On the Symmetry of Blast Waves

TL;DR

This paper reviews G. I. Taylor's solution to the point blast wave problem, using symmetry and differential systems to connect geometric symmetries with wave physics, and derives key laws governing blast wave behavior.

Contribution

It introduces a novel approach combining Lie group symmetry and exterior differential systems to analyze blast wave solutions and derive fundamental relations like Taylor's law.

Findings

Derivation of Taylor's two-fifths law using symmetry techniques

Demonstration of integrability of differential systems for blast wave variables

Connection between geometric symmetries and wave propagation physics

Abstract

This article presents a brief historical review of G. I. Taylor's solution of the point blast wave problem which was applied to the Trinity test of the first atomic bomb. Lie group symmetry techniques are used to derive Taylor's famous two-fifths law that relates the position of a blast wave to the time after the explosion and the total energy released. The theory of exterior differential systems is combined with the method of characteristics to demonstrate that the solution of the blast wave problem is directly related to the basic relationships that exist between the geometry (or symmetry) and the physics of wave propagation through the equations of motion. The point blast wave model is cast in terms of two exterior differential systems and both systems are shown to be integrable with local solutions for the velocity, pressure, and density along curves in space and time behind the blast wave.