## 5 Discussion  
Figures 6 and 8 report the first ever measurements by the TDBS technique in combination with optical reflectometry, AFM topography, and SEM of the variation in the longitudinal acoustic velocity and the longitudinal modulus across the rhombohedral-tetragonal structural transitions induced by the element substitution. It is worth mentioning here that, in Ref. [34], an attempt was reported to access this information in BDFO films even thinner than ours, by extracting from the TDBS signals the information on the transmission of the CAPs at the interface of the film with STO substrate. The transmission of CAPs across the interfaces are known to be accompanied by the detected BO amplitude changes [21], which are related to the conversion of the acoustic waves incident on the interface into reflected and transmitted waves of different amplitudes. So, via the acoustic reflection and transmission coefficients, the BO amplitudes contain information on the ratio of the acoustic impedances of the film and the substrate, potentially allowing the determination of the sound velocity in the films. Assuming that the amplitudes of the TDBS oscillations before and after the CAP incidence on the interface are proportional to the amplitudes of the incident and transmitted coherent phonons at BFs, the authors of Ref. [34] found that the acoustic transmission coefficient monotonically grows with increasing Dy substitution. They concluded on the monotonic variation of the BDFO elasticity across the structural phase transition. However, they did not take into account that the amplitude of the BOs is proportional not only to the amplitude of the coherent phonon but also to the photoelastic constant. The photoelastic constant for the BDFO films is unknown and, similar to other parameters, also depends on the doping concentration of the film. Thus, even the conclusion of monotonous growth across the structural phase transition of the acoustic transmission coefficient (see Figure 4 in Ref. [34]) could be hardly considered as a reliable one, as it was obtained without disentanglement of the dependencies of the BO amplitudes on the amplitudes of the CAPs and on the magnitudes of the photoelastic constants. The consequent conclusion derived in Ref. [34] on the softening of the BDFO in the structural transition from rhombohedral to tetragonal is twice doubtful in view of the opposite tendency revealed by us and the results reported in Ref. [33]. It is worth mentioning here that the general tendency of the rigidity growth with increasing Pr substitution, i.e., aside from the abovementioned anomaly, correlates with the observation that the sound velocities measured by us in BPFMO films on LAO substrate (column 7 in Table 3) are larger than in pure BFO films deposited on the same substrate [33]. An additional argument in favor of the reliability of the results reported by us in Sections 3.3 and 4.3 is the fact that the sharpness of the revealed anomaly in the elastic modulus is comparable to those measured earlier in these films for remnant magnetization and polarization (see Figure 5(b) in Ref. [13]). Moreover, the softening of the elastic modulus by more than \(\approx\)30% in the vicinity of the structural phase transition at \(\approx\)20 GHz frequency (see Table 1) initiated in our samples by Pr substitution is comparable in magnitude to one reported earlier at MHz frequencies near electrically induced rhombohedral-tetragonal phase transition [15]. Acoustic anomalies that are comparable in magnitude were also reported in heating-induced structural phase transitions [14]. If we compare the experimental results for BPFMO films with Pr substitutions of 10%, 15%, and 20% available both for Si and LAO substrates, they demonstrate higher acoustic velocities and expected higher rigidities for the films deposited on LAO substrate. This observation is in qualitative intuitive agreement with the fact that it was possible to scratch by the AFM tip apex the films deposited on Si but not on LAO.