Acoustic-gravity waves, generated at the base of the solar photosphere, offer excellent diagnostic potential of the solar atmosphere. By observing these waves at multiple heights and modeling their propagation through different atmospheric layers, we can infer key physical properties of the atmosphere. This approach enables estimation of quantities such as the sound speed, radiative cooling time, and the height separation between observing layers. These properties are important toward our understanding of the temperature structure of the solar atmosphere and energy deposition in the solar chromosphere. In this work, we demonstrate an improved modeling of cross-spectral phase difference in the solar atmosphere that takes into account wave reflection near the plasma beta~1 region in the lower chromosphere. Furthermore, we extend the modeling technique from surface-resolved observations of the Sun to disk-integrated observations of the Sun. As a proof of concept, we estimate atmospheric properties from disk-integrated data, demonstrating the potential for application to distant stars. We also identify signatures of unexpected atmospheric gravity waves in disk-integrated observations, possibly originating near the solar limb, and incorporate cross-power, convection, and trapped acoustic p-mode signals into the cross-spectral modeling framework.