  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.
