligned to the XFEL beam had an experimental rise time around 75 ps, we estimate that the true rise time was shorter than 75 ps. Appendix B Determination of the waveforms of decaying cylindrical shocks from the surface velocity data  
Due to the cylindrical symmetry and the pressure decay, a basic determination of the shock pressure from the shock velocity using the Rankine-Hugoniot relations for plane waves may not be accurate [42; 55]. For the determination of the initial pressure jump, Equation (3), which is derived for a plane shock, remains appropriate because the shock thickness is much smaller than the curvature of the shock. Using the upper estimate of the shock rise time from experimental data, 75 ps, and a typical shock velocity of 2 km/s for the present data, we estimate a shock thickness smaller than 150 nm, which is less than 0.01 of the shock radius of curvature of \(\sim\)15 \(\upmu\)m at the equatorial surface of a \(\sim\)30 \(\upmu\)m diameter drop. We also note that the assumption of a particle velocity equal to half of the surface velocity [33; 56], which is implicit in Equation (2), is likely to be appropriate because the peak shock pressures are below 1 GPa and no phase transitions occur up to this pressure. The fast decay of the shock requires a specific analysis for the determination of the pressure waveform, and here we chose a method based on Lagrangian wave analysis [41; 42] using [PERSON]'s phase velocity concept [40]. The main result is Equation (4), which allows the numerical integration of pressure from its peak value using the particle velocity history [57]. For plane shocks in a material that is isotropic and does not exhibit phase transitions, the phase velocity \(\mathrm{C_{C}}[P_{S}(t)]\) is equal to the Lagrangian phase velocity \(\mathrm{c_{L}}[P_{S}(t)]\), which is equal to the speed of sound in the compressed material multiplied by the ratio of the compressed material density and the zero-pressure density [38]. In a cylindrically symmetric wave, the phase velocity is different from the speed of sound even for linear, low-amplitude pressure waves. Basically, this occurs because in a cylindrically symmetric system the wave equation has outgoing wave solutions that are not plane waves, but are given by [58]\(P(r,t)=\mathrm{A}\ \mathrm{H_{0}}^{(2)}(\mathrm{kr})\mathrm{exp}(\mathrm{i} \omega t)\) where \(\mathrm{A}\) is the wave amplitude and \(\mathrm{H_{0}}^{(2)}\) is a Hankel function (a linear combination of Bessel functions).