Theory of embedded shock formation in rarefaction waves by homogeneous condensation

A theory of embedded shock formation by homogeneous condensation in the centered rarefaction wave of a shock tube is presented. The necessary and sufficient conditions for the existence of such embedded shock waves are exhibited in a parabolic approximation to the family of intersecting characteristics. In particular the coordinates of the embedded shock origin are derived explicitly by the construction of the envelope of the family. Predictions of the theory, including an estimate for the average embedded shock speed, are substantiated by comparison of the results obtained employing the classical nucleation theory and Hertz-Knudsen droplet growth law for the condensation model with those of typical experiments showing visualized shocks formed by homogeneous condensation during the expansion of water vapor in nitrogen (or air).