## 4 Discussion  
The L-RFA using the sweep-pulse excitation method is a technique used to search for resonance frequencies that match natural frequencies by adjusting the repetition rate of a low-power laser. Soft materials generally comprise frequencies that are several tens of Hertz lower than those of metals. However, in this study, forced vibrations of several kilohertz were considered. There are two reasons for this. First, soft materials exhibit lower Young's moduli than metals and dielectric materials, and are more readily subject to oscillation; thus, forced vibration can be more readily demonstrated. Second, forced vibration exhibits a lower decay time than resonance vibration, thus reducing the effect of changes in the excitation frequency during sweep-pulse excitation. As shown in Figure 5, the L-RFA measurements revealed that the differences in Young's moduli influenced the vibration spectrum of LIPTEW. In a simple spring vibration model, the spring constant is proportional to the Young's modulus; therefore, materials with higher Young's moduli vibrate less readily. However, the vibration was higher for the agarose sample with a Young's modulus of 0.76 MPa, as shown in Figure 5b, when compared with that with a Young's modulus 0.042 MPa, as shown in Figure 5a. This indicates that the damping coefficient was higher due to the softer material. This can be seen in the spectrograms, as shown in Figure 6, where the forced excitation energy due to laser irradiation at 0.76 MPa (Figure 6b) was highly distributed into high order vibration modes with high attenuation coefficients, whereas the agarose sample at 0.042 MPa (Figure 6a) had less presence. The agarose sample with a Young's modulus of 0.042 MPa, as shown in Figure 6a, revealed that most of the vibrational energy was concentrated at the frequency of forced excitation. However, when the Young's modulus was further increased, the vibration intensity decreased, as shown in Figure 5c. The noise component in the high-frequency band increased, thus resulting in a decrease in the S/N ratio of the vibration spectrum. This would be influenced by the resonance frequency and the damping behavior was changed. The resonance frequency \(f_{n}\) presents the relationship expressed by Equation (1) between the spring constant \(k\) and Young's modulus \(E\) in the spring vibration model. \[f_{n}\propto\sqrt{k}\propto\sqrt{E} \tag{1}\]  
Thus, the agarose sample in Figure 6c should indicate a resonance frequency that was higher than that in Figure 6a by a factor of approximately 55. The vibration damping effect was greater at frequencies higher than the resonance frequency. Therefore, it was assumed that the sample with a high Young's modulus did not exhibit a sufficient damping effect in the high-frequency band, thus resulting in a lower S/N ratio. The decrease in the S/N ratio at high Young's moduli has a negative influence because it prevents the acquisition of clear peaks in a simple spectral analysis. However, in the machine learning-based analysis used in this study, a high prediction accuracy was achieved by employing three types of multivariate nonlinear regression: center-of-gravity frequency, center-of-gravity intensity, and peak intensity. By introducing the center-of-gravity frequency and intensity as explanatory variables, the effect of increased noise in the high-frequency band at a high Young's modulus was reflected. In general, machine learning analysis should be based on a large amount of supervised data to obtain high accuracy. In this study, a higher accuracy was obtained by averaging the acquired data, as shown in Figure 7a, when compared with Figure 7b, training individual data. This suggests that sufficiently averaged data are required for soft materials because their morphologies are more unstable than those of metals and dielectrics, and the dispersion of the acquired data is larger. Additionally, it is more effective to select and combine the optimal explanatory variables among the multiple parameters.