## 4 Discussion  
_The energy carried by the shock wave._ The energy carried by the shock can be calculated by integrating the instantaneous wave intensity, \(I_{S}(t)\), which is given by:  
\[I_{S}(t)=P_{S}(t)u_{p}(t), \tag{5}\]  
Because the shocks launched in the droplets are, to a good approximation, cylindrically symmetric, their energies were expressed as linear energy densities. The linear energy density carried by the shock, from \(\mathrm{t_{shock}}\) until \(\mathrm{t_{acoustic}}\), is shown as a function of the deposited linear energy density in Figure 5a, and the ratio of the shock energy to the deposited energy is shown in Figure 5b. Since the shock wave energies shown in Figure 5 have been integrated only up to \(\mathrm{t_{acoustic}}\), they are less than the total energy carried by the shock. Also, the shocks already dissipated a fraction of their energy via irreversible heating of the liquid, when they had large amplitudes shortly after generation [8]. Nevertheless, the trends in the energy carried by shocks can be useful for designing XFEL experiments. Both the shock energy and the fractional amount of energy increased with the XFEL pulse energy, thus the energy of the shock increased faster than linearly with the pulse energy. Figure 4: Shock wave waveforms, calculated as described in the text from the fitted surface motion functions. For clarity, the data sets include the shocks with the smallest and largest amplitudes, and a subset of all available waveforms. (**a**) Waveforms at the surface of small drops, 14.95–15.25 \(\mathrm{\SIUnitSymbolMicro m}\) from the X-ray beam center. (**b**) Waveforms at the surface of large drops, 30.5 \(\mathrm{\SIUnitSymbolMicro m}\) from the X-ray beam. Figure 5: The energy carried by the shock waveforms up to \(\mathrm{t_{acoustic}}\). (**a**) Shock energy as a function of deposited energy. (**b**) Fraction of the absorbed energy carried by the shock waves. Within the range of parameters of our experiments (P\({}_{\text{S}}\) < 500 MPa), the shock waves were weak, in the sense that the energy lost by increasing the entropy of the shocked water remained small relative to the total energy of the shock [44]. We thus expect that the energy of the shock decayed slowly as it traveled between positions corresponding to the radii of the small and large droplets, and Figure 5 shows that at the same XFEL pulse energy, the calculated energy was approximately the same for \(\sim\)30 \(\upmu\)m and 61 \(\upmu\)m diameter drops. The larger energy calculated for the 61 \(\upmu\)m drops can be explained by the experimental uncertainty in energy, which arises from the uncertainty in pressure. We note that the maximum conversion ratio of absorbed energy to shock energy shown in Figure 5b, \(\sim\)12%, falls in between the conversion ratios observed for near-field and far-field shock waves generated by pulsed optical ablation in water [7,8]. _The short duration of the shock wave._ Two important characteristics of the cylindrical shocks generated by XFEL pulses in droplets are their brief duration and the likely decay of pressure to negative values (tension). The pressures in Figure 4a decayed to less than one tenth of their peak pressure by tacoustic. Since in our experiment the pressure decayed faster than the particle velocity, shocks that led to a maximum in drop diameter, and thus to the particle velocity crossing zero (see the example in Appendix A), had a negative pressure phase. Extrapolation of the waveforms in Figure 4a indicates that the pressure reached zero between 2 and 5 ns after the shock arrival, depending on the pulse energy. Multiplying these durations with the speed of sound in water (\(\sim\)1500 m/s) we can estimate the approximate length of the positive waveform (3 to 7.5 \(\upmu\)m), which is smaller than the 15 \(\upmu\)m radius of the drop. Because the positive part of the shock waveform was shorter than the drop radius, its reflections traveled inside the liquid until they no longer overlapped with the positive pressure phase of the incoming shock.