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Acoustic Waves in a Stratified Atmosphere: II. Three-Dimensional Hydrodynamics

We investigate analytically the propagation of linear waves in a three-dimensional, nonmagnetic, isothermal atmosphere stratified in plane-parallel layers. The motivation is to study oscillations in the non-magnetic chromosphere and to assess the limitations of one-dimensional simulations of the Kv bright point phenomenon. We consider an impulsively excited acoustic disturbance, emanating from a point source, and propagating outward as a spherical acoustic wave accompanied by an internal gravity wave. The waves amplify exponentially in the upward direction. A significant wave amplitude is therefore found only in a relatively narrow cone about the vertical. The amplitude of the wave decreases with time. Because of the lateral spread, the wave amplitude decays faster in 2D and 3D simulations than in 1D. The initial pulse, which travels at the sound speed, carries most of the energy injected into the medium. Subsequent wave crests leave the source region at ever-increasing phase speed, but slow to the sound speed as they approach the head of the wave. Important conclusions from the 3D solution that were not anticipated from the plane-wave solution are: 
1. The bulk of the energy is emitted in the upward (and downward) direction; much less goes into the horizontal direction. 
2. The wave profile narrows from the initial pulse through the amplitude maxima in the wake of the pulse. 
As a consequence of both points, the shock-heated regions in the wake of the initial pulse would weaken in strength and shrink in size.