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We establish the equivalence of free piston and delta shock, for theone-space-dimensional pressureless compressible Euler equations. The deltashock appearing in the singular Riemann problem is exactly the piston that maymove freely forward or backward in a straight tube, driven by the pressurelessEuler flows on two sides of it in the tube. This result not only helps tounderstand the physics of the somewhat mysterious delta shocks, but alsoprovides a way to reduce the fluid-solid interaction problem, which consists ofseveral initial-boundary value problems coupled with moving boundaries, to asimpler Cauchy problem. We show the equivalence from three differentperspectives. The first one is from the sticky particles, and derives theordinary differential equation (ODE) of the trajectory of the piston by astraightforward application of conservation law of momentum, which isphysically simple and clear. The second one is to study a coupledinitial-boundary value problem of pressureless Euler equations, with the pistonas a moving boundary following the Newton's second law. It depends on a conceptof Radon measure solutions of initial-boundary value problems of thecompressible Euler equations which enables us to calculate the force on thepiston given by the flow. The last one is to solve directly the singularRiemann problem and obtain the ODE of delta shock by the generalizedRankine-Hugoniot conditions. All the three methods lead to the same ODE.

Delta Shock as Free Piston in Pressureless Euler Flows