_\tau \approx \frac{1}{2}(\Delta_\tau^+ + \Delta_\tau^-), \end{equation} where $\Delta_\tau^+$ and $\Delta_\tau^-$ are the time steps for the forward and backward simulations, respectively. The time steps in the forward and backward simulations are chosen to satisfy a CFL condition for stability. We use the same CFL number for both simulations, which is set to 0.5 in our calculations. The CFL number is defined as \begin{equation} CFL = \frac{\text{max}(|u| + c) \cdot \Delta t}{\Delta x}, \end{equation} where $u$ is the velocity vector, $c$ is the speed of sound, $\Delta t$ is the time step, and $\Delta x$ is the spatial grid size. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Acknowledgments: % Funding sources: % Author contributions: % Competing interests: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5%%% % Appendices appear after the main body of the text. They are prefixed by an A and sequentially numbered: (A1), (A2), etc. Capitalize significant words in titles of appendices, figures, tables, etc. % Appendix sections can be concluded with a theorem-like environment (e.g., {proof}) or with a discussion of implications/conclusions (e.g., {discussion}). % Number appendices consecutively within each section or independently across sections (e.g., A1, A2, B1, C1). % Use \appendix* if there is only one appendix section. % Use * after section number to suppress numbering if desired (e.g., A*). % Omit “Appendix” before section title if only one appendix appears; use “Appendices” if more than one appendix appears; use “Appendix” before individual appendix titles that are not part of a numbered sequence within appendices; e.g., Appendix A*, Appendix B*. See also Section 4.4 in Manuscript Preparation Guidelines for Nature Research Journals at http://www.nature.com/npg/authors/prep/index.html#appendixes . % For equations in appendices, number consecutively with equation numbers in main text (i.e., 1A1, 1A2; 2A1, 2A2; etc.). Refer to equations in main text as Eqn 1 or Eqs 1–3 as appropriate; refer to equations in appendices as Eqn A1 or Eqs A1–A3 as appropriate . For figures and tables in appendices, refer to them as Fig A1 or Tables A1–A3 as appropriate . For examples without special numbering schemes refer to them simply as Example A1 . See also Section 4.4 in Manuscript Preparation Guidelines for Nature Research Journals at http://www.nature.com/npg/authors/prep/index