Shock Wave in the Beirut Explosion: Theory and Video Analysis

TL;DR

Problem: observe a weak spherical shock in the Beirut explosion and test nonlinear theory using publicly available footage. Approach: derive the Landau-Whitham scaling $l \\propto \\sqrt{\\ln R}$ for the overpressure layer via equal-angle construction and post-shock characteristics, then validate with frame-by-frame video measurements of $R$ and $l$. Findings: the data show a linear relation $l = a\\sqrt{\\ln R} + b$ with $R^2 \\approx 0.91$, indicating semi-quantitative agreement with theory. Significance: demonstrates a rare observable test of nonlinear shock dynamics and provides educational insight into shock-wave physics.

Abstract

Videos of the 2020 Beirut explosion offer a rare opportunity to see a shock wave. We summarize the non-linear theory of a weak shock, derive the Landau-Whitham formula for the thickness of the overpressure layer and, using frame-by-frame video analysis, we demonstrate a semi-quantitative agreement of data and theory.

Figures (4)

• Figure 1: Sketch of the structure of the blast wave. The lower pressure in the rarefaction layer causes water vapor to condense, resulting in the white cloud visible behind the shock front.
• Figure 2: Shock trajectory $R(t)$ and characteristics just behind (A) and ahead (B) of it in the $(c_0 t,R)$ plane. All three slopes are actually close to unity in a weak shock but have been exaggerated for clarity. The shock tangent bisects the angle between A and B (Eq. \ref{['eq5']}). Characteristics behind the shock are generally curved because the local velocity $u$ and sound speed $c$ varyWhitham:1950.
• Figure 3: Thickness $l$ of the high-pressure layer versus $\sqrt{\ln (R/1 \text{ meter})}$. Error bars are $1\,\sigma$, obtained by propagating a $\pm 5$-pixel picking uncertainty using 2.3 m/px scale ($\simeq 12 \text{ m}$ per measurement; (see App. \ref{['Conversion appendix']})). The solid line is a linear fit to $l=a\sqrt{\ln R}+b$, with the coefficient of determination of 0.91. The observed linearity is consistent with the Landau–Whitham scaling for layer expansion.
• Figure 4: Comparison of $R-l$ in pixels obtained from VHP to $R-l$ in meters obtained from V6 at the same times after the explosion ($0.867 < t < 1.067$ s). We estimate a $\pm$5 pixel uncertainty in our measurements from V6 and VHP. This uncertainty gives the horizontal error bars, and the vertical error bars are found by multiplying this uncertainty by Aouad's conversion factorAouad:2021aa of 0.565 m/px. A linear fit results in the conversion factor of $a=2.3\pm0.1$ m/px for VHP.