2021ApJ...908...23M

We address the shocks from acoustic pulses and wave trains in general one-dimensional flows, with an emphasis on the application to super-Eddington outbursts in massive stars. Using approximate adiabatic invariants, we generalize the classical equal-area technique in its integral and differential forms. We predict shock evolution for the case of an initially sinusoidal but finite wave train, with separate solutions for internal shocks and head or tail shocks, and demonstrate detailed agreement with numerical simulations. Our internal shock solution motivates improved expressions for the shock-heating rate. Our solution for head and tail shocks demonstrates that these preserve dramatically more wave energy to large radii and have a greater potential for the direct ejection of matter. This difference highlights the importance of the waveform for shock dynamics. Our weak-shock analysis predicts when shocks will become strong and provides a basis from which this transition can be addressed. We use it to estimate the mass ejected by sudden sound pulses and weak central explosions.