## Appendix A Empirical fitting of the droplet diameter data with an analytical function.
Far from the source (\(\mathrm{kr}\gg 1\)), \(\mathrm{H_{0}}^{(2)}(\mathrm{kr})\) is approximately equal to \((2/\pi)(\mathrm{kr})^{-1/2}\mathrm{exp}(-\mathrm{i}\mathrm{kr}+\pi/4)\) and the phase velocity \(\mathrm{C_{phase}}\) is the same as in a harmonic plane wave, \(\mathrm{C_{phase}}=\omega/\mathrm{k}\). However, near the origin (\(\mathrm{kr}<\) -10) the distance between the consecutive roots of the real part of the Hankel function is larger than \(\pi/\mathrm{k}\), thus the 'wavelength' of a cylindrical wave is longer than \(2\pi/\mathrm{k}\), and correspondingly near the symmetry axis \(\mathrm{C_{phase}}\) is larger than \(\omega/\mathrm{k}\). We derived the phase velocity to be used in Equation (4), \(\mathrm{C_{C}}[P_{S}(t)]\), by imposing that (i) both mass and momentum were conserved in the cylindrical wave and (ii) the pressure and density were related by an adiabatic equation of state. For the equation of state of water, we used a modified Tait equation of state (see the Supplemental Material of Reference [16]):  
\[(P+P_{\infty})/(P_{0}+P_{\infty})=(\rho/\rho_{0})^{\gamma}, \tag{1}\]  
where \(P\) and \(\rho\) are the absolute pressure and the density, \(P_{0}=1.01325\times 10^{5}\) Pa and \(\rho_{0}=996.7\) kg/m\({}^{3}\) are the pressure and density before the shock (i.e., water at atmospheric pressure and \(26.2\,^{\circ}\mathrm{C}\)). Equation (1) is referenced to the liquid density \(\rho_{0}\) at atmospheric pressure, which makes it more convenient to use than the more common form of the Tait equation [59], which is referenced to the liquid density at zero pressure. We used a value \(\gamma=7.15\), as employed in other studies [16]. The value of the constant \(P_{\infty}\), \(P_{\infty}\) = \(3.1258\times 10^{8}\) Pa, was determined by requiring that the speed of sound \(\mathrm{c_{0}}\) at \(26.2\,^{\circ}\mathrm{C}\), \(\mathrm{c_{0}}\) = \(1497.7\) m/s, is equal to the speed of sound \(\mathrm{c_{eos}}\) calculated from the equation of state (Equation (1)):  
\[\mathrm{c_{eos}}^{2}=(\partial P/\partial\rho)_{\mathrm{S}}=\gamma(P+P_{ \infty})/\rho, \tag{2}\]  
Following Fowles' derivation of the conservation relations for stress waves in spherical coordinates [43], we derived the conservation laws for density and pressure in cylindrical coordinates:  
\[\mathrm{d}\rho=(\rho^{2}r)/(\rho_{0}\tau_{0}\mathrm{C_{u}})\ \mathrm{d}u+( \rho u)/r\ \mathrm{d}t, \tag{3}\]  
\[\mathrm{d}P=(\rho_{0}\tau_{0}\mathrm{C_{P}})/r\ \mathrm{d}u, \tag{4}\]  
where \(\rho\), \(P\), \(u\) and \(r\) are the density, pressure, particle velocity, and radial position of the Lagrangian fluid element, \(\rho_{0}\), \(P_{0}\), and \(\tau_{0}\) the corresponding values before the shock arrival. The two phase velocities [40] are defined by \(\mathrm{C_{u}}=(\partial h/\partial t)_{\mathrm{u}}\) and \(\mathrm{C_{P}}=(\partial h/\partial t)_{\mathrm{P}}\), where \(h\) is the Lagrangian coordinate of the fluid element. Since the flow of the liquid after the shock discontinuity is adiabatic, and since no phase transitions occur during flow, we assumed that the phase velocities are equal: \(\mathrm{C_{u}}=\mathrm{C_{P}}=\mathrm{C_{C}}(P)\). At any given pressure \(P\), Equation (2) can be used to derive the density variation \(\mathrm{d}\rho_{\mathrm{eos}}=\mathrm{d}P/\mathrm{c_{eos}}^{2}\) corresponding to the differential change in pressure \(\mathrm{d}P\) given by Equation (4). The condition that the change in density should be the same when calculated by Equation (3) as when calculated from Equation (4) via the equation of state (i.e., \(\mathrm{d}\rho=\mathrm{d}\rho_{\mathrm{eos}}\)) leads to a quadratic equation in \(\mathrm{C_{C}}(P)\), from which we calculated the phase velocity. Finally, the shock waveforms \(P_{\mathrm{S}}(t)\) were calculated by numerical integration of Equation (4) from \(\mathrm{t_{shock}}\) until \(\mathrm{t_{acoustic}}\). The initial pressure was calculated using Equation (3).