tokens: therefore becomes is the change of phase angle. In other words, the wave propagates forradians before producing a shock, where the bracket means a time average along the wave front. The total propagation time is proportional to , so stronger and higher-frequency waves shock earlier. 3.3. Maturation of the Shockwave To estimate the point of intersection between the wave peak and shock front, we derive their respective trajectories. First, suppose a monochromatic wave with luminosity and frequency ω is led by a compressive edge (i.e., ). The wave peak initially lags behind the wave front by a distance . The peak propagates with a speed , where is the compressed local sound speed. Assuming properties of the gas do not vary significantly under compression (i.e., constant and ), we can write the compressed sound speed in terms of the peak pressure p w . The thermodynamic expression of the mean wave energy density allows to be expressed in terms of conserved wave properties, With the kinematic expression ( 3 ), the speed of the wave peak becomes Before shock formation (), the wave front simply travels at the local sound speed c s0 . For , the exact shock velocity requires numerical calculations to describe the arrival of the remaining wave pulse. We circumvent this calculation by approximating the shock front strength with the wave peak properties (). With the following shock jump condition,we approximate the shock Mach number to be Thus, the wave peak and shock front converge at the same location , onceis satisfied. 4. Hydrodynamic Simulations To test our analytical predictions of shock formation and provide examples of shock evolution and strengthening, we turn to numerical simulations. We construct 1D planar and spherical simulations in FLASH (Fryxell et al. 2000 ), a hydrodynamic adaptive mesh refinement (AMR) code. All quiescent structures begin in hydrostatic equilibrium with a gravitational acceleration g ( r ) that is independent of fluid perturbations (Cowling’s approximation). All simulations have inner reflecting and outer diode (outflow) boundary conditions. Two structures are considered: a planar isothermal atmosphere and a n = 3 stellar polytrope. The adiabatic index is held fixed in all of our simulations. 4.1. Planar Earth Atmosphere For planar shock formation, we consider a vertically stratified, initially isothermal atmosphere of gas (FLASH’s default Earth atmosphere) with in constant gravity. Lengths in this section should be compared to the 8.8 km scale height of the model, although the results can be scaled to any similar atmosphere. Upward-travelling waves are initialized by setting the isothermal atmosphere out of equilibrium; an example of the initial waveform and its evolution is shown in Figure 1 . The initially sinusoidal wave steepens (Phase 1) and forms a shock at the wave node, after which the wave peak approaches (Phase 2) and merges with the shock front. Figure 1. Deformation of a single sinusoidal impulse in a planar,
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