Context. We investigate the oscillatory behavior of the quiet solar chromosphere and its discrete components in terms of oscillation properties, i.e. network and internetwork. For this purpose, we use a time series of high resolution filtergrams at five wavelengths along the Hα profile, obtained by the Dutch Open Telescope. Aims. We aim to gain insight on the distribution of power in different period bands and its variation between network and internetwork. Our spectral resolution provides information on the vertical distribution of power, since the Hα line has both photospheric and chromospheric components. We investigate the effect of Hα mottles on chromospheric oscillations, since they are the most prominent feature of the Hα chromosphere and outline inclined magnetic fields. Methods. We use wavelet and phase difference analyses of Hα intensities and Doppler signals. Two-dimensional power maps in the 3, 5 and 7 min period bands as well as coherence and phase difference maps were constructed. Results. At photospheric heights, where the Hα ± 0.7 Å wing is formed, the 3 and 5 min power is enhanced around the network, and forms power halos. Higher in the chromosphere these areas are replaced by magnetic shadows, i.e. places of power suppression. Interestingly, the power maps show a filamentary structure in the network which correlates very well with mottles. These areas show positive phase differences at the 3 min period band. At the 5 min and 7 min period bands both positive and negative phase differences are obtained with an increased number of pixels with high coherence, indicating the existence of both upward and downward propagating waves. Conclusions. We attribute our findings to the interaction between acoustic oscillations and the magnetic fields that constitute the magnetic network. The network flux tubes diverge at chromospheric levels and obtain a significant horizontal component, which is betrayed by the presence of mottles. The variation of power reveals the discrete role of the magnetic field at different heights, which guides or suppresses the oscillations, depending on its inclination. Spectral resolution in Hα provides useful information on the coupling between the acoustic sub-canopy atmosphere and the magnetized chromosphere.