## 1. IntroductionThe solar chromosphere exhibits two signatures setting it apart from the underlying photosphere: an emission spectrum that characterizes its thermodynamic state, and oscillations that characterize its dynamical state. Both signatures are present in oscillations in and bright points, which arise in the emission peaks on the blue side of the central absorption features in the resonance lines of Ca II, the H and K lines in Fraunhofer's nomenclature; these features originate at a height of approximately 1 Mm above in the nonmagnetic chromosphere. The H and K lines are the strongest lines in the visible chromospheric spectrum and they show the three-minute oscillations. For these reasons they have often been the objects of observational studies (e.g., Liu 1974; Cram & Damè 1983; Lites, Rutten & Kalkofen 1993), and theoretical investigations (e.g., Fleck & Schmitz 1991; Kalkofen et al. 1994, hereafter KRBM; Sutmann & Umschneider 1995; Sutmann, Musielak & Ulmschneider 1998) with the aim of elucidating the excitation mechanism of the waves as well as the properties of the chromosphere. An empirical simulation by Carlsson & Stein (1997) employing a sophisticated radiation-hydrodynamic treatment showed that the characteristic features of the spectrum of the H line emerging from the chromosphere could be predicted from the observed velocity spectrum in the photosphere on the basis of propagating acoustic waves. Their model took the photospheric velocity field observed by Lites et al. (1993) and compared the computed time-dependent emergent H line profile with the observed profile. While the model gave an emergent H line intensity during the cool phase of the wave, as well as an overall temperature structure, that are strongly contradicted by the observations (Kalkofen, Ulmschneider & Avrett 1999), the intricate velocity and intensity variations in the line core during the bright phase of the wave are reproduced to high fidelity; only in the timing of the computed intensity variation relative to the photospheric motion (and probably also in the absolute intensity) are there larger discrepancies from the observations. The otherwise close agreement between simulations and observations is surprising since the simulations, like all other previous theoretical studies, assume plane acoustic waves. The reality of the wave propagation must be more complicated, however, suggesting an idealization of cylindrical symmetry. It is therefore interesting to investigate the dependence of the results on the geometry of the waves and the limitations of the assumed plane symmetry. We model the excitation of linear hydrodynamic waves and their outward propagation in a three-dimensional, stratified, isothermal atmosphere, assuming that the interaction that gives rise to an outward traveling wave can be modeled as a pressure pulse at some reference level in the photosphere. Unlike the 1D case, which allowed only acoustic waves to propagate, the 3D problem admits both acoustic and internal gravity waves. Our treatment of the hydrodynamic equations allows the separation of the two modes. We will focus mainly on the acoustic mode. Questions to be examined include (1) the lateral spreading of the energy as the wave propagates upward, (2) the decay of the energy flux from its high value in the initial pulse to later oscillations in the wake of the pulse and (3) the increase of the phase velocity from the sound speed at the head of the wave to infinite phase speeds in the asymptotic limit. The paper is structured as follows: We present the basic hydrodynamic equations in Sect. 2, describe the numerical results from the Fourier solution in Sect. 3, give the asymptotic analysis for late times in Sect. 4 and summarize the findings in Sect. 5. The appendices contain equations used in the Fourier solution of the problem. © European Southern Observatory (ESO) 2000 Online publication: January 31, 2000 |