## 3 Experimental Section
### Possible Mechanism of the Observed Elastic Behavior  
Although the mechanism behind the interfacial elastic dynamics responsible for the above-discussed resonance-like behaviors in the audible frequency range has not been understood, some arguments in association with the the phenomenon known as the interfacial blistering seem worth making. Thus, an attempt is made in this section to explain the experimental observation according to the following scenario. During the coating process, residual stresses are developed in the Ti layer. The residual stress either compresses or stretches the Ti layer parallel to the interface with the substrate and forms a pattern of compression and tension in the Ti layer. This pattern induces non-uniformity in the interfacial distance between the substrate and the Ti layer, generating a number of local, blister-like weak adhesions. The residual stress is not strong enough to cause delamination, but induces non-uniformity in the elastic strength. [PERSON] _et al._[25] discuss the effect of residual stress on thin aluminum films. Interfacial blistering is a major cause of delamination in the film-substrate interface and is usually discussed in the context of fracture mechanics. However, the dynamics of a blister in a thin-film's interface seems to have resonance in the audible frequency range. The details of interfacial blistering can be found elsewhere. In short, a blister can be formed in a film-substrate interfacial area where the adhesion is weaker than the surrounding area. [PERSON] [10] conducted a detailed study on the transition from the plate mode to the membrane mode dynamics in blister tests [12]. In blister tests, pressure is applied to the blistering area from the substrate side to inflate the film surface. When the height of the film-surface inflation reaches a critical value (relative to the film thickness), the elasticity of the film material takes over the longitudinal interfacial elasticity as the dominant restoring force mechanism of the blistering area; _i.e._, the elastic behavior of the area shifts from the plate mode to the membrane mode. In the present context, no pressure is applied to the film surface from the substrate side, and the structural condition of the interface is different from usual blister tests. However, if the applied dynamic load is at or close to the resonant frequency of the membrane-like elastic behavior, it is possible that the film surface bulges large enough to shift the elastic dynamics to membrane mode. When a membrane is subject to a dynamic load, the membrane exhibits various oscillation modes, such as the drumhead mode. Each mode has its own resonant frequency that is related to the tension of the membrane, the density of the membrane material and the size of the membrane. In the case of the drumhead mode of a rectangular membrane, the resonant frequency is given by the following equation. \[f_{mn}=\frac{1}{2}\sqrt{\frac{T}{\rho}[(\frac{m}{a})^{2}+(\frac{n}{b})^{2}]} \tag{12}\]  
Here, \(f_{mn}\) is the resonant frequency of the drumhead \(mn\) mode; \(T\) is the membrane tension; \(\rho\) is the density of the membrane material; \(a\) and \(b\) are the sides of the membrane associated with modes \(m\) and \(n\), respectively. its relaxation to some extent explains the observation of the long-term effect that, in some cases, the valley frequency is increased and other times decreased after long runs. To consider the possibility of the above mechanism in terms of the spatial profile of the adhesion weakness, we analyzed the peak ratio around the two valley frequencies observed in Figure 9 two-dimensionally. Figure 11 shows the peak-ratio valleys three-dimensionally for the treated and untreated specimens. Here, the horizontal axes are the driving frequency and the spatial coordinate axis perpendicular to the interferometric fringes. The valley in the untreated specimen appears circular as opposed to that in the treated specimen, which is more cylindrical.