Context. In previous work, we studied a prominence which appeared like a tornado in a movie made from 193 Å filtergrams obtained with the Atmospheric Imaging Assembly (AIA) imager aboard the Solar Dynamics Observatory (SDO). The observations in Hα obtained simultaneously during two consecutive sequences of one hour with the Multi-channel Subtractive Double Pass Spectrograph (MSDP) operating at the solar tower in Meudon showed that the cool plasma inside the tornado was not rotating around its vertical axis. Furthermore, the evolution of the Dopplershift pattern suggested the existence of oscillations of periods close to the time-span of each sequence. Aims: The aim of the present work is to assemble the two sequences of Hα observations as a full data set lasting two hours to confirm the existence of oscillations, and determine their nature. Methods: After having coaligned the Doppler maps of the two sequences, we use a Scargle periodogram analysis and cosine fitting to compute the periods and the phase of the oscillations in the full data set. Results: Our analysis confirms the existence of oscillations with periods between 40 and 80 min. In the Dopplershift maps, we identify large areas with strong spectral power. In two of them, the oscillations of individual pixels are in phase. However, in the top area of the prominence, the phase is varying slowly, suggesting wave propagation. Conclusions: We conclude that the prominence does not oscillate as a whole structure but exhibits different areas with their own oscillation periods and characteristics: standing or propagating waves. We discuss the nature of the standing oscillations and the propagating waves. These can be interpreted in terms of gravito-acoustic modes and magnetosonic waves, respectively. Movie attached to Fig. 2 is available at http://https://www.aanda.org